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COMPLEX CURVES IN ALMOST-COMPLEX

MANIFOLDS AND MEROMORPHIC HULLS

Sergei IVASHKOVICH – Vsevolod SHEVCHISHIN

[**Preface**]{}

[Chapter I. Local Properties of Complex Curves.]{}

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[Chapter II. Compactness Theorem.]{}

[**Lecture 10. Envelopes of Meromorphy of Two-Spheres. pp. 160–168**]{}

[**Appendix IV. Complex Points and Stein Neighborhods. pp. 169–172**]{}

[**Appendix V. Seiberg-Witten Invariants and Envelopes. pp. 173–181**]{}

[**References. pp. 182–186**]{}

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**P r e f a c e**

This are the notes of a course, given by the first author for the Graduiertenkollegs (= graduate students) at the Ruhr-University of Bochum, in December, 1997. Previously, shorter courses were given in Göteborg, Warsaw and Bonn.

These lectures pursued two main tasks:

[**First**]{} — to give a systematic and self-contained introduction to the Gromov theory of so-called pseudoholomorphic curves (a term which will be completely eliminated in these notes). This is handled in Chapters I,II,III.

[**Second**]{} — to explain our joint results on envelopes of meromorphy of real two-spheres in complex two-dimensional manifolds, which we obtained using Gromov theory. We do this in Chapter IV.

It was impossible, of course, to omit the original motivation of Gromov in the development of the subject:

- in Appendix III, §A3.5, we explain the Gromov’s idea how to attach an analytic disk to a Lagrangian submanifold in $\cc^n$;

- in §8.5 we give the proof of his “non-squeezing theorem";

- in §8.6 we study exeptional spheres in symplectic $4$-manifolds.

In Appendix V, following S. Nemirovski, we discuss one more approach to the problem of constructig envelopes of meromorphy of spheres. This approach uses Seiberg-Witten theory. As an example if his results

- we prove that a complex disk cannot be attached from the outside to the strictly pseudoconvex domain in $\cc^2$, which is diffeomorphic to the ball.

The authors would like to give their thanks to Alan Huckleberry, the head of Graduiertenkolleg program at the Ruhr-University and to SFB-236 for the constant support of our joint research. We would like also to give our thanks to Barbara Huckleberry for the uncountable number of grammatical corrections.

[Chapter I. Local Properties of Complex Curves.]{}

In this chapter we shall concentrate on local properties of complex curves in almost-complex manifolds.The first lecture is very basic and mainly defining. Its aim is to introduce a couple of notions and recall a few standard facts. In the second one we prove, following Vekua and Sikorav, the existence of $J$-complex curves (locally through any point and any direction) for any almost-complex structure $J$ of smoothness $C^{1,\alpha}$, $\alpha >0$. One more principal result here is the so-called [*first a priori estimate*]{} in §2.4. The third lecture, as well as [*Appendix 2*]{}, is devoted to the deeper study of complex curves in real 4-dimensional manifolds. We shall prove the positivity of intersections and the Adjunction Formula, showing that local properties of complex curves in nonintegrable almost-complex structures are similar to that in integrable ones.

[Lecture 1]{}

[Complex Curves in Almost-Complex Manifolds.]{}

[1.1. Almost-Complex Manifolds, Hermitian Metrics, Associated (1,1)-forms.]{}

Let $X$ be a real manifold, and $TX$ its tangent bundle.

Definition 1.1.1. [*A continuous endomorphism $J:TX\to TX$ such that $J^2=-\id $ is called an almost-complex structure on $X$. A pair $(X,J)$ is called an almost-complex manifold.* ]{}

Exercise. Prove that locally one can always find vector fields $e_1,e_2,...,e_{2n-1},e_{2n}$ such that $Je_{2k-1}=e_{2k}, k=1,...,n$. In particular, $X$ is even-dimensional.

Examples. $1^0.$ $(\rr^{2n},J\st )$. Denote by $e_1,...,e_{2n}$ the standard basis in $\rr^{2n}$. Define a [*standard*]{} complex structure $J\st $ in $\rr^{2n}$ by $J\st (e_{2k-1})=e_{2k}$.

If $(V,J)$ is an $\rr $-linear space endowed with an endomorphism $J$, such that $J^2=-\id $, then a structure of a $\cc $-linear space on $V$ is defined by $z\cdot v=(x+iy)\cdot v:=x\cdot v +y\cdot Jv$. In the case $(\rr^{2n},J\st )$ this gives $\cc^n$.

$2^0.$ Let $x_0\in (X,J)$. By an affine change of coordinates in some chart $U\ni x_0$ we can always suppose that $x_0=0$ and $J(0)=J\st $ in basis ${\d \over \d x_1},...,{\d \over \d x_{2n}}$. One can now put $y_j=x_{2j}$ in order to have $J(0) {\d \over \d x_j}={\d \over \d y_j}$.

*$3^0.$ Complex manifolds. By definition, a complex manifold is an almost-complex manifold $(X,J)$ such that in the neighborhood of any point $x_0\in X$ a coordinate system $x_1,y_1,...,x_n,y_n$ can be found such that $J {\d \over \d x_j}={\d \over \d y_j}$ everywhere in this neighborhood.*

Equivalently there is in a neighborhood of each point a $C^1$-diffeomorphism $\phi $ into $(\rr^{2n},J\st )$ such that its differential is $(J,J\st )$-linear. The latter means that $d\phi \scirc J=J\st \scirc d\phi $ as mappings $TX\to \rr^{2n}$.

Definition 1.1.2. [*Such an almost-complex structure is called complex.* ]{}

*$4^0.$ Riemann surfaces. Riemann surfaces are oriented, connected two-dimensional manifolds. Fix some Riemannian metric $g$ on such $S$. For $v\in TS$ define $Jv=\{$ rotation of $v$ in positive direction onto $90^\circ \} $. $J$ is an almost-complex structure on $S$. If $g_1=\lambda g$ is another, conformally equivalent to $g$ metric, then this construction leads to the same almost-complex structure.*

Vice-versa, given $J$ on $S$ , denote by $J^{\cc }$ the $\cc $-linear extension of $J$ onto the complexified tangent bundle $T^{\cc }S:=TS\otimes
\cc $. Here $J^{\cc }$ is defined by $J^{\cc }(v\otimes \lambda )=Jv\otimes
\lambda $ for $v\in TS$ and $\lambda \in \cc $. Take an eigenvector $w$ of $J^{\cc }$ in $T^{\cc }S$ with an eigenvalue $-i$. Write $w=u+iv$, with $u,v\in TS$ (more accurately $w=u\otimes 1 + v\otimes i$), and put $g(u,v)=0,
g(u)=g(v)=1$. This defines a Riemannian metric, which corresponds to $J$. The statement $J^{\cc }w=-iw=v-iu$ means that $Ju=v, Jv=-u$.

The freedom in choosing $w$ here permits us to define $g$ up to a real factor, , $J$ corresponds to a class of conformally equivalent metrics.

Exercise. Prove the last assertion. More precisely: if $\lambda w$ is another eigenvector of $J^{\cc }$ with an eigenvalue $-i$ and a metric $g_{\lambda }$ is constructed from $\lambda w$ as $g$ from $w$ above, then $g=\vert \lambda \vert^2g_{\lambda }$.

We shall see later in [*Corollary 2.2.2*]{} that any almost-complex structure on the Riemann surface is complex.

Definition 1.1.3.

*$J$-Hermitian metric on $X$ is a $\rr $-bilinear form $h:TX\times TX\to \cc $ such that*

\(a) $h(Ju,v)=\overline h(v,Ju)=ih(u,v)$;

\(b) $h(u,u)>0$ for $u\not= 0$.

Decompose $h(u,v)=g_h(u,v) -i\omega_h(u,v)$. Then :

Exercise. (a) $g_h$ is a Riemannian metric on $X$ and $\omega_h$ is an exterior 2-form.

\(b) $g_h(u,v)=\omega_h(u,Jv)$ and thus $h(u,v)=\omega_h(u,Jv) -
i\omega_h(u,v)$. More over, one has $\omega_h (Ju,Jv)=\omega_h (u,v)$.

\(c) $\omega_h(u,v)=-g_h(u,Jv)$.

Definition 1.1.4. The form [*$\omega_h$ is called a (1,1)-form associated to $h$.*]{}

Conversely,

Definition 1.1.5.

*An exterior $2$-form $\omega $ on $X$ is called $J$-calibrated if*

\(a) $\omega (Ju,Jv)=\omega (u,v)$;

\(b) $\omega (u,Ju)>0$ for $u\not= 0$.

Thus, $\omega $ clearly defines a $J$-Hermitian metric $h^{\omega }(u,v)=\omega (u,Jv) - i\omega (u,v)$.

A triple $(X,J,h)$, where $h$ is $J$-Hermitian is called an almost-Hermitian manifold. We shall not use this term in these notes.

Example. To explain the coefficient $-i$ chosen in the decomposition $h(u,v)=g(u,Jv) - i\omega (u,v)$ consider $(\rr^{2n}, \omega\st, J\st)$. Here $\omega\st = \sum_{j=1}^ndx_j\wedge dy_j$ is a standard symplectic form in $\rr^{2n}$. The Hermitian metric here is $h\st = \sum_{j=1}^ndz_j\otimes d\bar z_j = \sum_{j=1}^n (dx_j\otimes
dx_j + dy_j\otimes dy_j) - i\sum_{j=1}^n(dx_j\otimes dy_j - dy_j\otimes dx_j)
= g\st -i\omega\st $.

Thus $\omega\st = -\im h\st $ is a canonical expression for a Kähler form of the Hermitian metric.

Definition 1.1.6. [*We say that an exterior $2$-form $\omega $ on $X$ tames an almost-complex structure $J$ if $\omega (v,Jv)>0$ for $v\not= 0$.* ]{}

Remark. [Here and later we will use the convention that for forms $f$ and $h$ on the vector space $V$ the exterior product is defined as $f\wedge g: = f\otimes g - g\otimes f$, , without coefficient ${1\over 2}$. ]{}

1.2. Existence of Calibrating and Tame Structures.

Here we shall prove here the following simple but very important statement:

Proposition 1.2.1. [*Let $X$ be a real manifold and $\omega $ a nowhere degenerate exterior $2$-form on $X$, , $\omega^n\not=0$, where $2n=\dim_{\rr } X$. On the open subset $U\subset X$ a $\omega $-calibrating almost-complex structure $J$ is given. Then for any relatively compact open $U_1\comp U$ there exists a $\omega $-calibrating almost-complex structure $J_1$ on the whole $X$ such that $J_1\mid_{U_1}=J\mid_{U_1}$.* ]{}

Proof. Consider a Riemannian metric $g(u,v):=\omega (u,Jv)$ on $U$. Find a Riemannian metric $g_1$ on $X$ such that $g_1\mid_{U_1}=g\mid_{
U_1}$. Since $\omega $ is not degenerate, there exists a (unique!) isomorphism $A_1:TX\to TX$ such that $$\omega (u,v) = g_1(A_1u,v)\eqno(1.2.1)$$ for all $u,v\in TX$. Further, $\omega $ is antisymmetric, so $$g_1(A_1u,v) = \omega (u,v) = -\omega (v,u) = -g_1(A_1v,u) = -g_1(u,A_1v).$$ Thus $A_1^*=-A_1$. So $A_1^*A_1=A_1A_1^*=-A_1^2$ is positively definite and self-adjoint $g_1$. Let $Q_1:=\sqrt{-A_1^2}$ be a positive square root of $-A_1^2$. Put $$J_1=A_1Q_1^{-1}.\eqno(1.2.2)$$ $J_1$ is an almost-complex structure. Indeed $$J_1^2=A_1Q_1^{-1}A_1Q_1^{-1}=A_1^2(Q_1^{-1})^2=A_1^2(-A_1^{-2})=-\id .$$ Let us check that on $U_1$ one has $J_1=J$. In fact, on $U_1$ we have $g=g_1$ and so $$\omega (Ju,Jv) = \omega (u,v) = g(A_1u,v) = \omega (A_1u,Jv),$$ and thus $A_1=J$ on $U_1$. So $Q_1=\sqrt{-A_1^2}=\id $. From here we have $J_1=A_1Q_1^{-1}=A_1=J$ on $U_1$.

Remark. Note that at this point of the proof we construct a correspondence $P:g\to J$, which maps a Riemannian metric $g$ to an $\omega $-calibrated a.-c. structure $J$ with $g(u,v)=\omega (u,Jv)
$. This correspondence is obviously a continuous map from the space of metrics to the space of structures. Moreover, note that if $g$ appears as $g=g_J(u,v)=\omega (u,Jv)$ for some $\omega $-calibrated $J$, then $P(g_J)=J$.

Let us now check that $\omega $ is $J_1$-calibrated. First: $$\omega (J_1u,J_1v)=g_1(A_1J_1u,J_1v)=g_1(A_1^2Q_1^{-1}u,J_1v)=
g_1(Q_1u,-J_1v)=$$ $$=-g_1(Q_1u,J_1v)=-g_1(u,Q_1J_1v)=-g_1(u,A_1v)=-g_1(A_1v,u)=-\omega (v,u)=$$ $$=\omega (u,v).$$ Second: $$\omega (u,J_1u)=g_1(A_1u,J_1u)=g_1(A_1u,A_1Q_1^{-1}u)=$$ $$=g_1(u,u)>0$$ for nonzero $u$.

This remark leads to the following

Corollary 1.2.2.

*Let $(X,\omega )$ be as above. Then:*

1\. There exists a $\omega $-calibrating almost-complex structure on $X$.

2\. The space of $\omega $-calibrating almost-complex structures is contractible.

Proof. The first statement of this corollary is a particular case of the proposition when $U=\emptyset $.

To prove the second one, we fix some calibrating structure $J_0$. Denote by $g_{J_0}$ the corresponding Riemannian metric on $X$, , $g_{J_0}(u,v)=\omega (u,J_0v)$. The space of Riemannian metrics is a convex cone $\calc$ in $\hom_{\rr }(TX,T^*X)$. Therefore there exists a contraction $\Psi : \calc \times [0,1]\to \calc $ to $g_{J_0}$, , $\Psi (\cdot , 0)=\id $, $ \Psi (\cdot ,1 )\equiv g_{J_0}$ and $\Psi (g_{J_0},t)\equiv g_{J_0}$.

Consider the following map $\Phi :\calj_{\omega }^c\to \calc $ from the space of $\omega $-calibrating structures to the space of metrics: $$\Phi (J)(u,v)=g_{J}(u,v)=\omega (u,Jv).$$ In the proof of Proposition 1.2.1 (see Remark there) we showed that $$P\scirc \Phi =\id :\calj_{\omega }^c\to \calj_{\omega }^c.$$

Now $P\scirc \Psi (\cdot ,t)\scirc \Phi $ will be a contraction of $\calj_{\omega }^c$ to $J_0$.

Let us prove the following

Proposition 1.2.3. [*For any symplectic manifold $(X,\omega)$, the set $\calj_\omega$ of $\omega$-tame almost complex structures on $X$ is a non-empty contractible manifold*]{}.

The proposition follow immediately from the following result from linear algebra.

Lemma 1.2.4. [*Let $V$ be a (finite-dimensional) real vector space and $\omega$ a linear symplectic form on $V$. Then the set $\jj_\omega$ of $\omega$-tame linear complex structures on $V$ is a non-empty open contractible subset in the set $\jj$ of all linear complex structures on $V$*]{}.

Proof. [*Step 1*]{}. The linear version of the Darboux theorem states that there exists a basis $(\bfx_1, \ldots, \bfx_n, \bfy_1, \ldots, \bfy_n)$ of $V$ such that for corresponding linear coordinates $(x_1, \ldots, x_n, 
y_1, \ldots, y_n)$ on $V$ $$%\textstyle
\omega=\sum_{i=1}^n dx_i \wedge dy_i.
\eqno\rom(1.2.3)$$ Set $J_0 (\bfx_i) \deff \bfy_i$, $J_0 (\bfy_i) \deff -\bfx_i$. It is easy to see that $J_0$ is a linear complex structure on $V$. Moreover, the $\cc$-valued linear functionals $z_i \deff x_i + \isl\, y_i$ induce the isomorphism $\phi
:(V,J_0) \cong (\cc^n, J\st)$, $J\st \deff \isl$, of [*complex*]{} linear spaces, such that the given symplectic form $\omega$ on $V$ is mapped into the standard sympletic form $\omega\st$ on $\cc^n$ given by the same formula (A.T.1), $\phi^*\omega\st = \omega$. Thus we can conclude that there exists a $J_0$-Hermitian metric $h$ on $V$ with the following properties:

$g(v,w) \deff \re h(v,w)$ is a symmetric positively definite form on $V$ such that $h(v,w)= g(v,w) + \isl\,\omega(v,w)$ for any $v,w \in V$;

$g(J_0 v, J_0w) = g(v,w)$ and $\omega(J_0 v, J_0w) = \omega(v,w)$.

$g(v,w) = \omega(v, J_0w)$ and $\omega(v,w)=g(J_0v,w)$ for any $v,w 
\in V$.

In particular, $\omega(v, J_0w)$ is positive definite, $J_0 \in 
\jj_\omega$. This shows that $\jj$ is non-empty.

[*Step 2. For any $J \in \jj_\omega$ the operator $J + J_0$ is invertible*]{}. Really, otherwise we have $Jv = - J_0v$ for some non-zero $v\in V$ and then $0< \omega(v, Jv) = -\omega(v, J_0v) <0$, a contradiction.

For $J \in \jj$ with $J + J_0$ invertible we set $\sfL(J) \deff -(J - J_0) (J+
J_0)\inv$. The equivalent definitions are $$\matrix\format \r&\,\c\, & \l \\
\sfL (J) &=&  -(J - J_0) (J+J_0)\inv = 
-\bigl((J - J_0)J_0\bigr) \cdot \bigl( (J+J_0) J_0 \bigr)\inv \cr
\vph&=&(\bfone + JJ_0 )(\bfone - JJ_0  )\inv 
      = (\bfone - JJ_0)\inv (\bfone + JJ_0).
\endmatrix
\eqno\rom(1.2.4)$$

[*Step 3. If $J + J_0$ is invertible, then $W= \sfL(J)$ is $J_0$-antilinear, $W J_0= - J_0 W$*]{}. Really, $$\matrix \format \c &\,\c\, &\r &\,\c \,&\r &\,\c\, &\c
\vbox to 11pt{} \\
W J_0 &=& -(J - J_0) (J+J_0)\inv J_0 &=& 
(\bfone - JJ_0)\inv (\bfone + JJ_0) J_0 &=&
\cr 
&&
(\bfone -JJ_0)\inv (J_0- J)&=&
- (\bfone -JJ_0)\inv J(\bfone + JJ_0) &=& 
\cr
&&
(\bfone -JJ_0)\inv J\inv (\bfone +JJ_0) &=& 
(J - J^2 J_0)\inv (\bfone +JJ_0) &=&
\cr
&&
(J_0 + J)\inv (\bfone +JJ_0) &=& 
\bigl( (\bfone -JJ_0) J_0 \bigr)\inv (\bfone +JJ_0) &=& 
\cr
&&
J_0\inv(\bfone -JJ_0)\inv (\bfone +JJ_0) &=& 
- J_0(\bfone -JJ_0)\inv (\bfone +JJ_0) &=& - J_0 W
\endmatrix$$

[*Step 4. $J \in \jj_\omega$ implies $\bfone - W^\sft W \gg 0$ for $W= 
\sfL(J)$*]{}, the operator $\bfone - W^\sft W$ is positively definite, $g(v,v) > g(Wv,Wv)$ for any non-zero $v \in V$. The conjugation of $W \in 
\endo(V)$ is done using the metric $g$.

First we note that $J_0^\sft = J_0\inv =-J_0$. Since $\bfone -JJ_0$ is invertible, the positivity of $\bfone - W^\sft W$ is equivalent to the positivity of $(\bfone -JJ_0)^\sft(\bfone - W^\sft W) (\bfone -JJ_0)$. Computing this operator we obtain $$\matrix\format \c \vbox to 11.5pt{} \\
(\bfone -JJ_0)^\sft(\bfone - W^\sft W) (\bfone -JJ_0) = 
(\bfone -JJ_0)^\sft(\bfone -JJ_0) - 
\cr
(\bfone -JJ_0)^\sft\bigl( (\bfone +JJ_0) (\bfone -JJ_0)\inv \bigr)^\sft
\bigl( (\bfone +JJ_0) (\bfone -JJ_0)\inv \bigr)(\bfone -JJ_0)=
\cr
(\bfone -JJ_0)^\sft(\bfone -JJ_0) -
(\bfone +JJ_0)^\sft (\bfone +JJ_0) =
\cr
(\bfone +J_0 J^\sft)(\bfone - JJ_0) - (\bfone - J_0 J^\sft)(\bfone +JJ_0)=
\cr
\bfone +J_0 J^\sft - JJ_0  -  J_0 J^\sft JJ_0 -
\bfone + J_0 J^\sft - JJ_0 +  J_0 J^\sft JJ_0 =
\cr
2(J_0 J^\sft - JJ_0).
\endmatrix$$

Now, writing any non-zero $v\in V$ as $v= J_0w$, we obtain $$\matrix\format \l &\,\c\,\;& \l \vbox to 12.5pt{} \\
g(v, (J_0 J^\sft - JJ_0)v) &=&  g(J_0w, (J_0 J^\sft - JJ_0)J_0w)= 
\cr
g(J_0w, J_0J^\sft J_0w) + g(J_0w, Jw) &=& 
g(w, J^\sft J_0w) + g(J_0w, Jw) = 
\cr
g(Jw, J_0w) + g(J_0w, Jw) &=& 2\omega(w, Jw)>0.
\endmatrix$$

Thus $\sfL$ maps $\jj_\omega$ into the set $$\calw \deff 
\{\, W \in \endo(V)\;:\; W J_0 =- J_0W,\; \bfone -W^\sft W \gg0 \,\}.
\eqno\rom(1.2.5)$$

It is easy to see that $\calw$ is contractible. Thus it is sufficient to show that $\sfL: \jj_\omega \to \calw$ is a diffeomorphism. Explicit calculations show that the inverse map should be given by $$J= \sfK(W) \deff J_0 (\bfone + W) (\bfone - W)\inv= 
J_0 (\bfone - W)\inv (\bfone + W) = 
J_0 {\bfone + W \over \bfone - W}
\eqno\rom(1.2.6)$$

[*Step 5. Let $W\in \calw$. Then $J= \sfK(W) = J_0 {\bfone + W \over 
\bfone - W}$ is well-defined and $J^2 = -\bfone$*]{}. Really, the condition $\bfone -W^\sft W \gg0$ implies that $\bfone \pm W$ is invertible, and then $$\matrix\format \c &\,\c\,&\l  \vbox to 11.5pt{} \\
J^2 &=& 
-J_0 (\bfone + W) (\bfone - W)\inv J_0\inv (\bfone + W) (\bfone - W)\inv
 \cr
&=& - (\bfone + J_0 W J_0\inv) (\bfone - J_0 W J_0\inv)\inv
(\bfone + W) (\bfone - W)\inv
 \cr
&=& - (\bfone - W ) (\bfone + W )\inv
(\bfone + W) (\bfone - W)\inv \qquad  =- \bfone 
\endmatrix$$

[*Step 6. Let $W\in \calw$ and $J= \sfK(W)$. Then $J+ J_0$ is invertible and $J \in \jj_\omega$*]{}. In fact, $J + J_0 =  J_0\left( \bfone + {\bfone + 
W \over \bfone - W} \right) = {2 J_0 \over \bfone - W}$ is invertible. Now, repeating argumantations from [*Step 4*]{} the positivity of $\bfone - 
W^\sft W$ is equivalent to the positivity of $J_0 J^\sft - JJ_0$, which means the tameness of $J$.

The lemma is proved.

Remark. One can consider $\sfK:\calw \to \jj_\omega \subset \jj$ a generalized Cayley transformation defined on a bounded domain $\calw$ in the complex linear space $\barr \endo_\cc(V,J_0)$ of $J_0$-antilinear endomorphisms of $V$.

[1.3. Almost-Complex Submanifolds, Complex Curves, Energy and Area.]{}

Let $J$ be some linear complex structure on $\rr^{2n}$, , $J\in
\endo(\rr^{2n}), J^2=-\id $. Let $h$ be some $J$-Hermitian metric on $\rr^{2n}$, $\omega =\omega_h$ its associated $(1,1)$-form and $g=g_h$ corresponding Riemannian metric. Put $\sigma_k =  {1\over k!}\omega^k$.

Wirtinger Inequality. [*For any $g$-orthonormal system $v_1,...,v_{2k}$ in $\rr^{2n}$ one has $$\vert \sigma_k(v_1,...,v_{2k})\vert \le 1, \eqno(1.3.1)$$ with equality taking place iff the subspace $\<v_1,...,v_{2k}\>$ is $J$ - invariant.* ]{}

Remark. We shall also call $J$-invariant subspaces $J$-complex.

Proof. We shall prove (1.3.1) by induction on $k$.

Let $k=1$. Then $\vert \omega (v_1,v_2)\vert =\vert g(v_1,Jv_2)\vert
\le \Vert v_1\Vert_g\cdot \Vert Jv_2\Vert_g=$ $\Vert v_1\Vert_g\cdot \Vert v_2
\Vert_g$, with equality iff $v_1$ and $Jv_2$ are collinear. Thus the subspace $\<v_1,v_2\>$ is $J$-invariant.

Now let $V$ be a subspace of dimension $2k$. Put $\omega'=\omega\mid_V$. Find a $g$-orthonormal base $e_1,...,e_{2k}$ of $V$, s.t. $\omega'=
\lambda_1e^1\wedge e^2 + ... +\lambda_ke^{2k-1}\wedge e^{2k}$. By the case $k=1$ we have $\vert \omega' (e_{2p-1},e_{2p})\vert=
\vert \omega (e_{2p-1},e_{2p})\vert =\vert \lambda_p\vert \le 1$, with equality taking place iff $e_{2p}={+\over -}Je_{2p-1}$. So, for $\sigma'_k
=\sigma_k\mid_V$ we obtain $$\vert \sigma_k'(e_1,...,e_{2k})\vert = \vert {1\over k!}\omega^k(e_1,...,
e_{2k})\vert = \vert \lambda_1...\lambda_k\vert \le 1,$$ with equality iff $e_{2p}={+\over -}Je_{2p-1}$ for all $1\le p\le k$.

A submanifold $Y$ of an almost-complex manifold $(X,J)$ is called an almost-complex (or $J$-complex) if the tangent spaces to $Y$ are $J$-invariant.

Definition 1.3.1. [*Nowhere degenerate exterior two-form $\omega$ on $X$ is called symplectic if $d\omega =0$.*]{}

A pair $(X,\omega )$ is called a symplectic manifold.

Corollary 1.3.1. [*Let $(X,w,J)$ be a symplectic manifold with $w$ being $J$-calibrated. Then $J$-complex submanifolds are minimal and their volume is given by $$vol_{2k}(Y) = {1\over k!}\int_Yw^k. \eqno(1.3.2)$$*]{}

Proof. Let $Y$ be a $J$-complex submanifold of $X$, and $Y_1$ some submanifold of dimension $2k$ as well as $Y$. Suppose $\d Y=\d Y_1$ and $Y\cup (-Y_1) \sim 0$. Denote by $dy$ and $dy_1$ the volume forms of $Y$ and $Y_1$ with respect to metric $g(u,v)=\omega (u,Jv)$. Then we have $$0=    \int_{Y\cup (-Y_1)}\sigma_k = \int_{Y}\sigma_k  - \int_{Y_1}\sigma_k  =
\int_{Y}\sigma_k (TY)dy - \int_{Y_1}\sigma_k(TY_1)dy_1\ge$$ $$\ge \int_{Y}dy  - \int_{Y_1}dy_1=vol(Y)-vol(Y_1),$$ with equality taking place if and only if $\sigma_k(T_pY_1)=1$ for all $p\in Y_1$, , when $Y_1$ is also $J$-complex.

Definition 1.3.2. [*A $C^0\cap L^{1,2}$-map $u :(Y,J_Y)
\to (X,J_X)$ is called holomorphic if for a.-a. $x\in Y$ $$du_x\scirc J_Y(x)= J_X(u(x))\scirc du_x\eqno(1.3.3)$$ as mappings $T_xY\to T_{u(x)}X$.* ]{}

Exercise. [Check that for complex valued functions, , $u :(\rr^2,i)\to (\rr^2,i)$, (1.3.3) is a Cauchy-Riemann equation. ]{}

Sometimes one calls such $u$ a $(J_Y,J_X)$-holomorphic map. In the special case when $(Y,J_Y)$ is a Riemann surface $(S,J_S)$, one calls $(S,J_S,u)$ a parameterized $J_X$-complex curve. Its image $u(S)$ one simply calls a $J_X$-complex curve.

If some $J$-calibrated exterior $2$-form $\omega $ on an almost-complex manifold $(X,J)$ is chosen, one defines the $\omega $-area of a $J$-complex curve $u:(S,J_S)\to (X,J)$ as $$area[u(S)] = \int_Su^*\omega .\eqno(1.3.4)$$ Remark that if $\omega $ is symplectic, then by [*Corollary 1.3.1*]{} this area coincides with $g$-area for the metric $g(\cdot ,\cdot ) =
\omega (\cdot ,J\cdot )$ and $J$-complex curves are $g$-minimal surfaces.

Remark also that for a $J$-complex curve $u:(\Delta ,J\st )\to (X,J)$, parameterized by a standard unit disk, one has $$\int_{\Delta }u^*\omega = \int_{\Delta }\omega  (du({\d \over \d x}),
du({\d \over \d y}))dx\wedge dy =  -\int_{\Delta }{1\over 2}g(du({\d \over
\d x}),Jdu(i{\d \over \d x}))dx\wedge dy =$$ $$=-\int_{\Delta }{1\over 2}g(du({\d \over \d x}),J^2du({\d \over \d x}))dx
\wedge dy = \int_{\Delta }{1\over 2}g({\d u\over \d x}, {\d u \over \d x}))
dx\wedge dy = {1\over 2}\Vert {\d u\over \d x}\Vert^2_{L^2(\Delta ,X)}.$$ Thus $$\int_{\Delta }u^*\omega  = \Vert {\d u\over \d x}
\Vert^2_{L^2(\Delta ,X)}+\Vert {\d u\over \d y}\Vert^2_{L^2(\Delta ,X)}.
\eqno(1.3.5)$$ The right hand side of (1.3.5) is called an energy of a $C^0\cap L^{1,2}$-map (not necessary holomorphic!) from $\Delta $ to a Riemannian manifold $(X,g)$. Thus, (1.3.5) tells us that, for holomorphic maps, energy and the area of the image, taken with multiplicities, are equal.

Remark. Let $g$ be a Riemannian metric on $C$ compatible with $j_C$, $h$ a Riemannian metric on $X$, and $u:C
\to X$ a $J$-holomorphic immersion. Then $\norm{du}^2_{L^2(C)}$ is independent of the choice of $g$ and coincides with the area of the image $u(C)$ the metric $h_J(\cdot, \cdot) \deff \half(h(\cdot, \cdot) + h(J\cdot, J\cdot)
)$. The metric $h_J$ here can be seen as a “Hermitization” of $h$ $J$. The independence of $\norm{du}^2_{L^2(C)}$ of the choice of a metric $g$ on $C$ in the same conformal class is a well-known fact, see, , \[S-U\]. Thus we can use the flat metric $dx^2 + dy^2$ to compare the area and the energy. For a $J$-holomorphic map we get $$\norm{du}^2_{L^2(C)}= \int_C |\d_x u|_h^2 + |\d_y u|_h^2 =
\int_C |\d_x u|_h^2 + |J \d_x u|_h^2 = \int_C |du|_{h_J}^2 = 
\area_{h_J}(u(C)),$$ where the last equality is another well-known result, see, , \[G\]. Since we consider changing almost complex structures on $X$, it is useful to know that we can use any Riemannian metric on $X$ having a reasonable notion of area.

[1.4. Symplectic Surfaces.]{}

First we recall some elementary facts about orthogonal complex structures in $\rr^4$.

In $\rr^4$ with coordinates $x_1, y_1, x_2, y_2$ consider the standard symplectic form $\omega\st  = dx_1\wedge dy_1 + dx_2\wedge dy_2 $ and the standard complex structure $J\st$ defined by the operator $$J\st =
\left( \matrix
0&-1&0&0\cr
1&0&0&0\cr
0&0&0&-1\cr
0&0&1&0\cr
\endmatrix \right).$$

Let $\jj$ denote the set of all orthogonal complex structures in $\rr^4$ giving $\rr^4$ the same orientation as $J\st$. Orthogonality here means just that $J$ is an orthogonal matrix. The same orientaion means that for any pair $x_1, x_2$ of $J$-independent vectors the basis $x_1, Jx_1, x_2, Jx_2$ gives the same orientation of $\rr^4$ as $J\st$.

Exercise. [Prove that this orientation does not depend on the particular choice of $x_1, x_2$ and coincides with the orientation given by $\omega\st^2$. ]{}

The following lemma summarizes the elementary facts which we need for the sequel.

Lemma 1.4.1. *The elements of $\jj$ have the form $$J_{s} =
\left(
\matrix
0&-s &c_1&c_2\cr
s &0&c_2&-c_1\cr
-c_1&-c_2&0&-s \cr
-c_2&c_1&s &0\cr
\endmatrix \right),\eqno(1.4.1)$$ with $c_1^2 + c_2^2 + s^2 = 1$. One also has for $x\in \rr^4$ $$\omega\st  (x, J_s x) = s \Vert x\Vert^2 . \eqno(1.4.2)$$*

We remark that the set $\jj$ is a unit two-dimensional sphere $S^2$ in $\rr^3$ with coordinates $c_1, c_2,s $. We note also that the number $\omega\st (x, J_s x)$ does not depend on the choice of a *unit vector $x$. One also remarks that the standard structure corresponds to the north pole of $S^2$ and structures tamed by $\omega\st $ constitute the upper half-sphere.*

Exercise. 1. Note that $\jj = \{ J\in O(4): J^2=-E\} = \{ J: 
J^t=J^{-1}=-J\} $. Prove that condition $J^t=-J$ implies that $$J = \left(
\matrix
0&-s &c_1&c_2\cr
s &0&c_3&c_4\cr
-c_1&-c_3&0&-t \cr
-c_2&-c_4&t &0\cr
\endmatrix \right),$$ for some $s,t,c_i, i=1,...,4.$

[**2.**]{} Denote by $C$ the matrix $\matrix c_1& c_2 \cr c_3& c_4\endmatrix$. Prove that condition $J^2=-E$ implies that:

a\) $$C = \left(
\matrix
\pm c_1&c_2\cr
\mp c_2&c_1\cr
\endmatrix \right)$$ with $c_1^2+c_2^2=1-s^2$;

b\) $\vert t\vert =\vert s\vert $;

c\) and, finally, that $s=t$ and for $s\not= \pm 1$ the matrix ${1\over \sqrt{1-s^2}}\cdot C$ lies in $O_-(2)$ (the set of orthogonal $2\times 2$-matrices with determinant $-1$) iff the structure $$J_s := 
\left(
\matrix
0&-s &c_1&c_2\cr
s &0&c_3&c_4\cr
-c_1&-c_3&0&-s \cr
-c_2&-c_4&s &0\cr
\endmatrix \right)$$ defines the same orientation of $\rr^4$ as $J\st = J_1$.

[**3.**]{} Check finally that $\omega\st (x,J_sx)=s\cdot \Vert x\Vert^2$.

Let $M_1$ and $M_2$ be two smooth oriented surfaces in the unit ball $B\subset
\rr^4$ with zero as a common point. Let $v_1, w_1$ and $v_2, w_2$ be oriented bases of $T_0M_1$ and $T_0M_2$, respectively. Suppose that $M_1$ and $M_2$ intersect transversally at zero, [**]{}, $v_1, w_1, v_2, w_2$ is the basis of $\rr^4$. We say that they intersect positively if this basis gives the same orientation of $\rr ^4$ as the standard one.

Let $(X,\omega )$ be a manifold with nowhere degenerate exterior $2$-form $\omega $.

Definition 1.4.1. [*An immersion $u:S\to X$ of a real surface $S$ into $X$ is called $\omega $-positive if $u^*\omega $ never vanishes.* ]{}

If $\omega $ is symplectic we call such immersions [*symplectic*]{}.

Definition 1.4.2. [*An almost-complex structure $J$ is said to be tamed by an exterior $2$-form $\omega $ if $\omega (u,Ju)>0$ for any nonzero $u\in TX$*]{}.

In the following lemma we suppose for simplicity of proof that $\dim_{\rr }X
=4$.

Lemma 1.4.2.

*Let $M$ be a $\omega $-positive compact surface immersed into $(X,\omega )$ with only double positive local self-intersections, and let $U_1\subset \subset U$ be neighborhoods of $M$. Then for any given $\omega $-tamed a.-c. structure $J$ there exists a smooth family $\{ J_t\} _{t\in [0, 1]} $ of almost-complex structures on $X$ such that:*

a\) $J_0$ is the given structure $J$ on $X$;

b\) for each $t\in [0, 1]$ the set $\{ x\in X: J_t(x) \not= J_0(x) \} $ is contained in $U_1$;

c\) $M$ is $J_1$-holomorphic;

d\) all $\{J_t\}$ are tamed by the given form $\omega $, , $\omega(v,J_tv) > 0$ for every nonzero $v\in TX$.

Proof. Let $N$ be a normal bundle to $M$ in $X$ and $V_1$ a neighbourhood of the zero section in $N$. Shrinking $V_1$ and $U_1$ we can assume that $U_1$ is an image of $V_1$ under an $\exp$-map in $J$-Hermitian metric $h$ associated to $\omega $. More precisely, one should take the Riemannian metric associated to $h$, $g = \re h$. Shrinking $V_1$ and $U_1$ once more, if necessary, we can extend the distribution of tangent planes to $M =$ (zero section of $N$) to the distribution $\{ L_x\} _{x\in V_1}$ of $\omega $-positive planes on $V_1$. Here we do not distinguish between $\omega$ and its lift onto $V_1$ by $\exp$. Denote by $N_x$ the subspace in $T_xV_1$ which is $g$-orthogonal to $L_x$. Perturbing $g$ we can choose the distribution $\{ L_x\}$ in such a way that if $\exp(x)= \exp(y)$ for some $x\not= y$ from $V_1$ then $d\exp_x(L_x)=d\exp_y(N_y)$ and $d\exp_y(L_y)=d\exp_x(N_x)$. In particular, we suppose that $M$ intersects itself $g$-orthogonally. For every $x\in V_1$ choose an orthonormal basis $e_1(x), e_2(x)$ of $L_x$ such that $\omega_x(
e_1(x), e_2(x)) > 0$, and an orthonormal basis $e_3(x), e_4(x)$ of $N_x$ such that the basis $e_1(x), e_2(x), e_3(x), e_4(x)$ gives the same orientation of $T_xV_1$ as $\omega^2$.

Define an almost complex structure $J^1$ on $U_1$ by $J^1e_1 = e_2$, $J^1 e_3 =
e_4$. Then $J^1$ depends smoothly on $x$, even when $e_j(x)$ are not smooth in $x$. Furthermore, we have $\omega _x(e_1(x), J_x^1e_1(x)) > 0$ and thus from [*Lemma 1.1.1*]{} we see that the relation $\omega _x(e_3(x), J_x^1e_3(x)) > 0$ is also satisfied. This means that our $J^1$ is tamed by $\omega $. Note also that $M$ is $J^1$-holomorphic.

Denote by $\jj_x$ the sphere of $g$-orthogonal complex structures on $T_x
X$ as in [*Lemma 1.4.1*]{}. Let $\gamma_x$ be the shortest geodesic on $\jj_x$ joining $J(x)$ - our given integrable structure, with $ J_x^1$. Put $J_t(x)
= J^1_{\gamma _x(t\cdot \Vert \gamma _x\Vert \cdot \phi (x ))}$. Here $\Vert
\gamma_x \Vert $ denotes the length of $\gamma _x$, $\phi$ is smooth on $X$ with support in $U_1$ and identically one in the neighborhood of $M$. The curve $\{ J_t \}$ satisfies all the conditions of our lemma.

Lemma 1.4.3. [*Under the conditions of Lemma 1.4.2 the structure $J_1$ can be made complex in the neighborhood of $M$.* ]{}

The proof is left to the reader.

[1.5. Adjunction Formula for Immersed Symplectic Surfaces.]{}

Let us now prove the Adjunction Formula for *immersed symplectic surfaces. Let $u:\bigsqcup_{j=1}^dS_j\to (X,\omega)$ be a reduced compact symplectic surface (see Definition 1.4.1) immersed into a symplectic four-dimensional manifold. Let $g_j$ denote the genus of $S_j$ and $M_j=u(S_j)$. Put $M:=\bigcup_{j=1}^dM_j$ and denote by $[M]^2$ the homological self-intersection number of $M$. Define a geometrical self-intersection number $\delta $ of $M$ in the following way. Perturb $M$ to obtain a symplectic surface $\widetilde M$ with only transversal double points. Then $\delta$ will be the sum of indices of the intersection over those double points. Those indices can be equal to $1$ or $-1$.*

Exercise. Let $P$ be an oriented plane in $\rr^4$. Call $P$ symplectic if $\omega\st (P)>0$. Find two symplectic planes $P_1,P_2\in \rr^4$ which intersect transversally at origin with intersection index $-1$.

Let $J$ be some almost complex structure which is tamed by $\omega$, [*,*]{} $\omega(\xi, J\xi)>0$ for any nonzero $\xi \in TX$. Denote by $c_1(X, J)$ the first Chern class of $X$ with respect to $J$. Since, in fact, $c_1(X, J)$ does not depend on continuous changes of $J$ and since the set of $\omega$-tamed almost complex structures is contractible, we usually omit the dependence of $c_1(X)$ on $J$.

Lemma 1.5.1. [*Let $M=\bigcup_{j=1}^d M_j$ be a compact immersed symplectic surface in four-dimensional symplectic manifold $X$. Then $$\sum_{j=1}^d g_j = {[M]^2 - c_1(X)[M]\over2} + d - \delta .\eqno(1.5.1)$$*]{}

Proof. By replacing every $M_j$ by its small perturbation, we can suppose that $M_j$ has only transversal double self-intersection points. Let $N_j$ be a normal bundle to $M_j$ and let $\widetilde M_j$ denote the zero section of $N_j$. Also let $\exp_j $ be the exponential map from a neighborhood $V_j$ of $\widetilde M_j\subset N_j$ onto the neighborhood $W_j$ of $M_j$. Lift $\omega$ and $J$ onto $V_j$. Since $\widetilde M_j$ is embedded to $V_j$, we can apply [*Lemma 1.4.2*]{} to obtain the $\omega$-tame almost complex structure $J_j$ on $V_j$ such that $\widetilde M_j$ is $J_j$-holomorphic.

For every $j$ we now have the following exact sequence of complex bundles:

$$0\longrightarrow TS_j {\buildrel du\over \longrightarrow }
E_j \buildrel \pr \over \longrightarrow
N_j \longrightarrow 0 \eqno(1.5.2)$$

Here $E_j=(u^*TX)\ogran_{S_j}$ is endowed with complex structure given by $J_j$. Since $du$ is nowhere degenerate [*complex*]{} linear morphism, $N_j \deff
E_j/du(TM_j)$ is a complex line bundle over $S_j$. From (1.5.2) we get

$$c_1(E_j) = c_1(TS_j) + c_1(N_j). \eqno(1.5.3)$$

Observe now that $c_1(E_j) = c_1(X)[M_j]$ and that $c_1(TS) = \sum_{j=1}^d
c_1(TS_j) = \sum_{j=1}^d(2-2g_j) = 2d - 2\sum_{j=1}^d g_j$. Furthermore, $c_1(N_j)$ is the algebraic number of zeros of a generic smooth section of $N_j$. To compare this number with the self-intersection of $M_j$ in $X$, note that if we move $M_j$ generically to obtain $M_j'$, then the intersection number $int(M_j,
M_j')$ is equal to the algebraic number of zeros of generic section of $N_j$ plus two times the sum of intersection numbers of $M_j$ in self-intersection points, [*,*]{} $[M_j]^2 = c_1(N_j) + 2\delta_j$. So $$c_1(X)[M_j] = 2 - 2 g_j + [M_j]^2 - 2\delta _j . \eqno(1.5.4)$$ Now it only remains to take the sum over $j=1,\ldots, d$ and to remark that the intersection points of $M_i$ with $M_j$ for $i\not=j$ make a double contribution to $[M]^2$.

[Appendix 1]{}

[Chern Class and Riemann-Roch Formula]{}

[A1.1. First Chern Class.]{}

Let $L\to M$ be a complex line bundle over a real manifold $M$. One of the possible definitions consists of taking a real rank two bundle over $M$ with an operator $J\in \End (L)$, satisfying $J^2=-\id_L$. One can than locally find a frame $e_1(x), e_2(x)$ with $Je_1(x)=e_2(x)$. This gives a covering $\{ U_\alpha \} $ of $M$ together with isomorphisms of complex line bundles $\phi_{\alpha }:L\mid_{U_{\alpha }}\to U_{\alpha }
\times \cc $, , a standard definition of a complex line bundle. Sometimes we shall mark as $e_1^J, e_2^J, \phi_{\alpha }^J$ the corresponding objects to underline their dependence on $J$.

Denoting by $\cala=\cala_M$ and by $\cala^*=\cala_M^*$ the sheaves of complex valued (resp. complex valued nonvanishing) functions on $M$, we observe the following exact sequence

$$0\to \zz \buildrel{i}\over {\longrightarrow} \cala \buildrel{exp(2\pi i
\cdot )}\over {\longrightarrow} \cala^* \to 0. \eqno(A1.1.1)$$

Here $i$ is an imbedding of the sheaf of locally constant integer valued functions into $\cala$, end $exp(2\isl\pi \cdot ):f\to e^{2\isl\pi f}$. The sequence(A1.1.1) gives rise to the following long exact sequence of Čech cohomologies $$0=\sfh^1(M,\cala)\buildrel{\exp(2\isl\pi  \cdot)}\over {\longrightarrow}
\sfh^1(M,\cala^*\buildrel{\delta }\over {\longrightarrow} \sfh^2(M,\zz )
\to \sfh^2(M,\cala)=0. \eqno(A1.1.2)$$ Equalities $\sfh^1(M,\cala)=\sfh^2(M,\cala)=0$ follow from the fact that the sheaf $\cala$ admits a partition of unity.

Classes from $\sfh^1(M,\cala^*)$ are the defining cocycles of complex line bundles - one more possible definition of the line bundle. In terms of local trivializations $\{ \phi_{\alpha }\} $ such cocycles can be obtained as $\phi_{\alpha ,\beta }=\phi_{\alpha }\scirc \phi_{\beta }^{-1}
:(U_{\alpha }\cap U_{\beta })\times \cc \to (U_{\alpha }\cap U_{\beta })
\times \cc $, , $\phi_{\alpha ,\beta }\in \cala^*_{U_{\alpha }\cap 
U_{\beta }}$.

Definition A1.1.1. [*If $\{ \phi_{\alpha ,\beta }\}
\in \sfh^1(M, \cala^*)$ is a defining cocycle of a complex line bundle $L$, then $\delta (\{ \phi_{\alpha ,\beta }\} )\in \sfh^2(M, \zz)$ is called the first Chern class of $L$ and is usually denoted as $c_1(L)$.* ]{}

For the complex bundle $E$ of complex rank $r$ the first Chern class is defined as $c_1(\Lambda^rE)$. If $E=TX$, the tangent bundle to an almost-complex manifold $X$, then one simply writes $c_1(X)$ or $c_1(X,J)$ if an almost-complex structure is needed to be specified.

If an almost-complex structure $J$ on the real bundle $E$ varies continuously, then the corresponding trivializations $\{ \phi_{\alpha }\} $ above (on $\Lambda^r E$) can be obviously chosen to also vary continuously. Thus $c_1(E,J)$ varies continuously. But $c_1(E,J)\in \sfh^2(M, \zz )$, , takes values in a discrete group. So, it does not change at all. This simple but important observation together with [*Corollary 1.2.2*]{} leads to the following

Corollary A1.1.1. [*Let $\omega $ be a nondegenerate exterior two-form on the even-dimensional real manifold $X$. Then $c_1(X,J)$ does not depend on the choice of $\omega $-calibrating (and even $\omega $-compatible) almost-complex structure $J$.* ]{}

*A1.2. Riemann-Roch Formula and index of $\dbar $-type operators.*

For the holomorphic bundle $E$ of complex rank $r$ over a compact Riemann surface $S$ denote by $\calo_E$ the sheaf of its holomorphic sections. In a usual way one defines the cohomology groups $\sfh^0(S, \calo_E$ and $\sfh^1(S, \calo_E)$. Denote $h^i=dim_{\cc }\sfh^i$. By $g$ we denote the genus of $S$ and $$c_1 = \int_Sc_1(E).$$ These numbers are related by the classical

Riemann-Roch Formula.

*For a holomorphic bundle $E$ over a Riemann surface $S$ one has*

$$h^0-h^1 =  c_1 + r\cdot (1 - g). \eqno(A1.2.1)$$

This formula can be interpreted as the formula for the index of some operators acting on the spaces of smooth sections of $E$. On the sheaf of smooth sections of a holomorphic bundle over a Riemann surface, or more generally over a complex manifold, one can define $\dbar $-operators. Those are $\cc $-linear operators $\dbar :\Gamma^{1,p}(S,E)\to \Gamma^p_{0,1}(S,E)$ satisfying

$$\dbar (f\cdot \sigma ) = \dbar_Sf\otimes \sigma + f\cdot \dbar \sigma .
\eqno(A1.2.2)$$

Here by $\Gamma^{1,p}(S,E)$ we denote the Sobolev space of $(1,p)$- smooth sections of $E$, and by $\Gamma^p_{0,1}(S,E)$ the space of $(0,1)$ $L^p$-integrable forms with coefficients in $E$. $\dbar_S$ is a canonical $\dbar $-operator on $S$.

If one additionally fixes some Hermitian metric on $E$, then such an operator is determined uniquely if one imposes the additional condition to preserve the scalar product, see \[G-H\] Ch.0 for details.

The operator $\dbar $ being elliptic is Fredholm and its index is defined as $\ind \dbar := \dim \ker \dbar \allowbreak - \dim \coker \dbar $. Remark that $\ind \dbar = h^0-h^1$ and so by the Riemann-Roch formula $$\ind \dbar = c_1 + r\cdot (1 - g). \eqno(A1.2.3)$$

Definition A1.2.1. [*An $\rr $-linear operator $D:\dbar :\Gamma^{1,p}(S,E)\to \Gamma^p_{0,1}(S,E)$ which can be represented as $\dbar + R$ with $R\in C^0(S, \hom _{\rr }(E,\Lambda^{0,1}
\otimes E))$ shall be called a $\dbar $-type operator.* ]{}

This is again an elliptic (Fredholm) operator, which is homotopic to $\dbar $. So by the homotopy invariance of the index we have that for any $\dbar $-type operator $D$ $$\ind_\rr D = 2\cdot (c_1 + r\cdot (1 - g)). \eqno(A1.2.4)$$ The reader should take into account that since $D$ is real, the real dimensions in the last formula are considered. That is why the number $2$ appears.

[Lecture 2]{}

[Local Existence of Curves]{}

[2.1. Sobolev Imbeddings, Cauchy-Green Operators, Calderon-Zygmund Inequality.]{}

For a natural $k$ and a real $p\ge 1$ the Sobolev space $L^{k,p}
(\Delta ,\cc^n)$ consists of functions $f\in L^p(\Delta ,\cc^n)$ such that their derivatives up to the order $k$ are also in $L^p(\Delta ,\cc^n)$. One puts $\Vert f\Vert_{k,p}:= \Sigma_{0\le \vert i\vert \le k}\Vert D^if\Vert_p$.

For $0<\alpha <1$ one considers also the Hölder spaces $C^{k,\alpha }(\Delta , \cc^n)$. $C^{k,\alpha }(\Delta , \cc^n)$ is the space of $f\in C^k(\Delta )$ such that $$\Vert f\Vert_{k,\alpha }:=
\Vert f(x)\Vert_{C^k}  + \sup_{x\not= y}{\Vert D^kf(x)-D^kf(y)\Vert
\over \vert x-y\vert^{\alpha }}<\infty.$$ Where $D^kf$ denotes the vector of derivatives of $f$ of order $k$.

One has the following important

Sobolev Imbeddings.

**

\(a) For $1\le p<2$ and $1\le q\le {2p\over 2-p}$

$$L^{k,p}(\Delta , \cc^n) \subset  L^{k-1,q}(\Delta , \cc^n); \eqno(2.1.1)$$

\(b) for $2<p\le \infty $ and $0\le \alpha \le 1-{2\over p}$

$$L^{k,p}(\Delta , \cc^n) \subset C^{k-1,\alpha }(\Delta , \cc^n).
\eqno(2.1.2)$$ Moreover, the imbedding (2.1.1) is a bounded operator and imbedding (2.1.2) is compact.

Existence of such imbeddings means, in particular, the existence of universal constants $C_q$ and $C_{\alpha }$ s.t. $\Vert f\Vert_{L^{k-1,q}(
\Delta )}\le C_q\cdot \Vert f\Vert_{L^{k,p}(\Delta )}$ and $\Vert f\Vert_{C^{k-1,\alpha }(\Delta )}\le C_{\alpha }\cdot 
\Vert f\Vert_{L^{k,p}(\Delta )}$. This is the most frequently used form of Sobolev Imbedding Theorem.

In the Schwartz spaces $\cals (\cc ,\cc^n)$ and $\cals'(\cc ,\cc^n)$ consider the operators $\d ={\d \over \d z}$, $\dbar = {\d \over \d \bar z}$ and $T=T_{CG}={1 \over 2\pi iz}*(\cdot )$, $\bar T={1 \over 2\pi i\bar z}*
(\cdot )$. Note that Cauchy-Green operators $T$ and $\bar T$ act from $\cals $ only to $\cals'$. Nevertheless one has the following identities on $\cals $, on $L^p(\cc )$ and on $\cals'$: $$\dbar \scirc T=T\scirc \dbar = \id , \eqno(2.1.3)$$ and $$\d \scirc \bar T=\bar T\scirc  \d = \id .$$

Recall also the

Calderon-Zygmund Inequality. [*For all $1<p<\infty $ there is a constant $C_p$ such that for all $f\in L^p(\cc ,\cc^n )$ $$\Vert (\d \scirc T)(f)\Vert_{L^p}\le C_p\cdot \Vert f\Vert_{L^p}
\eqno(2.1.4)$$ and $$\Vert (\dbar \scirc \bar T)(f)\Vert_{L^p}\le C_p\cdot
\Vert f\Vert_{L^p}.$$* ]{}

This implies that for $f\in L^p(\cc ,\cc^n )$ and $g=Tf$ (or $g=\bar Tf$) one has $g\in L^p_\loc(\cc ,\cc^n )$ and $\Vert dg\Vert_{L^p}(\cc ,\cc^n )\le (1+C_p)
\Vert f\Vert_{L^p(\cc ,\cc^n)}$. For the proof of (2.1.4) we refer to \[Mc-Sa\].

Properties (2.1.3) and (2.1.4) also imply that the Cauchy-Green operator $T$ is a bounded linear operator from $L^{k,p}(\Omega , \cc^{n})$ to $L^{k+1,p}(\Omega , \cc^{n})$ if $\Omega \comp \cc $. The same is true in Hölder spaces $C^{k,\alpha }$.

We shall repeteadly use

Hölder Inequality.

*Let $p,q>1$ and let ${1\over p}+{1\over q}=
{1\over r}$. Then for all $f\in L^p(\Delta )$, $g\in L^q(\Delta )$ we have that $fg\in L^r(\Delta )$ and*

$$\Vert fg\Vert_{L^r(\Delta )}\le \Vert f\Vert_{L^p(\Delta )}\cdot 
\Vert g\Vert_{L^q(\Delta )}.\eqno(2.1.5)$$

The behavior of $L^p$-norms under dilatations is also frequently used.

Lemma 2.1.1.

*Let $h\in L^p(\Delta )$, $\tau >0$ and $\pi_{\tau }:
\Delta\to \Delta $ denotes the contraction $\pi_{\tau }:z\to \tau z$. Put $\pi_{\tau }^*h(z)=h(\pi_{\tau }(z))=h(\tau z)$ and $\Delta_{\tau }=
\pi_{\tau }(\Delta )=\{ z\in \cc : \vert z\vert <\tau \} $. Then*

$$\Vert \pi_{\tau }^*h\Vert _{L^p(\Delta )} = \tau^{-{2\over p }}\cdot 
\Vert h\Vert_{L^p(\Delta_{\tau })}.\eqno(2.1.6)$$

Proof. [ $$\Vert \pi_{\tau }^*h\Vert _{L^p(\Delta )} = \bigl(\int_{\Delta }\vert h(\tau z)
\vert^pdz\wedge d\bar z\bigr)^{1\over p} =  \bigl(\int_{\Delta_{\tau } }
\vert h(w)\vert^p{1\over \tau^{2}}dw\wedge d\bar w\bigr)^{1\over p} =
\tau^{-{2\over p}}\cdot \Vert h\Vert_{L^p(\Delta_{\tau })},$$ where $w=\tau z$. ]{} [2.2. Local Existence of Curves]{}

Proposition 2.2.1. [*(Local existence of curves)*]{} [*Let $(X,J)$ be an almost-complex manifold with $J$ of class $C^{k,\alpha }, k\ge 1, \alpha
>0$, and $x_0\in
X$. Then for every $v\in T_{x_0}X$ small enough there exists a $J$-complex curve $u:(\Delta ,0)\to (X,x_0)$ such that $du(0)({\d \over \d x})=v$.* ]{}

Proof.

Take a chart $B\ni x_0$ with $x_0=0$ and $J(0)=J\st$. Denote by $z=x+iy$ the coordinate in $\Delta $, and by $u_1,...,u_{2n}$ the coordinates in $B$. The Cauchy-Riemann equation for $u:(\Delta ,J_{\Delta })
\to (B,J)$ has, in our local coordinates, the form $${\d u\over \d y} + J(u)du(J_{\Delta }{\d \over \d y}) = 0,$$ where $J_{\Delta }$ denotes the canonical a.-c. structure in $\Delta $, , a multiplication by $i$. Using the fact that $J_{\Delta }({\d \over \d y})=-{\d \over
\d x}$, one obtains $${\d u\over \d y} - J(u){\d u\over \d x} = 0.$$ This is equivalent to $${\d u\over \d \bar z} - Q(u){\d u\over \d z} = 0, \eqno(2.2.1)$$ where $Q(u)=[J\st + J(u)]^{-1}\scirc [J\st - J(u)]$.

After rescaling, , considering $Q_t(u):=Q(tu)$ and $u_{t,\tau }(z)
:= t^{-1}u(\tau z)$, we can assume that $\Vert Q\Vert_{C^1}$ is sufficiently small. Note also, that $(2.2.1)$ is equivalent to the holomorphicity of $h=(\id - T[Q(u){\d \over \d z}])u$.

Consider a $C^1$-mapping $$\Phi :]-1,1[ \times C^{1,\alpha }(\Delta ,B)\to C^{1,\alpha }(\Delta ,\cc^n)$$ given by $$\Phi (t,u) = (\id - T[Q(tu){\d \over \d z}])u. \eqno(2.2.2)$$ Note that $\Phi (0,u) = \id (u) = u$. Thus, the Implicit Function Theorem tells us that there exists a $t_0>0$, such that for $\vert t\vert <t_0$ the map $\Phi (t,\cdot )$ is a $C^1$-diffeomorphism of the neighborhood of zero in $C^{1,\alpha }(\Delta ,B)$ onto the neighborhood of zero $V$ in $C^{1,\alpha }(\Delta , \cc^n)$.

For $w\in \cc^n$ small enough, a holomorphic function $h_w(z):=z\cdot w$ belongs to $V$. Put $u_{t,w}=\Phi (t,\cdot )^{-1}(h_w)$. Then $tu_{t,w}$ is $J$-holomorphic. In fact, $h_w=\Phi (t,\cdot )[{1\over t}u_{t,w}]=
{1\over t}[\id - TQ(u_{t,w}){\d \over \d z}]u_{t,w}$. Moreover $u_{0,w}=
h_w$, so than $du_{0,w}(0)=w$-linear map from $\cc $ to $\cc^n$. This shows that for $t>0$ small enough $w\to du_{t,w}(0)({\d \over \d x})$ is a diffeomorphism between the neighborhoods of zero in $\cc^n$.

We immediately obtain the following

Corollary 2.2.2. [*Every almost complex structure on a Riemann surface is complex.*]{}

[2.3. Generalized Calderon-Zygmund Inequality. ]{}

Consider a continuous linear complex structure $J(z)$ on the trivial bundle $\cc \times \rr^{2n}\to \cc $, , $J(z)$ is a continuous family of endomorphisms $\rr^{2n}\to \rr^{2n}$ with $J(z)^2=-Id$. Define an operator $\dbar_{J}:\cals' (\cc ,\rr^{2n})\to \cals'(\cc ,\rr^{2n})$ by the formula $$(\dbar_{J}f)(z)={1\over 2}[\d_xf(z) + J(z)\d_yf(z)].$$ If $J_0$ is another continuous complex structure on $\cc \times \rr^{2n}$, then for $f\in L^p(\cc ,\rr^{2n})$ it holds that $$\Vert (\dbar_J\scirc T - \dbar_{J_0}\scirc T)f\Vert_{L^p(\cc )}\le
\Vert J - J_0\Vert_{L^{\infty }(\cc )}\cdot \Vert d(Tf)\Vert_{L^p(\cc )}
\le$$ $$\le \Vert J - J_0\Vert_{L^{\infty }(\cc )}(1+C_p)\Vert f\Vert_{L^p(\cc )}.
\eqno(2.3.1)$$ If we take $J_0(z)\equiv J\st $, the standard structure in $\cc^n$, then as was remarked above, $\dbar_{J_0}\scirc T: L^p(\cc ,\cc^n)\to L^p(\cc,\cc^n)$ is an identity. So from (2.3.1) we see that there exists $\eps_p={1\over 1+C_p}$ such that if $\Vert J - J\st \Vert <\eps_p$, then $\dbar_{J}\scirc T: L^p(\cc,
\cc^n)\to L^p(\cc , \cc^n)$ is an isomorphism. Moreover, since $\dbar_J\scirc T = \dbar_{J\st}\scirc T + (\dbar_J-\dbar_{J\st})\scirc T$, we have $$(\dbar_J\scirc T)\inv = (\id + (\dbar_J-\dbar_{J\st})\scirc T)^{-1} =
\Sigma_{n=0}^{\infty }(-1)^n[(\dbar_J -
\dbar_{J\st})\scirc T]^n.\eqno(2.3.2)$$ This shows, in particular, that $(\dbar_J\scirc T)\inv $ does not depend on $p>1$. Now we shall prove the following statement, which can be viewed as a generalization of the Calderon-Zygmund estimate.

Lemma 2.3.1. [*For any $u\in L^{1,2}(\cc ,\rr^{2n})$ with compact support, any continuous $J$ with $\Vert J-J\st \Vert_{L^{\infty }(\cc ,End(\rr^{2n}))}<
\eps_p$ the condition $\dbar_Ju\in L^p(\cc ,\rr^{2n})$ implies $$\Vert du \Vert_{L^p(\cc )}\le C\cdot \Vert \dbar_Ju\Vert_{L^p(\cc )}
\eqno(2.3.3)$$ with some $C=C(p,\Vert J-J\st \Vert_{L^{\infty }(\cc )})$.* ]{}

Proof. Put $v=u-T\scirc \dbar_{J\st}u $. Then $\dbar_{J\st}v=0$. So $v$ is holomorphic and descends at infinity. Thus $v=0$, which implies $u=(T\scirc \dbar_{J\st})u$. By the Calderon-Zygmund inequality, to estimate $\Vert du\Vert_{L^p(\cc )}$ it is sufficient to estimate $\Vert \dbar_{J\st}u
\Vert_{L^p(\cc )}$. We have $(\dbar_J\scirc T)\scirc \dbar_{J\st}u =
\dbar_Ju
\subset L^p(\cc )\cap L^2(\cc )$. From (2.3.2) we obtain $\dbar_{J\st}u\in
L^p(\cc )\cap L^2(\cc )$ with an estimate $$\Vert \dbar_{J\st }u\Vert_{L^p(\cc )}\le \sum_{n=0}^{\infty }\Vert
(\dbar_J-\dbar_{J\st})\scirc T\Vert_p^n\cdot \Vert \dbar_Ju\Vert_{L^p(\cc )}
\le
C\cdot \Vert \dbar_Ju\Vert_{L^p(\cc )},$$ where $C=C(p,\Vert J-J\st\Vert_{L^{\infty }})=\sum_{n=1}^{\infty } 
\left(\vert J-J\st\vert_{L^{\infty }}(1+C_p)\right)^n$, provided that $\Vert J-J\st\Vert_{L^{\infty }}<\eps_p={1\over 1+C_p}$.

This yields (2.3.3).

Corollary 2.3.2. [*If $u\in (L^{1,2}\cap C^0)(\Delta ,\rr^n)$ is $J$-holomorphic for a continuous a.-c. structure $J$ belongs to $L^{1,p}_\loc
(\Delta ,\rr^2)$ for all $2<p<\infty $.* ]{}

[2.4. First A priori Estimate.]{}

Let $(X,J_0)$ be an almost complex manifold, and the tensor $J_0$ is supposed to be of class $C^0$ (continuous) only. Let $K\comp X$ be some compact in $X$. We fix some Riemannian metric $h$ on $X$ and denote by $\mu (J_0)$ the module of continuity of $J_0$. All norms and distances are taken with respect to $h$.

Recall that for a map $u:X\to Y$ into a metric space an oscillation of $u$ on a subset $D\subset X$ is defined as $\osc (D,u):=\sup \{ d_Y(u(x),u(y)):
x,y\in D\} $.

Lemma 2.4.1.

*(First a priori estimate). For every $2< p<\infty $ there exists an $\eps_1 =\eps_1(p,\mu (J_0),K,h)$ and $C_p=C(p,\mu (J_0),K,h)$ such that for any $J\in C^0$, $\vert\vert J - J_0\vert\vert_{C^0(K)}<\eps_1 $ and every $J$-holomorphic map $u\in C^0\cap L^{1,2}(\Delta ,X)$ with $u(\Delta )\subset K$, satisfying $\Vert du\Vert_{L^2(
\Delta )}<\eps_1 $ the following holds $$\Vert du\Vert_{L^p({1\over 2}\Delta )}\le C_p\cdot
\Vert du\Vert_{L^2(\Delta )}.\eqno(2.4.1)$$*

Proof.

This will be done in several steps.

*Step 1. We prove first the inequality (2.4.1) for the case when $K\comp U\subset \cc^n$, $h$ Euclidean metric, $J_0$ the standard complex structure in $\cc^n=\rr^{2n}$, and $\Vert J-
J_0\Vert_{L^{\infty }}< \eps_p$ for $\eps_p$ from [*Lemma 2.3.1*]{}.*

Note that in this case the condition that $\Vert du\Vert_{L^2(\Delta )}$ should be small is not needed.

To prove the first step consider a $J$-holomorphic map $u:\Delta \to (\rr^{2n},J)$, $u(\Delta )\subset K$ and $\Vert J-J\st \Vert
<\eps_p$. Define on $\Delta \times \rr^{2n}$ a linear complex structure $J(z)=(u^*J)(z)$. Then $u$ defines a $J$-holomorphic section of $(\Delta \times \rr^{2n}, J)$ with $\Vert J-J\st \Vert_{L^{\infty } (\Delta )}
< \eps_p$. Extend $J$ onto $\cc \times \rr^{2n}$ with the same estimate.

Let $\psi $ be a non-negative cut-off function supported in $\Delta_{3/4}$ and equal to one on $\Delta_{1/2}$. Put $u_1=u\psi $. Then $u_1\in 
L^{1,2}(\Delta)$ (because $u\in L^{1,2}(\Delta)$) and $\dbar_Ju_1=u\dbar_J\psi
\in L^p(\cc )$ with $\Vert \dbar_Ju_1\Vert_{L^p(\Delta )}=\Vert u\dbar_J\psi 
\Vert_{L^p(\Delta )} \le C\Vert du\Vert_{L^2(\Delta )}$ for any $p$ by the Sobolev imbedding $L^{1,2}(\Delta ,\cc )\to L^p(\Delta ,\cc )$ for all $p<\infty $. Now the generalized Calderon-Zygmund estimate (2.3.3) applies in obtaining an estimate of *Step 1.*

Using the Sobolev imbedding $L^{1,p}\subset C^{1-{2\over p}}$ and obvious properties of $L^p$-norms with respect to dilatations, one easily derives from [*Step 1*]{} the following

*Step 2. *Fix $2<p<\infty $. There exists $\eps_2=\eps_2(p,\mu (J_0),K,h)>
0$ such that for every $J$ with $\Vert J-J_0\Vert_{L^{\infty }}<\eps_p$ and any $J$-holomorphic $u$ with $\osc(u,\Delta (x,r))\le \eps_2$ on any $\Delta (x,r)\subset \Delta$ it holds that $$\osc(u,\Delta (x,{r\over 2}))\le  Cr^{{2\over p}-1}\cdot
\vert \vert du\vert \vert_{L^2(\Delta (x,r))},\eqno(2.4.2)$$**

and $$\Vert du\Vert_{L^p(\Delta (x,r/2))}\le C_p\cdot 
\Vert du\Vert_{L^2(\Delta (x,r))}.$$

In fact, $$\osc(u,\Delta (x,{r\over 2}))\le C_1\vert \vert du\vert \vert_
{L^p(\Delta (x,r/2))}=C(\int_{\Delta (0,{r\over 2})}\Vert du\Vert^pdxdy)^{1
\over p}=$$ $$=C(\int_{\Delta (0,{1\over 2})}\Vert du(rw){1\over r}\Vert^p)^{1\over p}=
d(rw)d\overline{(rw)})^{1\over p}=r^{2-p\over p}\Vert du_r\Vert_{L^p(\Delta
(0,{1\over 2}))}$$ for $u_r(w)=u(rw)$.

Furthermore, if $\eps_2>0$ is such that $\Vert J(x) - J(y)\Vert 
<\eps_p$ for $h(x,y)<\eps_2$, $x,y\in K$, then applying Step 1 to $u_r$ we get

$$r^{2-p\over p}\Vert du_r\Vert_{L^p(\Delta (0,{1\over 2}))}\le Cr^{2-p\over p}
\Vert du_r\Vert_{L^2(\Delta (0,1))} = 
Cr^{2-p\over p}\Vert du\Vert_{L^2(\Delta (0,r))}.$$

Let $\alpha \ge 0$ be the following continuous function: $\alpha \equiv 1$ for $t\le {1\over 2}$ and $\alpha (t)\equiv 0$ for $t\ge 3/4$. On the interval ${1\over 2}\le t\le {3\over 4}, \alpha (t)=3-4t$. Put for $x\in \Delta $

$$f(x):=\max \{ t: 0\le t \le {1\over 8}, \osc(u,\bar\Delta(x,t\cdot \alpha
(\vert x\vert )))\le \eps_2\}.$$

Clearly $f$ is continuous and $f\equiv {1\over 8}$ for $ {3\over 4}\le
\vert x \vert <1$.

*Step 3. *There exists an $\eps_1=\eps_1(p,J_0,K,h)>0$ such that $f(x)\equiv {1\over 8}$.**

Suppose that there is an $x_0$ with $f(x_0)=\min \{ f(x):x\in \Delta \}
< {1\over 8} $. It is clear that $f(x_0)>0$.

Take the disc $\Delta (x_0,a), a:=f(x_0)\alpha (\vert x_0\vert )$. Note that $$\osc(u,\Delta (x_0,a))=\eps_2\eqno(2.4.3)$$ Condition (2.4.2) together with the Sobolev embedding $L^{1,4}(\Delta )\subset C^{0,{1\over 2}}(\Delta )$ tells us (because $\osc(u,\Delta (x_0,a))=\eps_2$) that $\osc(u,\Delta (x_0,{a\over 2}))$ $\le C\cdot \vert \vert du\vert \vert_{L^2(\Delta (x_0,a))}$. Take a point $x_1 \in \Delta (x_0,a)$ on the distance not more then ${3\over 4}a$ from $x_0$. We have from $f(x_0)={a\over \alpha (\vert x_0\vert )}$ that $f(x_1)\ge {a\over \alpha (\vert x_0\vert )}$ and thus $f(x_1)\alpha (\vert x_0\vert )\ge a$. At the same time $\alpha (\vert
x_1\vert )\ge \alpha (\vert x_0\vert ) - 3a$, so $f(x_1)\alpha (\vert x_1
\vert )\ge a-3a\cdot f(x_1)\ge {a\over 2}$, because $f(x_1)\le {1\over 8}$. That means that $\osc(u,\Delta(x_1,{a\over 2}))\le \eps_2$ and thus $\osc(u,\Delta(x_1,{a\over 4}))\le C\cdot \vert \vert du\vert \vert_{L^2
(\Delta (x_1,{a\over 2}))}$. Thus, $\osc(u,\Delta(x_0,a))\le 4C\cdot \vert \vert du\vert \vert_{L^2(\Delta )}$. If $\eps_1 $ were taken smaller than ${\eps_2\over 4C}$, then we would obtain a contradiction with (2.4.3). Step 3 is proved.

This means that $\osc(u,\Delta(x,{1\over 8}))\le \eps_2$ for any $x\in
{1\over 2}\Delta $. Therefore, Step 2 with $r={1\over 8}$ gives us the conclusion of the Lemma.

This statement can be used to prove once more the following statement, which was already proved in Corollary 2.3.2:

Corollary 2.4.2. [*A $J$-holomorphic map $u:\Delta
\to (X,J)$ is $L^{1,p}$-continuous for any $p<\infty $, provided $J$ is of class $C^0$.* ]{}

Proof. Note that $u_{\eps }(z):=u(\eps z)$ is also $J$-holomorphic and $\Vert du_{\eps }\Vert_{L^2(\Delta (0,1))} =
\Vert du\Vert_{L^2(\Delta (0,\eps ))}$. For $\eps $ small enough we shall have $\Vert du_{\eps }\Vert_{L^2(\Delta (0,1))} =
\Vert du\Vert_{L^2(\Delta (0,\eps ))}<\eps_1 $ - from Lemma 2.4.1. Now estimate (2.4.1) gives us $L^{1,p}$-continuity of $u_{\eps }$ and thus of $u$ in the neighborhood of zero.

2.5. Convergence outside of a finite set of points.

Another immediate consequence of the a priori estimate (2.4.1) is the following

Corollary 2.5.1. [*Let $\{ J_n\}$ be a sequence of continuous almost complex structures on $X$ such that $J_n\to J$ in $C^0$-topology on $X$. Let $u_n\in C^0\cap L^{1,2}_{loc }(\Delta ,X)$ be a sequence of $J_n$-holomorphic maps such that $\Vert du_n\Vert_{L^2(\Delta )}\le \eps_1$. Then there exists a subsequence $u_{n_k}$ which $L^{1,p}_\loc$-converge to a $J$-holomorphic map $u_\infty$ for all $p\ge 2$.*]{}

Proof. The main estimate together with the Sobolev imbedding $L^{1,p}
\subset C^{1-{2\over p}}$ gives us a subsequence $\{ u_{n_k}\} $ which converges to $u_\infty$ in $C^{\alpha }(\Delta )$ for any $\alpha
<1$. Take $\phi \in C_0^{\infty }(\Delta )$, $\phi\mid_{\Delta (r)}
\equiv 1$ and consider $\dbar_{J_{n_k}}(\phi u_{n_k})=\d_x(\phi u_{n_k})
+ J_{n_k}(u_{n_k})\d_y(\phi u_{n_k})=$ $(\d_x\phi +J_{n_k}(u_{n_k})
\d_y\phi)u_{n_k}$, which is $C^0$ and thus $L^p$-convergent for any $p\ge 2$. The a priori estimate for $\dbar_J$-operator (Calderon-Zygmund inequality 2.3.1) gives us $L^{1,p}_\loc$-convergency of $\{ u_{n_k}\} $.

This corollary implies another one:

Corollary 2.5.2. [*Let $\{ J_n\}$ be a sequence of continuous almost-complex structures on $X$ such that $J_n\to J\in C^0$ in $C^0$-topology on $X$. Let $u_n\in C^0\cap L^{1,2}_{loc }(\Delta ,X)$ be a sequence of $J_n$- holomorphic maps such that $\Vert du_n\Vert_{L^2(\Delta )}\le C$. Then there exists a subsequence $\{ u_{n_k}\} $ and a finite set of points $\{ x_1,...,x_l\} $ in $\Delta $, such that $\{ u_{n_k}\} $ is $L^{1,p}_\loc$-convergent on compacts in $\Delta \setminus \{ x_1,...,x_l\} $ for all $p\ge 2$. Moreover $l\le {3C\over \eps_1}$.* ]{}

Proof. For any $n\in \nn $, cover our disk $\Delta $ by a finite number of disks of radii ${1\over n}$ in such a way that no point in $\Delta $ belongs at the same time to more than $3$ of them. Then there is at most $[{3C\over \eps_1}$ disks, on which the energy of our maps (after going to a subsequence) is more then $\eps_1$. Thus, on the complement to those disks the [*Corollary 2.5.1*]{} applies.

Taking $n$ bigger and bigger (and passing to a subsequence) we obtain the desired result.

One can prove also the following regularity statements about $j$-holomorphic maps:

Lemma 2.5.3.

*a) If the structure $J$ is Lipschitz then $J$-holomorphic maps are of class $C^1$;*

2\) If $J\in C^k$ then $J$-holomorphic maps are in $C^{k,\alpha }$ for all $\alpha <1$.

[Lecture 3]{}

[Positivity of Intersections of Complex Curves.]{}

*3.1. Unique Continuation Lemma.*

We start with a unique continuation-type lemma for $\dbar$-unequalities, namely with [*Lemma 3.1.1*]{} (compare with \[Ar\] and \[Hr-W\]). In the proof we shall use the following special version of the theorem of Harvey and Polking \[Ha-Po\]:

Theorem of Harvey-Polking. [*Let $f:\Delta
\to \cc^n$ be locally $L^2$-integrable. Assume that for some $g \in
L^1_\loc (\Delta, \cc^n)$ the equation $\dbar f =g$ holds (in the weak sence) in the punctured disc $\check\Delta$. Then $\dbar f =g$ holds in the whole disc $\Delta$.* ]{}

Lemma 3.1.1.

*Suppose that the function $u\in L^2_\loc(\Delta,
\cc^n)$ is not identically $0$, $\dbar u\in L^1_\loc(\Delta,\cc^n)$ and satisfies [*a.e.*]{} the inequality $$\vert \dbar u \vert \le h\cdot \vert u \vert\eqno(3.1.1)$$ for some nonnegative $h\in L^p_\loc(\Delta)$ with $2<p<\infty$. Then*

$u\in L^{1, p}_\loc(\Delta)$, in particular $u\in C^{0,\alpha}_\loc
(\Delta)$ with $\alpha\deff1-{2\over p}$;

for any $z_0\in\Delta$ such that $u(z_0)=0$ there exists $\mu\in\nn$—the multiplicity of zero of $u$ in $z_0$—such that $u(z)=(z-z_0)^\mu \cdot g(z)$ for some $g\in L^{1, p}_\loc (\Delta)$ with $g(z_0)\not=0$.

Proof. Statement is easily obtained by increasing the smoothness argument. Let us for the methodological reasons give it in full details.

*Step 0. $u$ is in $L^{p_0}_\loc$ for $p_0:=p>2$.*

From (3.1.1) and the Hölder inequality we see that $h\vert u\vert \in L_\loc^{{1\over {1\over 2}+{1\over p}}}=
L^{{2p\over p+2}}_\loc$. Therefore one obtains $\bar\partial u\in L^{2p\over p+2}_\loc(\Delta)$. Consequently, $u\in L^{1, {2p\over p+2}}_\loc(\Delta)$ due to ellipticity of $\dbar$ and because ${2p\over p+2} >1$. By the Sobolev imbedding we have that $u\in L_\loc^{{2{2p\over p+2}\over 2-{2p\over p+2}}}=L^p_\loc(\Delta)$.

Thus we proved that $u\in L^{p_0}_\loc$ for $p_0:=p>2$.

*Step 1. $u$ is in $L^{p_1}_\loc$ for $p_1:={2p\over 4-p}>p_0$.*

Again, $\vert \dbar u\vert \le h\vert u\vert\in 
L_\loc^{{1\over {1\over p}+{1\over p}}}=L_\loc^{{p\over 2}}$. Therefore $u\in L_\loc^{1,{p\over 2}}\subset L_\loc^{{p\over 2-{p\over 2}}}= 
L_\loc^{{2p\over 4-p}}$.

*Step n. There exists $r>1$ such that $u\in L_\loc^{p_n}$, where $p_n\ge rp_{n-1}$ for all $n$.*

$u\in L^{p_{n-1}}$ implies $\vert \dbar u\vert \le h\vert u\vert 
\in L^{{1\over {1\over p}+{1\over p_{n-1}}}}=L^{{pp_{n-1}\over p+p_{n-1}}}$. Therefore $u\in  L^{1,{pp_{n-1}\over p+p_{n-1}}}\subset 
L^{{2pp_{n-1}\over 2p+2p_{n-1}-pp_{n-1}}}=L^{p_n}$ with $p_n={1\over {1\over p}
+{1\over p_{n-1}}-{1\over 2}}$. One easily checks that ${p_n\over p_{n-1}}\ge 
r>1$, which does not depend on $n$.

*Step $\infty $. $u$ is in $C^{0,\alpha }$ with $\alpha =1-{2\over p}$.*

When $p_n\to \infty $ $\vert \dbar u\vert \le h\vert u\vert \in 
L^{{1\over {1\over p}+{1\over p_n}}}=L^q$, where $q\sim p$, because ${1\over 
p_n}\sim 0$. Therefore $u\in L^{1,q}$ with $q>2$ close to $p$. Sobolev Imbedding tells in this case that $L^{1,q}\subset C^{0,\beta }$ with $\beta 
=1-{1\over q}$. Finally we see that $\dbar u\in L^p$, therefore $u\in C^{0,
\alpha }$.

Now suppose that $u(z_0)=0$. Then, due to the Hölder continuity, we have $\vert u(z) \vert \le C\vert z-z_0 \vert^\alpha$ for $z$ close enough to $z_0$ and consequently $u_1(z) \deff u(z) / (z-z_0)$ is from $L^2_\loc(\Delta)
$. The theorem of Harvey-Polking provides that $\dbar u_1\in
L^1_\loc(\Delta)$ and $u_1$ also satisfies inequality (3.1.1). In particular, $u_1$ is also continuous. Iteration of this procedure gives the possibility of defining the multiplicity of zero of $u$ in $z_0$ provided we show that after a finite number of steps we obtain the function $u_N$ with $u_N(z_0)\not= 0$. To do this we may assume that $z_0=0$. Let $\pi_\tau (z)\deff\tau\cdot z$ for $0<\tau<1$. Then $u_\tau\deff
\pi_\tau^*(u)$ satisfies the inequality $\vert \dbar u_\tau \vert \le
\tau\pi_\tau^*h \cdot \vert u_\tau \vert$. Since $$\Vert \tau\pi_\tau^*h \Vert_{L^p(\Delta)} =
\tau^{1-2/p}\cdot \Vert h \Vert_{L^p(\pi_\tau(\Delta)),}$$ we can also assume that $\Vert h \Vert_{L^p(\Delta)}$ is small enough. Fix a cut-off function $\varphi\in C^{\infty }_0(\Delta)$ which is identically $1$ in ${1\over2}\Delta$, the disk of radius $1\over2$. Then $$\Vert \varphi u \Vert_{L^p(\Delta)}\le
C_1\cdot \Vert \varphi u \Vert_{L^{1, {2p\over2+p}}(\Delta)}\le
C_2\cdot \Vert \dbar(\varphi u) \Vert_{L^{2p\over2+p}(\Delta)}\le$$ $$\le C_2\cdot (\Vert \varphi \dbar u \Vert_{L^{2p\over2+p}(\Delta)}+
\Vert \dbar\varphi  u \Vert_{L^{2p\over2+p}(\Delta)})\le
%%$$ $$\le
C_2\cdot ( \Vert \varphi h u  \Vert_{L^{2p\over2+p}(\Delta)}
+  \Vert \dbar\varphi u \Vert_{L^{2p\over2+p}(\Delta)} \le$$ $$\le  C_3(\Vert \varphi u \Vert_{L^p(\Delta)}\cdot \Vert h \Vert _
{L^p(\Delta )} + \Vert u\Vert _{L^p(\Delta \setminus {1\over2}\Delta )}
\cdot \Vert \dbar\varphi \Vert _{L^2(\Delta \setminus {1\over2}\Delta )}).$$

Here we used the fact that the support of $\dbar\varphi$ lies in $\Delta\bss{1\over2}\Delta$. Since $\Vert h  \Vert_{L^p(\Delta)}$ is small enough we obtain the estimate $$\Vert u \Vert_{L^p({1\over3}\Delta)}
\le C\cdot \Vert u \Vert_{L^p(\Delta\setminus {1\over2}\Delta)}$$ with the constant $C$ independent of $u$. Thus if the multiplicity of zero of $u$ in $z_0=0$ is at least $\mu$, then $$\Vert z^{-\mu}u \Vert_{L^p({1\over3}\Delta)}
\le C\cdot \Vert z^{-\mu}u \Vert_{L^p(\Delta\bss{1\over2}\Delta))}$$ which easily gives $$\Vert u \Vert_{L^p({1\over3}\Delta)} \le C
\left(\hbox{$2\over3$}\right)^{-\mu}
\cdot \Vert u \Vert_{L^p(\Delta\bss{1\over2}\Delta))}.$$ Now one can easily see that either $u$ has isolated zeros of finite multiplicity, or $u$ is identically zero. Yet the last case is excluded by the hypothesis of the lemma.

Lemma 3.1.2. [*Under the hypothesis of [*Lemma 3.1.1*]{} suppose additionally that $u$ satisfies [*a.e.*]{} the inequality $$\vert \dbar u(z) \vert \le \vert z-z_0 \vert^\nu
h(z)\cdot \vert u(z) \vert,\eqno(3.1.2)$$ with $z_0\in \Delta$, $\nu\in\nn$, and $h\in L^p_\loc(\Delta)$, $2<p<\infty$. Then $$u(z)=(z-z_0)^\mu\bigl(P^{(\nu)}(z) + (z-z_0)^\nu g(z)\bigr),
\eqno(3.1.3)$$ where $\mu\in\nn$ is the multiplicity of $u$ in $z_0$, defined above, $P^{(\nu)}$ is a polynomial in $z$ of degree $\le \nu$ with $P^{(\nu)}(z_0)
\not=0$, $g\in L^{1, p}_\loc(\Delta,\cc^n)\hookrightarrow C^{0,\alpha}$, $\alpha=1-{2\over p}$, and $g(z)=O(\vert z-z_0\vert^\alpha)$.* ]{}

**Proof. The proof uses the same idea as in the previous lemma. Define $u_0(z)
\deff \msmall{u(z)\over (z-z_0)^\mu}$ and $h_1(z) \deff h(z)\cdot \vert u_0(z)
\vert$. Due to [*Lemma 3.1.1*]{}, $u_0\in C^{0,\alpha}$, $u_0(z_0)\not=0$, $h_1\in L^p_\loc$, and $u_0$ satisfies [*a.e.*]{} the inequality $\vert \dbar
u_0(z) \vert \le \vert z-z_0 \vert^\nu h_1(z)$.**

Set $a_0=u_0(z_0)$. Since $u_0(z)-a_0 = O(\vert z-z_0 \vert^\alpha)$, we have $u_1\deff\msmall{u_0(z) - a_0 \over z-z_0}\in L^2_\loc$. Applying the theorem of Harvey-Polking once more, we obtain $\vert \dbar u_1 \vert
\le \vert z-z_0 \vert^{\nu-1} h_1$, and consequently $u_1\in C^{0,\alpha}$, $u_1(z)-u_1(z_0)=O(\vert z-z_0 \vert^\alpha)$. Repeating this procedure $\nu$ times, we obtain the polynomial $$P^{(\nu)}(z)=a_0 + (z-z_0)a_1 + \cdots + (z-z_0)^\nu a_\nu$$ with $$a_k\deff\lim_{z\to z_0}
\msmall{ u(z)-\sum_{i=0}^{k-1} (z-z_0)^i a_i \over (z-z_0)^k },
\qquad 0\le k\le\nu,$$ and the function $$g(z)\deff \msmall{{u(z)-P^{(\nu)}(z) \over (z-z_0)^\nu }},$$ which satisfies the conclusion of the lemma.

Corollary 3.1.3. [*Let $J$ be a Lipschitz-continuous almost complex structure in a neighborhood $U$ of $0\in \cc^n$ such that $J(0)=J\st$, the standard complex structure in $\cc^n$. Suppose that $u:\Delta\to U$ is a $J$-holomorphic $C^1$-map with $u(0)=0$. Then there exist uniquely defined $\mu\in\nn$ and a (holomorphic) polynomial $P^{(\mu-1)}$ of degree $\le\mu-1$ such that $u(z)=z^\mu\cdot P^{(\mu-1)} +z^{2\mu -1}v(z)$ with $v(0)=0$ and $v\in L^{1,p}(\Delta,\cc^n)$ for any $p<\infty$.* ]{}

Proof.

In fact, by the hypothesis of the lemma $du + J(u)
\scirc du \scirc J_\Delta =0$ and hence $$\vert \dbar u \vert  =
\bigl\vert \msmall{1\over2}( du + J\st \scirc du \scirc J_\Delta)\bigr\vert =
\bigl\vert \msmall{1\over2} (J\st - J(u))\scirc du \scirc J_\Delta \bigr\vert
\le \Vert J \Vert_{C^{0,1}(B)} \cdot \vert u \vert  \cdot
\vert du \vert.
\eqno(3.1.4)$$

Thus by [*Lemma 3.1.1*]{}, $u(z) = z^\mu w(z)$ with some $\mu \in\nn$ and $w\in L^{1,p}_\loc(\Delta, \cc^n)$ for any $p<\infty$. If $\mu =1$ we are done. Otherwise $ du(z)/z^{\mu-1} =
\mu\,w\,dz + z\,dw \in L^p_\loc(\Delta, \cc^n)$ for any $p<\infty$. Thus, the corollary follows now from [*Lemma 3.1.2*]{}.

Corollary 3.1.4. [*Let $u:S\to (X, J)$ be a $J$-holomorphic map. Then for any $p\in X$ the set $u\inv(p)$ is discrete in $S$, provided that $J$ is Lipschitz-continuous.* ]{}

Proof.

Take an [*integrable*]{} complex structure $J_1$ in some neighborhood $U$ of $p\in X$, such that $J(p)=J_1(p)$. Take also $J_1$-holomorphic coordinates $w_1,\ldots, w_n$ in $U$, such that $w_i(p)=0$. Then the statement follows easily from [*Corollary 3.1.3.* ]{}

*3.2. Inversion of $\dbar_{J}+R$ and $L^{k,p}$-topologies.*

Now suppose that $J\in C^0(\Delta, \endo_\rr(\rr^{2n}))$ satisfies the identity $J^2\equiv -1$,, $J$ is a continuous almost complex structure in the trivial $\rr^{2n}$-bundle over $\Delta$. Define the $\rr$-linear differential first order operator $\dbarj:L^{1, p}(\Delta,\rr^{2n}) \to L^p(\Delta,\rr^{2n})$ by setting $$\dbarj(f)=\msmall{1\over2}\left(
\msmall{\d f \over \d x} + J \msmall{\d f\over \d y} \right).\eqno(3.2.1)$$ For example, for $J\equiv J\st$, the standard complex structure in $\rr^{2n}=
\cc^n$, the operator $\dbarj$ is a usual Cauchy-Riemann operator $\dbar$. The operator $\dbarj$ is elliptic and possesses nice regularity properties in Sobolev spaces $L^{k, p}$ with $1<p<\infty$ and Hölder spaces $C^{k,\alpha}$ with $0<\alpha<1$. The following two statements are typical for (nonlinear) elliptic PDE and produces a result which we need for the purpose of this paper.

Lemma 3.2.1.

*Let $J$ be $C^k$-continuous with $k\ge0$ and $\dbarj$ be as above. Also let $R$ be an $\endo (\rr^{2n})$-valued function in $\Delta$ of class $L^{k,p}$ with $1<p<\infty$. If $k=0$, we also assume that $p>2$. Suppose that for $\dbarj f +Rf \in L^{k,p}$ for some $f\in L^{1,
1}(\Delta,\rr^{2n})$. Then $f\in L^{k+1,p}_\loc (\Delta,\rr^{2n})$ and for $r<1$ $$ i) f \_[L\^[k+1, p]{}((r))]{} C\_1(J, R\_[L\^[k,p]{}]{}, k, p, r) ( f + Rf \_[L\^[k, p]{}()]{} + f \_[L\^1()]{}) $$*

If, in addition, $J$ and $R$ are $C^{k,\alpha}$-smooth with $0<\alpha<1$, then $f\in C^{k+1,\alpha}_{\loc }(\Delta,\rr^{2n})$ and $$ ii) f \_[C\^[k+1,]{}((r))]{} C\_2( J , R \_[C\^[k,]{}]{},k, , r)( f + Rf \_[C\^[k,]{}()]{}+ f \_[L\^1().]{}) $$ If, additionally, $\norm{ J-J\st }_{C^k(\Delta)} + \norm{R }\Vert_{L^{k,p}(\Delta)}$ (resp., $\Vert J-J\st \Vert_{C^{k,\alpha}(\Delta)}+ 
\Vert R \Vert_{C^{k,\alpha}(\Delta)}$) is small enough, then there exists a linear bounded operator $T_{J, R}:L^{k,p}(\Delta,\rr^{2n}) \to L^{k+1, p}(\Delta,\rr^{2n})$ $($resp., $T_{J, R}:C^{k, \alpha}(\Delta,\rr^{2n}) \longrightarrow
C^{k+1,\alpha}(\Delta,\rr^{2n})$ such that $(\dbarj + R)\scirc T_{J, R}\equiv \id$ and $T_{J,
R}(f)\vert_{z=0}=0$.

Proof. The estimates and of the theorem are obtained from the ellipticity of $\dbarj$ by standard methods. See \[Mo\] for the case of general elliptic systems or \[Sk\] for the direct proof. We note only that for $J$ being only continuous the constant $C_1$ in crucially depends on the modulus of continuity of $J$, whereas in all the other cases of the theorem the dependence is essentially only on the corresponding norm of $J$.

Let $T$ be a composition ${\d\over\d z} \scirc G$ where $G(f)$ is the solution of the Poisson equation $\Delta u=f$ with the Dirichlet boundary condition on $\d\Delta$. Then for $J\equiv J\st$ and $R\equiv0$ we can define $T_{J\st,0}(f) \deff T(f)- T(f)(0)$. In the general case we set $$T_{J, R}=\sum_{n=0}^\infty (-1)^nT_{J\st, 0}\scirc
\bigl((\dbarj-\dbar_{J\st}+R)\scirc T_{J\st, 0}\bigr)^n.$$ Due to the Sobolev imbedding theorem, the series converges in an appropriated norm, provided $\Vert J-J\st \Vert + \Vert R \Vert$ is small enough.

Remark.

The last statement of this lemma Implies that for $\Vert J-J\st \Vert + \Vert R\Vert $ small enough $L^{k+1,p}(\Delta ,\rr^{2n})
=H_{\dbar_J+R}\oplus T_{J,R}(L^{k,p}(\Delta ,\rr^{2n}))$ where $H_{\dbar_J+R}=\ker (\dbar_J+R)$. This will be used later in the proof of Lemma 6.3.4.

Corollary 3.2.2.

*Let $k\in \nn$, $q>2$, and $J$ a $C^k$-smooth almost complex structure in $X$. Also let $(S, J_S)$ be a complex curve. Suppose that $L^{1,q}$-map $u:S\to X$ satisfies the equation $$du + J \scirc du \scirc J_S = 0.$$ Then $u$ is $L^{k+1,p}$-smooth for any $p<\infty$. If, in addition, $J$ is $C^{k,\alpha}$-smooth with $0<\alpha<1$, then $u$ is $C^{k+1,\alpha}$-smooth.*

Let $J^{(n)}$ (resp. $J^{(n)}_S$) be a sequence of almost complex structures on $X$ (resp. on $S$), which $C^k$-converges to $J$ (resp. to $J^{(n)}_S$) and let $u_n:S \to X$ be a sequence of $(J^{(n)}_S, J^{(n)})$-holomorphic maps. Then the $C^0$-convergence $u_n \longrightarrow u$ implies the $L^{k+1,p}$-convergence for any $p<\infty$, and $C^{k+1, \alpha}$-convergence if $(J^{(n)}_S, J^{(n)})$ converge to $(J_S, J)$ in $C^{k,\alpha}$, $0<\alpha<1$.

Proof. The map $u$ is continuous and in local coordinates $w_1,\ldots, w_{2n}$ on $X$ and $z=x+iy$ on $S$ the equation has the form $$du(z) + J(u(z)) \scirc du \scirc J_M = 0,$$ which is equivalent to $\dbar_{J\scirc u} u =0$. Using [*Lemma 3.2.1*]{} and induction in $k'=0\ldots k$, one can obtain the regularity of $u$.

Similarly, for $J^{(n)}$ and $u^{(n)}$ satisfying the hypothesis of the corollary one gets $$\dbar_{J\scirc u}(u^{(n)}-u)=
(\dbar_{J\scirc u} - \dbar_{J^{(n)} \scirc u^{(n)}})u^{(n)}
\longrightarrow 0 \quad\hbox{in $L^{k',p}_\loc$
(resp.\ in $C^{k',\alpha}_\loc$)}$$ by induction in $k'=0\ldots k$.

Remark. The corollary implies that for a compact Riemann surface $S$ the topology in the space $\calp$ of $(J_S, J)$-holomorphic maps $u:S\to
X$ is independent of the particular choice of the functional space $L^{k', p}(S,X)\supset \calp$ with $1\le k' \le k+1$, $1< p<\infty$, and $kp>2$, provided $J_S$ and $J$ are changing $C^k$-smoothly. In the same way, [*Lemma 3.2.1*]{} implies that for $J\in C^k$ with $k\ge1$ the differential structure on $\calp$ is also independent of the particular choice of a functional space $L^{k',p}(S,X)$ with $1\le k' \le k$, $1<p<\infty$ and $k'p>2$.

Lemma 3.2.3.

*For any natural numbers $\mu>\nu\ge0$ and real numbers $p>2$, $\alpha<{2\over p}$ and $\gamma>0$ there exists $C =
C(\mu, \nu, \alpha, p, \gamma)>0$ with the following property. Let $J$ be an almost complex structure in $B \subset \cc^n$ with $J(0)=J\st$ and let $u:\Delta \to B$ be a $J$-holomorphic map of the form $u(z)=z^\mu(P(z) +z^\nu v(z))$, where $P(z)$ is some (holomorphic) polynomial of the degree $\nu$ and $v\in L^{1,p}(\Delta, \cc^n)$ with $v(0)=0$. Suppose that $\Vert J-J\st
\Vert_{C^1(B)} \le \gamma$ and $\Vert u\Vert_{L^{1,p}(\Delta)} \le \gamma$. Then for any $0<r<{1\over2}$ $$\bigl\Vert z dv \bigr\Vert_{C^0(\Delta(r))} +
\bigl\Vert z dv \bigr\Vert_{L^{1,p}(\Delta(r))}
\le
C \cdot r^\alpha\cdot \Vert u \Vert_{L^{1,p}(\Delta)},
\eqno(3.2.2)$$*

In particular, $du(z) = d(z^\mu P(z)) + o(|z|^{\mu+ \nu -1 +\alpha})$ for any $\alpha<1$.

Proof. For $0<r<{3\over4}$ we define the map $\pi_r:B\to B$, setting $\pi_r(w) \deff r^\mu w$. We also set $J^{(r)} \deff \pi_r^*J$, $u^{(r)}(z)
\deff \pi_r^{-1} \scirc u(rz)$, $P^{(r)}(z)\deff P(rz)$ and $v^{(r)}(z)=
r^\nu v(rz)$. Then $J^{(r)}$ is an almost complex structure in $B$ with $\Vert J^{(r)} -J\st \Vert_{C^1(B)} \le r^\mu
\Vert J-J\st\Vert_{C^1(B)}$, and $u^{(r)} \equiv z^\mu (P^{(r)}(z) +
z^\nu v^{(r)}(z))$ is a $J^{(r)}$-holomorphic.

Without losing generality, we can suppose that $\alpha>0$. Set $\beta \deff
1+\alpha - {2\over p}$ and $q\deff {2\over1-\beta}$. Then $\alpha<\beta<1$, $\beta + {2\over p} -1=\alpha$, and $q>2$. [*Lemma 3.1.3*]{} implies that $\Vert v \Vert_{C^{0,\beta}(\Delta(2/3))} + \Vert dv \Vert_{L^q(\Delta(2/3))}
\le C_1 \cdot \Vert u \Vert_{L^{1,p}(\Delta)}$. Here the constant $C_1$, as well as the constants $C_2,\ldots,C_6$ below, depend only on $\mu, \nu, p, \alpha$, and $\gamma$, but are independent of $r$. Consequently, $$\Vert u^{(r)} - z^\mu P^{(r)}(z) \Vert_{L^{1,q}(\Delta)}
\le C_2\cdot r^{\nu + \beta} \cdot
\Vert u \Vert_{L^{1,p}(\Delta)}.$$ Furthermore, due to [*Corollary 3.2.2*]{}, we have $\Vert u^{(r)}
\Vert_{L^{2,p}(\Delta)} \le C_3\cdot \Vert u \Vert_{L^{1,p}(\Delta)}$. Thus $$\Vert \dbar_{\!J\st}( u^{(r)} - z^\mu P^{(r)}(z) ) \Vert_{L^{1,p}(\Delta)} =
\Vert (\dbar_{\!J\st} -\dbar_{\!J^{(r)} \scirc u^{(r)} } )
( u^{(r)}) \Vert_{L^{1,p}(\Delta)}
\le$$ $$\le C_4\cdot r^\mu \cdot
\Vert u \Vert_{L^{1,p}(\Delta)}.$$ Applying [*Lemma 3.2.1*]{}, we obtain $$\bigl\Vert z dv^{(r)} \bigr\Vert_{L^{1,p} (\Delta(2/3))} \le
C_5 \cdot r^{\nu + \beta} \cdot \Vert u \Vert_{L^{1,p}(\Delta)},
\eqno(3.2.3)$$ which is equivalent to $$\bigl\Vert z dv \bigr\Vert_{L^{1,p}(\Delta(2r/3))} \le
C_5 \cdot r^{\beta + 2/p -1} \cdot \Vert u \Vert_{L^{1,p}(\Delta)}.$$ On the other hand, $(3.2.3)$ implies that $$\bigl\Vert z dv^{(r)} \bigr\Vert_{C^0 (\Delta(2/3))} \le
C_6 \cdot r^{\nu + \beta} \cdot \Vert u \Vert_{L^{1,p}(\Delta)},$$ and consequently $$\bigl\Vert z dv \bigr\Vert_{C^0(\Delta(2r/3))} \le
C_6 \cdot r^\beta \cdot \Vert u \Vert_{L^{1,p}(\Delta)}.$$

*3.3. Perturbing Cusps of Complex Curves.*

For understanding the rest of this lecture the reader should consult §6.2 and §6.3 for the definition and some elementary properties of Gromov’s $\dbar $-operator.

Lemma 3.3.1. [*For a given $p$, $2<p<\infty$, and $\gamma>0$ there exist constants $\epsi=\epsi(p,\gamma)$ and $C=C(p,\gamma)$ with the following property. Suppose that $J$ is a $C^1$-smooth almost complex structure in the unit ball $B \subset \cc^n$ with $\Vert J-J\st \Vert_{C^1(B)}
\le \epsi$ and $u\in  L^{1,p}(\Delta, B(0,{1\over2}))$ is a $J$-holomorphic map with $\Vert u \Vert_{L^{1,p}(\Delta)}
\le \gamma$. Then for every almost complex structure $\tilde J$ in $B$ with $\Vert \tilde J-J \Vert_{C^1(B)} \le \epsi$ there exists a $\tilde J$-holomorphic map $\tilde u: \Delta \to
B$ with $$\Vert \tilde u-u \Vert_{L^{1,p}(\Delta)} \le
C\cdot \Vert \tilde J-J \Vert_{C^1(B)}$$ such that $\tilde u(0) = u(0)$.* ]{}

Proof. Let $J_t$ be a curve of $C^1$-smooth almost complex structures in $B$ starting at $J_0=J$ and depending $C^1$-smoothly on $t\in [0,1]$. Consider an ordinary differential equation for $u_t \in L^{1,p} (\Delta, B)$ with the initial condition $u_0=u$ and

$$\msmall{du_t\over dt} = - T_{J_t \scirc u_t,\, R_t}
\biggl(\msmall{dJ \over dt} \scirc du_t \scirc J_\Delta \biggr),$$

where $R_t$ is defined by the relation $D_{J_t, u_t}=\dbar_{J_t, u_t}
+ R_t$. (See [*paragraph 6.2*]{}). Since $J_0$ and $R_0$ satisfy the hypothesis of [*Lemma 3.2.1*]{} and $R_t$ depends $L^p$-continuously on $J_t\in C^1$ and $u_t \in
L^{1,p}$, the solution exists in some small interval $0\le t \le
t_0$. For such a solution using (6.3.1) one has $$\left\Vert \msmall{du_t\over dt} \right\Vert_{L^{1,p}} +
\left\Vert \msmall{dR_t\over dt} \right\Vert_{L^p}
\le C\cdot (\Vert u_t\Vert_{L^{1,p}} +
\Vert J_t  - J\st \Vert_{C^1} )
\cdot \left\Vert \msmall{dJ_t\over dt} \right\Vert_{C^1}.$$ This implies the existence of the solution of our equation for all $t\in [0,1]$, provided $$\int_{t=0}^1
\left\Vert \msmall{dJ_t\over dt} \right\Vert_{C^1}
dt \le \epsi.$$

In this paragraph we suppose that some $p$ with $2<p<\infty$ is fixed.

Lemma 3.3.2.

*For every $\gamma >0$ and every pair of integers $\nu\ge1$, $\mu\ge1$ there exists an $\epsi=\epsi(\mu,\nu,\gamma)>0$ such that for an almost complex structure $J_0$ in $B$ with $\Vert J_0-J\st \Vert_{C^1(B)}\le \epsi$, $J_0(0)=J\st$, and for a $J_0$-holomorphic map $u_0: \Delta \to B(0, {1\over2})$ with $\Vert u_0
\Vert_{L^{1,p}(\Delta)} \le\gamma $ and with the multiplicity $\mu$ at $0\in \Delta$ the following holds:*

If $\nu\le2\mu-1$, then for every almost complex structure $J$ in $B$ with $J(0)=J\st$ and for every $v\in L^{1,p}(\Delta,\cc^n)$ with $\Vert v \Vert_{L^{1,p}(\Delta)} + \Vert J-J\st \Vert_{C^1(\Delta)}
\le\epsi$ there exists $w\in L^{1,p}
(\Delta,\cc^n)$ with $w(0)=0$, satisfying $$\dbarj(u_0+z^\nu(v+w))=0\eqno(3.3.1)$$ and $$\Vert w \Vert_{L^{1,p} (\Delta)}\le
C\cdot \left(
\Vert v \Vert_{L^{1,p}(\Delta)} +
\Vert J-J_0 \Vert_{C^1(\Delta)}
\right).\eqno(3.3.2)$$

If $\nu\ge2\mu$, then for every $v\in L^{1,p}(\Delta,\cc^n)$ with $\Vert v \Vert_{L^{1,p}(\Delta)}\le\epsi$ there exists $w\in
L^{1,p}(\Delta,\cc^n)$ with $w(0)=0$, satisfying $$\bar\partial_{J_0}(u_0+z^\nu(v+w))=0\eqno(3.3.3)$$ and $$\Vert w \Vert_{L^{1,p} (\Delta)}\le
C\cdot \Vert v \Vert_{L^{1,p}(\Delta)}.
\eqno(3.3.4)$$

Proof. If $\nu\le 2\mu -1$, we fix a curve $J_t$, $t\in [0,1]$, of $C^1$-smooth almost complex structures in $B$, which starts in $J_0$, finishes in $J$ and satisfies the conditions $J_t(0)=J\st$ and $\int_{t=0}^1\Vert \dot J_t \Vert_{C^1(B)} dt \le 2\epsi$, where $\dot J_t \deff dJ_t/dt$. If $\nu\ge2\mu$, we simply set $J_t\equiv J_0$.

As in the previous lemma, we want to find a needed $w$ by solving for $t\in [0,1]$ and $w_t\in L^{1,p}(\Delta,\cc^n)$ the equation $$z^{-\nu}\dbar_{J_t}(u_0 + z^\nu(t\cdot v + w_t)) =0.$$ However, this time we need to consider the fact that now we are dealing with different (almost) complex structures on $B$, namely $J\st$ and $J_t$ for any fixed $t\in [0,1]$. Thus, we write the last equation in the more correct form: $$(x + y J\st)^{-\nu}\dbar_{J_t}
(u_0 + (x + y J\st)^\nu(t\cdot v + w_t)) =0.
\eqno(3.3.5)$$

The differentiation of (3.3.5) with respect to $t$ gives $$(x + y J\st)^{-\nu}D_{u_t, J_t}
((x + y J\st)^\nu(v + \dot w_t))
+(x + y J\st)^{-\nu}\dot J_t
\scirc du_t \scirc J_\Delta
=0,
\eqno(3.3.6)$$ where $u_t\deff u_0+ (x + y J\st)^\nu(t\cdot v + w_t)$ and $J\st$ also denotes the pull-back of the standard complex structure on $E\deff u_t^*TB\cong \cc^n$.

First we show that the operator $(x + y J\st)^{-\nu} \scirc
D_{u_t, J_t} \scirc (x + y J\st)^\nu$ has the form $\dbar_{J^{(\nu)}_t} + R^{(\nu)}_t$ for an appropriate (almost) complex structure $J^{(\nu)}_t$ in $E\cong \cc^n$ and $\rr$-linear operator $R^{(\nu)}_t$. In fact, the explicit formula (6.2.5) for $D_{u, J}$ shows that taking the standard connection $\nabla$ in $TB\cong \cc^n$ and identifying $\Lambda^{(0,1)}\Delta
\cong \cc$ one gets $$D_{u_t, J_t}(v)=\msmall{1\over2}\left( \msmall{\d \over \d x}v +
J_t \msmall{\d \over \d x}v \right) + R^{(0)}_t(v)$$ with $R^{(0)}_t\in C^0(\Delta, \endo_\rr(\cc^n))$. Moreover, the formula (6.3.1) implies that $$\Vert R^{(0)}_t \Vert_{L^p(\Delta)} \le  \Vert J_t \Vert_{C^1(B)}
\cdot \Vert du_t \Vert_{L^p(\Delta)}.$$ Hence $$2(x + y J\st)^{-\nu} \scirc D_{u_t, J_t}
\scirc (x + y J\st)^\nu   =  \Bigl[\msmall{ \d \over \d x} +
(x + y J\st)^{-\nu} \scirc J_t \scirc (x + y J\st)^\nu
\msmall{\d \over \d y}  \Bigr]+$$ $$+ \Bigl[ \nu \cdot(x + y J\st)^{-\nu} \scirc
(1 + J_t \scirc J\st) \scirc (x + y J\st)^{\nu -1}
+ 2(x + y J\st)^{-\nu} \scirc R^{(0)}_t \scirc (x + y J\st)^\nu
\Bigr]=$$ $$=2\dbar_{J^{(\nu)}_t} + 2\,R^{(\nu)}_t.$$ One has the obvious identities $(J^{(\nu)}_t)^2 \equiv -1$, $$\Vert J^{(\nu)}_t -J\st \Vert_{C^0(\Delta)}=
\Vert (x + y J\st)^{-\nu} \scirc (J_t -J\st)
\scirc (x + y J\st)^\nu \Vert_{C^0(\Delta)} =
\Vert J_t -J\st \Vert_{C^0(B)},$$ and $$\Vert (x + y J\st)^{-\nu} \scirc R^{(0)}_t \scirc (x + y J\st)^\nu
\Vert_{L^p(\Delta)} =\Vert R^{(0)}_t \Vert_{L^p(\Delta)} .$$ Furthermore, $1 + J_t(0) J\st=0$; hence $\bigl\Vert \> |z|^{-1}(1 +J_t J\st)
\bigr\Vert_{C^0(\Delta)} \le \Vert J_t -J\st \Vert_{C^1(B)}$. This gives the estimate $$\Vert R^{(\nu)}_t \Vert_{L^p(\Delta)} \le \bigl(C_1 \cdot \nu +
C_2 \cdot \Vert du_t \Vert_{L^p(\Delta)} \bigr) \cdot
\Vert J_t -J\st \Vert_{C^1(B)}.$$

To show the existence of the solution of (3.3.5) for all $t\in [0,1]$, we need the estimate $$\left\Vert z^{-\nu}\cdot \dot J_t (u_0 + z^\nu w)
\scirc d(u_0 + z^\nu w) \right\Vert_{L^p(\Delta)}
\le C_1(\mu, \nu) \cdot \Vert u_0  \Vert_{L^{1,p}(\Delta)}^2
\cdot \Vert \dot J_t \Vert_{C^1(B)}
\eqno(3.3.7)$$ for all sufficiently small $w\in L^{1,p}(\Delta, \cc^n)$. The estimate trivially holds for $\nu\ge 2\mu$, because in this case $\dot J_t\equiv 0$. Otherwise for $\lambda \deff \min\{\mu, \nu\}$ from [*Corollary 3.1.3.*]{} one obtains $$\left\Vert z^{-\lambda}(u_0 + z^\nu w)
\right\Vert_{L^\infty (\Delta)} +
\left\Vert z^{-\lambda +1} d(u_0 + z^\nu w)
\right\Vert_{L^p(\Delta)}
\le C_2(\mu, \nu)\cdot
\left\Vert u_0 \right\Vert_{L^{1,p}(\Delta)}
\eqno(3.3.8)$$ which gives the estimate (3.3.7).

Now consider the ordinary differential equation for $w_t \in L^{1,p}
(\Delta, B)$ with the initial condition $w_0\equiv 0$ and $$\msmall{dw_t\over dt} = - T_{J^{(\nu)}_t,\, R^{(\nu)}_t}
\Bigl(z^{-\nu}\cdot \dot J_t \scirc du_t \scirc J_\Delta  + v \Bigr).
\eqno(3.3.9)$$ As in [*Lemma 3.3.1*]{} one has the estimate $$\biggl\Vert \msmall{dw_t\over dt} \biggr\Vert_{L^{1,p}(\Delta)} +
\biggl\Vert \msmall{dR_t^{(\nu)}\over dt} \biggr\Vert_{L^p(\Delta)}
\le$$ $$\le C\cdot (\Vert u_t\Vert_{L^{1,p}(\Delta)} +
\Vert J_t - J\st \Vert_{C^1(B)} )
\cdot \left( \bigl\Vert \msmall{dJ_t\over dt} \bigr\Vert_{C^1(B)}
+ \bigl\Vert v \bigr\Vert_{L^{1,p}(\Delta)} \right),$$ which implies the existence of the solution of (3.3.9) for all $t\in [0,1]$.

*3.4. Primitivity.*

We need to study first the notions of primitivity and distinctness of complex curves in $C^1$-smooth nonintegrable structures. Let us start with the following

Definition 3.4.1. Two $J$-holomorphic maps $u_1: (S_1,j_1) \to X$ and $u_2: (S_2,j_2) \to X$ with $u_1(a_1)=u_2(a_2)$ for some $a_i\in S_i$ are called [*distinct in $(a_1,a_2)$*]{} if there are no neighborhoods $U_i\subset S_i$ of $a_i$ with $u_1(U_1) =
u_2(U_2)$. We call $u_i: (S_i,j_i) \to X$ [*distinct*]{} if they are distinct for all pairs $(a_1,a_2)\in S_1 \times S_2$ with $u_1(a_1)=u_2(a_2)$.

Definition 3.4.2. A $J$-holomorphic map $u: (S,j) \to X$ is called [*primitive*]{} if there are no disjoint non-empty open sets $U_1, U_2\subset S$ with $u(U_1) =u(U_2)$. Note that $u$ must be nonconstant.

An important regularity property of a $J$ - complex curve with $J\in C^1$ is contained in the following

**Theorem 3.4.1. *Let $(S_1, j_1)$ and $(S_2, j_2)$ be smooth connected complex curves and $u_i : (S_i,j_i) \to (X,J)$ non-constant $J$-holomorphic maps with $J \in C^1$. If there are non-empty open sets $U_i \subset S_i$ with $u_1(U_1)= u_2(U_2)$, then there exists a smooth [*connected*]{} complex curve $(S, j)$ and a $J$-holomorphic map $u: (S,j) \to (X,J)$ such that $u_1(S_1) \cup u_2(S_2) =u(S)$ and $u:S \to X$ is primitive.***

Moreover, maps $u_i :S_i \to X$ factorize through $u: S \to
X$, i.e., there exist holomorphic maps $g_i : (S_i,j_i) \to (S,j)$ such that $u_i = u \scirc g_i$.

**Proof.**

[*Step 1.*]{} *For every $J$-holomorphic imbedding $u:\Delta \to (X,J)$ with $J\in C^1$ and for every $p\in \Delta$ there exist a neighborhood $U \subset X$ of $u(p)$, a $C^1$-smooth integrable complex structure $J_1$ in $U$ and $J_1$-holomorphic coordinates $(w_1, \ldots, w_n)=(w_1,w')$ ($w'=(w_2, \ldots, w_n))$ in $U$ such that $u(\Delta) \cap U = \{ (w_1, w')\in U: w'=0\}$ and $J|_{u(\Delta)} = J_1|_{u(\Delta)}$. In particular $w_1|_{u(\Delta)}$ is a holomorphic coordinate in $u\inv(U) \subset \Delta$ and*

$$|J(w_1, w')-J_1(w_1, w')| \le C(|w'|).
\eqno(3.4.1)$$

For the proof take any $J$-Hermitian metric $h$ on $X$. Let $N_1$ denote the normal bundle $u_1^*TX/ du(TS_1)$ with fiber $N_{1,z}$ over $z\in S_1$. Then the exponential map $\phi: \xi \in N_{1,z} \mapsto
\exp_{u_1(z)} (\xi)$ is a local diffeomorphism in some neighborhood $V$ of the zero section of $N_1$ over $S_1$. Note that along the zero section we can identify $\phi$ with $u_1$.

Equip $N_1$ with a canonical holomorphic structure of the quotient $u_1^*TX/
du(TS_1)$. Since $h$ was chosen $J$-Hermitian, the operator of the complex structure $J_{N_1}$ coincides along zero section with the structure $\phi^*J
\equiv u_1^*J$. Note that $J_{N_1}$ is [*integrable*]{} in $V$.

[*Step 2. (“Unique Continuation Lemma”).*]{} *Let the open subset $U \subset X$, $C^1$-smooth complex structure $J_1$ in $U$, and $J_1$-holomorphic coordinates $(w_1, \ldots, w_n)=(w_1,w')$ be as in the previous step. Set $C \deff \{ (w_1, w')\in U: w'=0\}$. Let $v: \Delta \to U$ be a $J$-holomorphic map. Then either $v(\Delta) \subset
C$ or $v\inv(C)$ is discrete.*

To show this, we use $(w_1, w')$ as complex coordinates and consider $U$ as an open subset of $\cc^n$, so that $J_1= J\st$. Set $v' \deff w' \scirc v$, i.e., $v'$ is obtained from $v$ by forgetting the first component. Then $v\inv(C)=v^{\prime\,-1}(0)$. Furthermore, $$|\dbar_{J\st}v'(z)| \le |\dbar_{J\st}v(z)| = |\dbar_{J_1}v(z)| =
|\dbar_Jv(z)-\dbar_{J_1}v(z)| \le$$ $$\le \norm{dv}_{L^\infty} \cdot
|J(v'(z))- J_1(v'(z))|,$$ which means that locally $|\dbar_{J\st}v'| \le C |v'|$. By [*Lemma 3.1.1*]{} either $v^{\prime\,-1}(0)$ is discrete, or $v'$ is identically zero.

Remarks. 1. Note that, in the latter case, $v:\Delta \to C$ is holomorphic.

**2. The “Unique Continuation Lemma” remains true for Lipschitz $J$ (i.e., $J\in C^{0,1}$) but not for Hölder and continuous $J$ (i.e., $J\in C^{0,\alpha}$ with $0\le\alpha<1$). To see this, we note that if $J\in C^{0,\alpha}$ with $0<\alpha\le1$, then any $J$-holomorphic map $u: \Delta \to (X,J)$ is $C^1$-smooth by standard regularity for the elliptic equation $\dbar_J u=0$. If $u$ is an immersion, then $du: T\Delta \to u^*TX$ is an imbedding of complex (not holomorphic!) bundles, and $N \deff u^*TX/ du(T\Delta)$ has a canonical complex structure. Equip $N$ with any holomorphic structure compatible with the complex one. Then [*Step 1*]{} from above goes through and we can locally construct $J_1$, which coincides with $J$ along some neighborhood of a given point $p\in \Delta$. Instead of (1) we obtain $$|J(w_1, w')-J_1(w_1, w')| \le C(|w'|^\alpha).
\eqno(3.4.2)$$ Thus the “similarity principle” and [*Step 2*]{} are applicable if $\alpha=1$ (Lipschitz case) but fail if $\alpha<1$ (Hölder case).**

Exercise. [Construct a counter example to “unique continuation lemma” for the Hölder $J$ in the following way. Take two different smoothly imbedded disks which coincide along some open subset, and try to construct a complex structure $J$ to make both $J$ - complex . It appears that for appropriately chosen disks one succeeds with $J \in C^{0,\alpha}$ for at least some $0<\alpha<1$. ]{}

[*Step 3*]{}. [*Proof of the Theorem in the case when $u_i: (S_i, j_i) 
\to
(X,J)$ are immersions.*]{}

Consider set $\cala$ of those $V$ such that $V$ is an open subset in one of $S_i$ and $u_i: V \to X$ is injective. Equip every $V \in \cala$ with the complex structure and with a $J$-holomorphic map $\ti u:
V\to X$ induced from $S_i$. Define the equivalence relation $\sim$ on the disjoint union $\sqcup_{V\in \cala}V$, identifying points $p'\in V'$ and $p'' \in V''$ if there exist $V\in \cala$ and $p\in V$ such that $\ti u(V) \subset \ti u(V') \cap \ti u(V'')$ and $\ti u(p) = \ti u(p') =
\ti u(p'')$.

Set $S \deff \bigl(\sqcup_{V\in \cala}V\bigr) / \sim$ with projection $\pi :
\sqcup_{V\in
\cala}V \to S$. Induce the quotient topology on $S$ declaring $\{ \pi(V) :V\in \cala\}$ as the basis of the topology. Then $\ti u$ induces the map $u: S \to X$ which is holomorphic. We shall show that $S$ and $u$ obey the desired properties. The main point at this step is to show that the quotient topology on $S$ is Hausdorff.

Let $p'$ and $p''$ be two distinct points on $S$. If $u(p') \not= u(p'')$, then there exist disjoint neighborhoods $u(p') \in U' \subset X$ and $u(p'') \in U'' \subset X$ and their pre-images $V' \deff u\inv(U')$ and $V'' \deff u\inv(U'')$ are disjoint neighborhoods of $p'$ and $p''$. It remains to consider the case when $u(p') = u(p'')$.

Let $\ti p' \in \ti V'$ and $\ti p''\in \ti V''$ be representatives, i.e., $\ti V', \ti V'' \in \cala$, $\pi(\ti p')=p'$, and $\pi(\ti p'')=p''$. Shrinking $\ti V'$, if needed, we may assume that for $\ti u(\ti V')$ there exists a neighborhood $U \subset X$, an (integrable) complex structure $J_1$ in $U$ and $J_1$-holomorphic coordinates $(w_1, w')$ in $U$ with the properties of [*Step 1*]{}. Shrinking $\ti V''$, we may assume that $\ti u(\ti V'')$ is also contained in $U$. Using [*Step 2*]{}, we may additionally assume that $u(p')=u(p'')$ is the only intersection point of $\ti u(\ti V')$ and $\ti u(\ti V'')$. Then $\pi(\ti V')$ and $\pi(\ti V'')$ are disjoint neighborhoods of $p'$ and $p''$. This shows that $S$ is Hausdorff.

By assumption, maps $u_i: S_i \to X$ are immersions. Thus every $S_i$ can be covered by open sets $V \subset S_i$ such that restrictions $u_i|_V$ are imbeddings. Every such $V$ belongs to the atlas $\cala$, and the canonical projection $\pi: V \to S$ is a holomorphic imbedding. By the definition of the equivalence relation $\sim$, these local maps $\pi: V\subset S_i \to S$ can be glued together to holomorphic maps $g_i: S_i \to S$. One can see that the constructed $(S,j)$, $u:S \to X$, and $g_i: S_i \to S$ have the desired properties. **Non immersed case. [*Here we consider the case where $u_i: (S_i,j_i) \to (X,J)$ are not necessarily immersions.*]{}**

From [*Proposition 6.3.2*]{} of [*Lecture 6*]{}, maps $u_i$ define bundle homomorphisms $du_i : TS_i \to E_i :=
u_i^*(TX)$ which are holomorphic w.r.t. the canonical holomorphic structure on $E_i$. This implies that the zero set $A_i:=\{y \in S_i:  du_i(y)=0 \}$ is discrete in $S_i$. In particular, every $u_i$ is an immersion outside $A_i$.

Set $\check S_i \deff S_i \bs A_i$. Then $u_i: \check S_i \to X$ are immersions, and the previous arguments remain valid. Let $(\check S, \check \jmath) $, $\check u:\check S \to X$, and $\check g_i: \check S_i \to \check S$ be the corresponding objects.

Numerate points in $A_1$ in any order, $A_1 = \{\, a_1, a_2, \ldots \,\}$.

[*Step 4.*]{} [*We state that for $k=1,2,\ldots\,$ there exist Riemann surfaces $S^{(k)}$ and holomorphic maps $g_1^{(k)}: S_1^{(k)} \to S^{(k)}$ such that $S_1^{(k)} \deff
\check S_1 \cup \{a_1, \ldots a_k\}$, $g_1^{(k)}|_{\check S} =\check g_1$, and $S^{(k)} = \check S \cup g_1^{(k)}(\{a_1, \ldots a_k\})$.* ]{}

In other words, at the $k$-th step we add to $S_1^{(k-1)}$ the point $a_k$, and extend $g_1^{(k-1)}
: S_1^{(k-1)} \to S^{(k-1)}$ through $a_k$ holomorphically, adding, if needed, some new point to $S^{(k-1)}$.

The proof of this statement proceeds inductively. For a given $k$ take a sufficiently small neighborhood $V_k \subset S_1$ of $a_k$ which contains no other points of $A_1$. Set $\check V_k \deff V_k
\bs \{a_k\}$. If the restriction $g_1^{(k-1)}|_{\check V_k}: \check V_k \to
S^{(k-1)}$ extends to a holomorphic map $g_1^{(k)}|_{V_k}: \check V_k \to
S^{(k-1)}$, one has nothing to do but set $S^{(k)}\deff S^{(k-1)}$.

Otherwise we set $S^{(k)}\deff S^{(k-1)} \sqcup \{b_k\}$ and $g_1^{(k)}(a_k)
= b_k$. We must prove existence of appropriate topology and complex structure on the constructed $S^{(k)}$. We give a proof of these properties in full detail.

Choose $C^2$-smooth local complex (not holomorphic) coordinates $(w_1, \ldots, w_n)$ in a neighborhood of $u_1(a_k) \in X$ such that a complex structure $J'$ defined by $(w_1, \ldots,
w_n)$ coincides with $J$ in $u_1(a_k)$. Let $z$ be a local complex coordinate on $S_1$ in a neighborhood of $a_k$. We may assume that $(w_1, \ldots, w_n)$ (resp. $z$) are centered in $u_1(a_k)$ (resp. in $a_k$). [*Lemma 3.1.1*]{} provides that in these coordinates the map $u_1 : S_1 \to X$ has the form $u_1(z) = z^l \cdot (v_1(z), \ldots,v_n(z))$ with $\vec v=(v_1,
\ldots, v_n) \in L^{1,p}$ and $\vec v(0) \not=0$.

Choose a coordinate $w_p$ such that the corresponding component $v_p$ does not vanish at $z=0$, $v_p(0) \not=0$. Then there exists the $l$-th root of this $v_p$, i.e., $v_p(z) = (f(z))^l$ for some continuous non-vanishing function $f(z)$. Since $v_p(z)$ is $L^{1,p}$-smooth, so is $f(z)$. Shrinking our neighborhood $V_k$, if needed, we may assume that $f(z)$ is not vanishing in $V_k$. Thus for an appropriate neighborhood $U$ of $u_1(a_k)$ there exists a $C^2$-smooth map $w_p:U \to \cc$ such that $h_p \deff w_p\scirc u_1 (z)=
(zf(z))^l$ for some $L^{1,p}$-smooth non-vanishing function $f(z)$ defined in a neighborhood of $a_k$. We may additionally assume that $$|f(z)-f(0)|\le \eps <\!\!<1 \qquad\hbox{for all $z\in V_k$}.
\eqno(3.4.3)$$ Note that for every sufficiently small $b \in \cc$ we have at least one solution $z\in V_k$ of the equation $h_p(z)=b$ with $|z  f(0)|^l \le
2|b|$. In order to see this, one considers $h_p$-images of circles $\gamma_t \deff
\{\, |zf(0)|^l = t|b|\,  \}$. Then due to (3.4.3), the image $h_p(\gamma_{1\over2})$ lies in the disk $\Delta(|b|)$, whereas $h_p(\gamma_2)$ lies outside of $\Delta(|b|)$. Homotopy argument shows that $b\in \psi(\gamma_t)$ for some $t\in ]{1\over2}, 2[$. Similarly, one obtains the estimate $$c\cdot |h_p(z)|^{1\over l} \le |z| \le C\cdot |h_p(z)|^{1\over l}
\qquad\hbox{for $z\in V_k$}
\eqno(3.4.4)$$ with some constants $c$ and $C$. Take a sufficiently small disk $\Delta(r)$ in $\cc$ (range of $h_p(z)= \omega_p
\scirc
u_1(z)$) and consider the set $V'_k \deff \{ z\in V_k: |z|^l < {2r \over
|f(0)|^l}, \, |h_p(z)|< r\}$. Consider a sequence $z_i \in V'_k$ such that $h_p(z_i)$ converges to some $b\in \Delta(r)$. Then some subsequence of $z_i$ converges to $z^*$ with $|z^*| \le {2r \over |f(0)|^l}$. But then $|h_p(z^*)|=
|z^*f(z^*)|^l =|b|$ which implies $|z^*|^l = {|b| \over  |f(z^*)|^l} <{2r
\over |f(0)|^l}$, so that $z^* \in V'_k$. This shows that the map $h_p: V_k
\to \Delta(r)$ is proper. By (4), $h_p(z)=0$ for $z\in V'_k$ implies $z=0$. Thus, for $\check V'_k \deff \{z\in V'_k : z\not=0\} = V'_k \bs\{a_k\}$ the map $h_p: \check V'_k \to \check\Delta(r)$ is also proper.

Recall that the bundle homomorphism $du_1: TS_1 \to E_1\deff u_1^*TX$ is holomorphic with respect to the canonical holomorphic structure on $E_1$. Thus, in $V'_k$ we can represent $du_1$ in the form $du_1(z)= z^{l-1}\cdot
s(z)$ with $s(z)$ also holomorphic and $s(0)\not=0$. This implies that for $z\not=0 \in V'_k$ sufficiently close to $0$ the image $du_1(T_zS_1)$ of the tangent space $T_zS_1$ is close to a real 2-dimentional space $\cc s(0)\subset E_1|_0=T_{u_1(a_k)}X$. This is a complex line in $T_{u_1(a_k)}X$ generated by $s(0)$. We may assume that the coordinates $(w_1, \ldots, w_n)$ were chosen in such a way that the $p$-th component of the vector $s(0)$ is non-zero. This means that the linear map $dw_p|_{du_1(a_k)}: T_{u_1(a_k)}X \to \cc$ is not degenerated $\cc s(0) \subset T_{u_1(a_k)}X$. Thus, for any $b$ sufficiently close to $u_1(a_k)$, and for any real 2-dimensional space $L \subset T_bX$ sufficiently close to $\cc s(0)$ the linear map $dw_l|_b: T_bX \to \cc$ is not degenerated on $L$.

Hence we can conclude that $h_p: \check V'_k \to \check \Delta(r)$ is not degenerated provided $r$ was chosen sufficiently small. This means that $h_p: \check V'_k \to \check \Delta(r)$ is a covering. Note that $\pi_1(\check
\Delta(r)) = \zz$ and that $\pi_1(\check V'_k)$ contains $\zz$ as a subgroup generated by a small circle $\{ |z| = \rho\}$. From inclusion $(u_1)_*:
\pi_1(\check V'_k) \hookrightarrow \pi_1(\check \Delta(r))$ we conclude that $\pi_1(\check V'_k) =\zz$, and that $h_p: \check V'_k \to \check
\Delta(r)$ is a finite covering. Since a small circle $\{ |z| = \rho\}$ around $a_k \in V'_k$ is a generator of $\pi_1(\check V'_k)$, we conclude that $V'_k$ is a disk and $\check V'_k$ is a punctured disk.

Now recall that the map $h_p:\check V'_k \to \check \Delta(r)$ can be seen as a composition of $g^{(k-1)}: \check V'_k \to S^{(k-1)}$ with a coordinate function $w_p$ restricted on $\check u_1(\check V'_k)$. Set $\check W_k \deff
g^{(k-1)} (\check V'_k)$ and consider a holomorphic map $g^{(k-1)}: \check
V'_k \to \check W_k$. Then this map is again proper and non-degenerate and therefore is also a finite covering. This implies that $\check W_k$ is also a punctured disk.

Let $\check\psi: \check W_k \to \check \Delta$ be a biholomorphism which can be seen as a local chart on $\check W_k \subset S^{(k-1)}$. Define holomorphic structure on $S^{(k)}= S^{(k-1)} \sqcup \{b_k\}$ in the following way. Interpret $b_k$ as a puncture point of $\check W_k$, set $W_k \deff \check W_k\sqcup \{b_k\}$, extend $\check \psi$ to the map $\psi: W_k \to \Delta$ setting $\psi(b_k) \deff 0$. Extend the topology on $S^{(k)}$ in such a way that $\psi: W_k \to \Delta$ becomes a homeomorphism. Declare $\psi$ to be a holomorphic coordinate on $S^{(k)}$. Note that the composition $\check\psi \scirc g^{(k-1)}: \check V'_k \to \check \Delta$ is bounded; thus it extends to a holomorphic map from $V'_k$ to $\Delta$. Consequently, $\check\psi\inv: \check \Delta \to \check W_k \subset S^{(k-1)}$ cannot be holomorphically extended to $\psi\inv: \Delta \to S^{(k-1)}$, because otherwise this would mean holomorphic extensibility of $g^{(k-1)}: S_1^{(k-1)}
\to S^{(k-1)}$ through $a_k$. This property insures that $S^{(k)}$ is Hausdorff. Thus constructed, $S^{(k)}$ and $g^{(k)}: S_1^{(k)} \to S^{(k)}$ obey the desired properties.

Setting $S' \deff \cup_k S^{(k)}$ we obtain the minimal extension of $\check S$ such that the map $\check g_1: \check S_1 \to \check S$ extends holomorphically to $g'_1: S_1 \to S'$. A similar construction should be done to extend $\check g_2 : \check S_2 \to S'$ to a holomorphic map $g_2: S_2 \to S$. This finishes the proof of the [*theorem* ]{}.

In the spirit of [*Theorem 3.4.1*]{} one obtains the following result.

Proposition 3.4.2. *Let $(X,J)$ be an almost complex manifold with $J \in C^1$. Then any primitive $J$-holomorphic map $u: (S,j) \to X$ with connected irreducible $(S,j)$ is injective outside some countable subset $A\subset S$. Moreover, $A$ can be chosen in such a way that for any compact $K\Subset S$ the intersection $A \cap K$ is finite.*

Proof. We have shown that $A_1 \deff \{a\in S: du(a)=0\}$ is discrete; therefore it is countable. Set $\check S:= S\setminus A_1$. Consider the set $A_2 \deff \{ (a,b) \in \check S\times \check S : u(a)=u(b) \}$. We state that $A_2$ is discrete in $\check S\times \check S\setminus \Delta $, where $\Delta $ is a diagonal. Otherwise there must exist a sequence $(a_n,b_n)$ converging to some pair $(a^*,b^*) \in \check S\times
\check S$. Take a sufficiently small neighborhood $V \subset \check S$ of $a^*$ and find a neighborhood $U \subset X$ of $u(a^*)$, which obeys the properties of [*Step 1*]{} from the previous proof. Note that obviously $u(b^*) = u(a^*)$. Now we obtain a contradiction with argumentations from [*Step 2*]{} above. This shows that $A_2$ must be discrete in $\check S\times \check S\Delta $.

Thus it is also countable. Let $A_3$ be the projection of $A_2$ onto the first $\check S$. Then $A:=A_1\cup A_2$ obeys the desired properties.

Remark. In fact, positivity of the intersection of $J$ - complex curves with $J\in C^1$ provides that there exists a [*discrete*]{} subset $A\subset
S$ fulfilling the condition of [*Proposition 3.4.2*]{}. We will prove this in next section.

*3.5. Positivity of Intersection.*

Let us first recall the notion of the intersection number (index) of two surfaces in $\rr^4$. Let $M_1$ and $M_2$ be two-dimensional, oriented, smooth surfaces in $\rr^4$ passing through the origin. We suppose further that both $M_1,M_2$ intersect the unit sphere $S^3$ transversally by curves $\gamma_1$ and $\gamma_2$, respectively, and that $\gamma_1$ and $\gamma_2$ do not meet one another. Let $\tilde M_i$ be small perturbations of $M_i$ making them intersect transversally.

Definition 3.5.1. [*The intersection number of $M_1$ and $M_2$ is defined to be the algebraic intersection number of $\tilde M_1$ and $\tilde M_2$. If $M_1,M_2$ intersect only at zero, we also say that the number just defined is the intersection index of $M_1$ and $M_2$ at zero. It will be denoted by $\delta_0(M_1,M_2)$ or $\delta_0$.* ]{}

Remark. [This number is independent of the particular choice of perturbations $\tilde M_i$. We shall use the fact that the intersection number of $M_1$ and $M_2$ is equal to the [*linking number*]{} $l(\gamma_1,
\gamma_2)$ of the (reducible in general) curves $\gamma_i$ on $S^3$, see \[Rf\]. ]{}

In the following theorem the structure $J$ is supposed to be of class $C^1$.

Theorem 3.5.1. *Let $u_i:\Delta\to(\rr^4, J)$, $i=1, 2$ be two primitive distinct $J$ - complex disks such that $u_1(0)=u_2(0)$. Set $M_i \deff u_i
(\Delta)$. Let $Q = M_1\cap  M_2$ be the intersection set of the disks. If $J$ is $C^1$-smooth, then the following is true.*

The set $\{\, (z_1,z_2) \in \Delta \times \Delta: 
u_1(z_1)= u_2(z_2)\,\}$ is a discrete subset of $\Delta\times \Delta $. In particular, $u_1(\Delta) \cap
u_2(\Delta)$ is a countable set;

The intersection index in any such point of $Q$ is strictly positive. Moreover, if $\mu_1$ and $\mu_2$ are the multiplicities of $u_1$ and $u_2$ in $z_1$ and $z_2$, respectively, with $u_1(z_1)=u_2(z_2)=p$, then the intersection number $\delta_p$ of branches of $M_1$ and $M_2$ at $z_1$ and $z_2$ is at least $\mu_1\cdot \mu_2$;

$\delta_p=1$ iff $M_1$ and $M_2$ intersect at $p$ transversally.

Proof. [*Case 1. The map $u_1$ has no critical points*]{}. Thus $u_1: \Delta \to \rr^4$ is an immersion. Take any $(a_1,a_2) \in \Delta \times \Delta$ with $u_1(a_1)=u_2
(a_2)$. In a neighborhood of $u_1(a_1)\in \rr^4$ we find local coordinates $(w_1, w_2)$ with the properties from [*Step 1*]{} of the proof of [*Theorem 3.4.1*]{}. We may assume that in these coordinates the map $u_1$ has the form $u_1(z)=(z,0)$ and that $a_1$ is the point with $z=0$. Choose the local coordinate $z$ on the second disk such that $z=0$ is the point $a_2$. Consider representation of the map $u_2: \Delta \to \rr^4$ in coordinates $(w_1,w_2)$ such that $u_2(z)= (w_1(z),w_2(z))$. From the inequality (3.4.1) and [*Lemma 3.1.1*]{} we conclude that $w_2(z)=
z^\nu b(z)$ for some $b\in L^{1,p}$ with $b(0)\not =0$. This implies that the only intersection point of images of small neighborhoods of $a_1$ and $a_2$ is $u_1(a_1)=u_2(a_2)=p$, and its intersection index is $\nu$. This provides the properties – for the case when one of the maps $u_1$ or $u_2$ is an immersion.

[*Case 2. Both $u_1$ and $u_2$ are immersions outside $0\in \Delta$, but $du_1(0)=0=du_2(0)$. Moreover, $u_1(0)=u_2(0)$.*]{}

First we collect some information about the local behavior of $u_i$ at $0\in
\Delta$. We consider only the map $u_1$; the same procedure should be done for $u_2$.

Without loss of generality we may assume that the point $u_1(0)=u_2(0)$ is the center of coordinates in $\rr^4$. We may also assume that the standard complex structure $J\st$ in $\rr^4$ coincides with the given structure $J$ in the point $P$. Let $(w_1,w_2)$ be the standard complex coordinate in $(\rr^4,
J\st) =\cc^2$. To avoid possible confusion, denote by $\ti u=\ti u(z)$ the representation of $u$ in coordinates $(w_1,w_2)$. Then $\ti u_1(z)= z^{\mu_1}
v_1(z)$ with $v_1 \in L^{1,p}$ and $v_1(0) \not=0$ by the “similarity principle”.

Now consider the bundle $E_1:= u_1^*T\rr^4$ over $\Delta$. Two complex structures in $\rr^4$, $J$ and $J\st$, induce two complex structures in $E_1$, $u_1^*J$ and $u_1^*J\st$. We always equip $E_1$ with the co-structure $u_1^*J$; otherwise the contrary is noted explicitly. In particular, $E_1$ equipped with $u_1^*J$ has the canonical holomorphic structure such that the homomorphism $du_1: T\Delta \to E_1$ is holomorphic. Consequently, $du_1(z)=
z^{\nu_1} s_1(z)$ for some holomorphic section $s_1$ of $E_1$ with $s_1(0)
\not =0$.

On the other hand, since the tangent bundle $(T\cc^2, J\st)$ is trivial, the complex bundle $(E_1, u_1^*J\st)$ has natural trivialization. Denote this trivialization by $\psi: (E_1, u_1^*J\st) \to \cc^2$, where $\cc^2$ stands for the trivial bundle over $\Delta$. Then the homomorphism $\psi(z): E_1|_z
\to \cc^2$ is an $\rr$-linear isomorphism in general, and complex linear exactly for those $z\in\Delta$, for which $J(u_1(z))= J\st(u_1(z))$. In particular, this holds for $z=0$. This implies $\psi(z^\nu s(z))= z^\nu
\psi(s(z)) + O(|z|^{\nu+1})$. From our construction of $psi$ follows the equality $\psi \scirc du_1(z) = d\ti u_1(z)$. Thus $d\ti u_1(z)= z^{\nu_1}
\cdot \psi \scirc s_1(z) + O(z^{\nu_1+1})$. Comparing this asymptotic expansion with $\ti u_1(z)= z^{\mu_1} v_1(z)$ we conclude that $\nu_1=
\mu_1-1$ and $s_1(0)= \mu_1 v_1(0)$.

From now on we do not distinguish the map $u_1$ from its representation $\ti u_1$ in coordinates $(w_1, w_2)$. Note that we can use the same coordinates $(w_1, w_2)$ and the structure $J\st$ considering the map $u_2$. Thus we get the asymptotic relations $$\eqalignno{
u_i(z) &= z^{\mu_i} v_i(0) + O(|z|^{\mu_i+ \alpha}) &(3.5.1)
\cr
du_i(z) &= z^{\mu_i-1} v_i(0) + O(|z|^{\mu_i-1+\alpha}) &(3.5.2)
}$$ with some Hölder exponent $\alpha>0$. This implies the transversality of small $J$ - complex disks $u_i(\Delta
(\rho))$ to small spheres $S^3_r$. More precisely, there exist radii $\rho>0$ and $R>0$ such that for any $0<r<R$ the $J$-curves $u_i(\Delta (\rho))$ intersect the sphere $S^3_r\deff \{ |w_1|^2+ |w_2|^2=r^2 \}$ transversally along smooth immersed circles $\gamma_i(r)$. In fact, the asymptotic relation (4) provides that for any $\theta \in [0,2\pi]$ there exists at least one solution of the equation $\bigl|u_i(\rho_{i} e^{\isl\theta}) \bigr| =r$ with $\rho_i< \rho$, and that any such solution $\rho_i$ must be close to $\left(
{r \over |v_1(0)|}\right)^{1/\mu_i}$. Then one uses (5) to show that the set $\ti\gamma_i(r) \deff \{ z\in\Delta: \bigl|u_i(z) \bigr| =r \}$ is, in fact, a smooth imbedded curve in $\Delta$, parameterized by $\theta\in [0,2\pi]$, and that $u_i: \ti\gamma_i(r) \to S^3_r$ is an immersion with the image $\gamma_i(r)$.

Taking an appropriate small subdisk and rescaling, we may assume that $\rho=1
=R$. Note that the points of the self- (resp. mutual) intersection of $\gamma_i(r)$ are self-( or resp. mutual) intersection points of $u_i
(\Delta)$. Let us call $r \in ]0,1[$ non-exceptional if curves $\gamma_i(r)
\subset S^3_r$ are imbedded and disjoint. Thus $r^* \in ]0,1[$ is exceptional if $S^3_{r^*}$ contains intersection points of $u_i (\Delta)$.

The “unique continuation lemma” of Step 2 of the proof of [*Theorem 3.4.1*]{} provides that any such intersection point is isolated in the punctured ball $\check B \deff \{ 0<
|w_1|^2 + |w_2|^2 <1 \}$. This implies that either there exist finitely many exceptional radii $r^*\in ]0,1[$, or that they form a sequence $r^*_n$ converging to $0$.

Denote $M_i(r) \deff u_i (\Delta) \cap B(r)$. For non-exceptional $r$ we can correctly define the intersection index of $M_1(r)$ with $M_2(r)$.

By [*Corollary 3.1.3*]{}, the maps $u_i$ can be represented in the form $u_i(z)=z^{\mu_i}\cdot v_i(z)$ with $v_i\in L^{1,p}(\Delta,\cc^2)$ such that $v_1(0) \not= 0 \not= v_2(0)$. As above, we consider two cases.

[*Case 1. The vectors $v_1(0)$ and $v_2(0)$ are not collinear.*]{}

It is easy to see that, in this case, $0\in\cc^2$ is an isolated intersection point of $u_1(\Delta)$ and $u_2(\Delta)$ with multiplicity exactly $\mu_1\cdot \mu_2$. The asymptotic formula (3.4.4) provides that if vectors $v_1(0)$ and $v_2(0)$ are complex linear independent, then $0\in \rr^4$ is an isolated intersection point of $Q =u_1(\Delta) \cap u_2(\Delta)$ with the index $\mu_1 {\cdot}
\mu_2$. Thus we have only finitely many intersection points $p\in Q$. Since all other points $p\in Q$ are smooth, the intersection index in every such point is positive. Thus in the case of non-collinear $v_1(0)$ and $v_2(0)$ for any non-exceptional $r>0$ the intersection index of $M_1(r)$ and $M_2(r)$ is positive.

[*Case 2. The second case is when the vectors $v_1(0)$ and $v_2(0)$ are collinear.*]{}

The idea is to “turn” the map $u_2$ a little and to reduce the case to the previous one. So let $T\in \endc(\cc^2)$ be a linear unitary map which is close enough to identity such that $T(v_2(0))$ is not collinear to $v_1(0)$. Define $J^T\deff T\inv \scirc J \scirc T$ so that $\Vert J^T -J \Vert_{C^1(B)} \le \Vert T - \id \Vert$. Applying [*Lemma 3.3.1*]{} with $u_0=z^{\mu _2}v_2$ and $v=0$, we find $w\in L^{1,p}(
\Delta,\cc^2)$ with $w(0)=0$ such that $\dbar_{J^T} (z^{\mu_2}(v_2+w)) =0$. The map $\tilde u_2\deff T(z^{\mu_2}(v_2+w))$ is the required “turned” $J$-holomorphic map. Since such a “turn” can be made small enough, the intersection number of $u_i(\Delta)$ in the ball $B_{r^-}$ does not change.

This implies that the intersection number of $u_1(\Delta)$ and $u_2(\Delta)$ in any ball $B_r$ is not less than $\mu_1\cdot \mu_2$. Another conclusion is that $0\in\cc^2$ is an isolated intersection point of $u_j(\Delta)$. Otherwise we could find a sequence $r_i\msmall{\searrow}0$ with at least one intersection point of $u_j(\Delta)$ in every spherical shell $B_{r_i} \!\backslash B_{r_{i+1}}$; therefore the intersection number of $u_j(\Delta)$ in the balls $B_{r_i}$ would be strictly decreasing in $i$.

Thus the statements (i) and (ii) are proved. The proof of (iii) is now obvious and follows from the observation that $\mu _1 = \mu _2 =1$ in this case.

Corollary 3.5.2. [*Let $u_i:S_i\to(X, J)$, $i=1, 2$ be compact irreducible $J$ - complex curves such that $u_1(S_1)=M_1\not=u_2(S_2)=M_2$. Then they have finitely many intersection points and the intersection index in any such point is strictly positive. Moreover, if $\mu_1$ and $\mu_2$ are the  multiplicities of $u_1$ and $u_2$ in such a point $p$, then the intersection number of $M_1$ and $M_2$ in $p$ is at least $\mu_1\cdot \mu_2$.* ]{}

[Appendix 2]{}

[The Bennequin Index and Genus Formula.]{}

*A2.1. Bennequin Index of a Cusp.*

Let $u : (\Delta , 0)\to (\cc^2, J, 0) $ be a germ of a nonconstant complex curve in zero. Without loss of generality we always suppose that $J(0)=J\st$. Taking into account that zeros of $du$ are isolated, we can suppose that $du$ vanishes only at zero. Furthermore, let $w_1, w_2$ be the standard complex coordinates in $(\cc^2, J\st)$. According to [*Lemma 3.1.3*]{} we can write our curve in the form

$$u(z) = z^\mu\cdot a + o(\vert z\vert^{\mu +\alpha}),
\qquad a\in \cc^2 \setminus \{ 0\},\> 0<\alpha<1. \eqno(A2.1.1)$$

For $r>0$ define $F_r\deff TS^3_r \cap J(TS^3_r)$ to be the distribution of $J$-complex planes in the tangent bundle $TS^3_r$ to the sphere of radius $r$. $F_r$ is trivial, because $J$ is homotopic to $J\st = J(0)$. By $F$ we denote the distribution $\cup_{r>0}F_r\subset \cup_{r>0}TS^3_r \subset 
TB^*$, where $TB^*$ is the tangent bundle to the punctured ball in $\cc^2$.

**Lemma A2.1.1. *The (possibly reducible) curve $\gamma_r = M\cap S^3_r$ is transversal to $F_r$ for all sufficiently small $r>0$.***

**Proof. In fact, due to [*Lemma 3.2.3*]{} one has $du(z) = \mu
z^{\mu-1} \cdot a\,dz + o(\vert z\vert^{\mu -1+\alpha})$. Since $J\approx J\st$ for $r$ sufficiently small, $T\gamma_r$ is close to $J\st  n_r$, where $ n_r$ is the field of normal vectors to $S^3_r$.**

On the other hand, for sufficiently small $r$, the distribution $F_r$ is close to the one of $J\st$ complex planes in $TS^3_r$, which is orthogonal to $J\st n_r$.

This fact permits us to define the Bennequin index of $\gamma_r$. Namely, take any nonvanishing section $\vec v$ of $F_r$ and move $\gamma_r$ along the vector field $v$ to obtain a curve $\gamma'_r$. We can make this move for a small enough time, so that $\gamma'_r $ does not intersect $\gamma_r$. Following Bennequin \[Bn\], we have

Definition A2.1.1. Define the *Bennequin index $b(\gamma_r)$ to be the linking number of $\gamma_r$ and $\gamma'_r$.*

This number does not depend on $r>0$, taken sufficiently small, because $\gamma_r$ is homotopic to $\gamma_{r_1}$ for $r_1<r$ within the curves transversal to $F$, see \[Bn\]. It is also independent of the particular choice of the field $\vec v$. Thus for the standard complex structure $J\st$ in $B\subset \cc^2$ we use $\vec v\st(w_1,w_2)\deff  (-\bar w_2, \bar w_1)$ for calculating the Bennequin index of the curves on sufficiently small spheres. For an arbitrary almost complex structure $J$ with $J(0)=J\st$ we can find the vector field $\vec v_J$, which is defined in a small punctured neighborhood of $0\in B$, is a small perturbation of $\vec v\st$, and lies in the distribution $F$ defined by $J$.

Denote by $B_{r_1, r_2}$ the spherical shell $B_{r_2}\setminus B_{r_1}$ for $r_1<r_2$.

Lemma A2.1.2. *Let $\Gamma $ be an immersed $J$ - complex curve in the neighborhood of $\overline B_{r_1, r_2}$ such that all self intersection points of $\Gamma $ are contained in $B_{r_1, r_2}$ and for every $r_1\le r\le r_2$ all components of the curve $\gamma_r = \Gamma \cap 
S^3_r$ are transversal to $F_r$. Then $$b(\gamma_{r_2}) = b(\gamma_{r_1}) + 2\cdot \sum\nolimits
_{x\in Sing(\Gamma) }\delta_x,
\eqno(A2.1.2)$$ where the sum is taken over self-intersection points of $\Gamma $.*

Proof. Move $\Gamma $ a little along $v_J$ to obtain $\Gamma^\varepsilon$. By $\gamma^\varepsilon_{r_1},
\gamma^\varepsilon_{r_2}$ denote the intersections $\Gamma^\varepsilon \cap
S^3_{r_1},\Gamma^\epsi\cap S^3_{r_2}$, which are of course the moves of $\gamma_{r_j}$ along $v_J$. We have $l(\gamma_{r_2},\gamma
_{r_1}^\epsi) - l(\gamma_{r_1},\gamma_{r_1}^ \epsi) =
{\sf int}(\Gamma ,\Gamma^\epsi)$, where $l(\cdot ,\cdot )$ is the linking number and ${\sf int}(\cdot ,\cdot )$ is the intersection number.

Now let us calculate $int(\Gamma ,\Gamma^\epsi)$. From [*Theorem 3.5.1*]{} we know that there are only a finite number $\{ p_1,\ldots, p_N\} $ of self-intersection points of $\Gamma $. Take one of them, say $p_1$. Let $M_1,
\ldots, M_d$ be the disks on $\Gamma $ with a common point $p_1$ and otherwise mutually disjoint. More precisely we take $M_j$ to be irreducible components of $\Gamma \cap B_{p_1}(\rho)$ for $\rho >0$ small enough. Remark that $M_j$ are transversal to $v_J$, so their moves $M_j^\varepsilon$ don’t intersect them, [*i.e.,*]{} $M_j\cap M_j^\epsi = \emptyset $. Note also that ${\sf int}(M_k, M_j) = {\sf int}(M_k, M^\epsi_j)$ for $\varepsilon >0$ sufficiently small. So $int(M_k, M_j) =
int(M_k, M^\epsi_j) + int(M^\epsi_k, M_j)$. Finally $\delta_{p_1} = \sum_{1\le k<j\le d} int(M_k, M_j) =
int(\Gamma \cap B_{p_1}(\rho ),\Gamma^\epsi\cap
B_{p_1}(\rho ))$. This means that $int(\Gamma ,\Gamma^\epsi)
= 2\cdot \sum_{j=1}^N\delta_{p_j}$.

*A2.2. Proof of Adjunction Formula.*

In $\S \S 3.4, 3.5$ we have proved that compact $J$ - complex curve with a finite number of irreducible components $M = \bigcup_{i=1}^d M_i$ has only a finite number of nodes ([*i.e.,*]{} self-intersection points) points, provided $J$ is of class $C^1$.

For each such point $p$ we can introduce, according to [*Definition 3.5.1*]{}, the self-intersection number $\delta_p(M)$ of $M$ at $p$. Namely, let $S_j$ be a parameter curve for $M_j$, [**]{} $M_j$ is given as an image of the $J$-holomorphic map $u_j : S_j\to M_j$. We always suppose that the parameterization $u_j$ is primitive, [*i.e.,*]{} they cannot be decomposed like $u_j=v_j\scirc r$ where $r$ is a nontrivial covering of $S_j$ by another Riemann surface. Denote by $\{ x_1,\ldots, x_N\}$ the set of all pre-images of $p$ under $u : \bigsqcup_{i=1}^d S_j\to X$, and take mutually disjoint disks $\{ D_1,\ldots, D_N\}$ with centers $x_1,\ldots, x_N$ such that their images have no other common points different from $p$. For each pair $D_i,
D_j$, $i\not= j$, define an intersection number as in [*Definition 3.5.1*]{} and take the sum over all different pairs to obtain $\delta_p(M)$.

Now put $\delta = \sum_{p\in D(M)}\delta_p(M)$, where the sum is taken over the set $D(M)$ of all nodes of $M$, [*i.e.,*]{} points which have at least two pre-images.

Consider now the set $\{ p_1,\ldots, p_L\} \subset \bigcup_{j=1}^d S_j = S$ of all cusps of $M$, [*i.e.,*]{} points where the differential of the appropriate parameterization vanishes. Take a small ball $B_r$ around $u(p_i)$ and a small disk $D_{p_i}$ around $p_i$. Let $\gamma^i_r\deff u(\Delta_{p_i}) \cap \partial B_r$ and $b_i$ be the Bennequin index of $\gamma^i_r$, defined in [*Definition A2.1.1*]{}.

Definition A2.2.1. The number $\varkappa_i\deff (b_i +1)/2$ is called the [*conductor of the cusp $a_i=u(p_i)$*]{}.

Also let $\varkappa\deff \sum_{i=1}^L \varkappa_i$. We are now able to state the genus formula.

**Theorem A2.2.1. *Let $M =\cup_{j=1}^d M_j$ be a compact $J$ - complex curve in an almost complex surface $(X, J)$ with the distinct irreducible components $\{M_j\}$, where $J$ is of class $L^{1,p}$. Then***

$$\sum_{j=1}^d g_j = {[M]^2 - c_1(X, J)[M]\over2} + d -\delta - \varkappa ,
\eqno(A2.2.1)$$

where $g_j$ are the genera of parameter curves $S_j$.

Proof. The main line of the proof is the reduction of a general case to the case where $M$ is immersed, which was proved in $\S 1$.

Let $u: \bigsqcup_{j=1}^d S_j \longrightarrow X$ be a $J$-holomorphic map, which parameterizes the curve $M$. Also let $\{x_1,\ldots,x_n\}$ be the set of cusp-points of $M$, [*i.e.,*]{} the images of critical points of $u$. Our reduction procedure is local, and we make our constructions in a neighborhood of every point $x_j$ separately. Therefore, from now on we fix such a point $x$. Due to [*Corollaries 2.2.3*]{} and [*3.4.2*]{} there exists a neighborhood $U$ of $x$ which contains no other cusp-points and no other self-intersection points. [*Theorem 3.5.1 i)*]{} implies that by taking the neighborhood $U$ small enough, we may assume that any component of $M\cap U$ goes through $p\in U$. Without losing generality we may also assume that $U$ is the unit ball $B
\subset \cc^2$, that $x$ corresponds to the center $0$ of $B$, and $J(0)=J\st$.

Denote by $\Gamma_j$ the irreducible components of $M \cap B$ and let $u_j :\Delta \to B$ be a parameterization of $\Gamma_j$ such that $u_j(0)=0
\in B$. Denote $\mu_j \deff \ord_0 du_j +1$; thus $0\in \Delta$ is the cusp-point for $u_j$ $\mu_j\ge2$.

[*Step 1. Rescaling procedure*]{}.

Take some cusp-component $\Gamma_j$. Due to [*Corollary 3.1.3*]{} the map $u_j$ has the form $u_j(z)= z^{\mu_j} \cdot a_j + O\bigl(|z|^{\mu_j+\alpha}
\bigr)$ with the constant $a_j\not=0\in\cc^2$. Moreover, [*Lemma 3.2.3*]{} provides that $$\bigl\vert du_j(z) - \mu_j z^{\mu_j-1} a_j \cdot dz \bigr\vert
\le C(\Vert u_j \Vert_{L^{1,p}(\Delta)}, \alpha, p) \cdot
|z|^{\mu_j-1+\alpha},$$ for any $0<\alpha<1$ and $2<p<\infty$.

For $0<r\le1$ we consider the maps $\pi_r:B\to B$, $\pi_r(w)\deff r^{\mu_j}
\cdot w$, the rescaled maps $u^{(r)}_j:\Delta \to B$, $\pi_r\scirc
u^{(r)}_j(z) = u_j (rz)$, and the rescaled almost complex structures $J^{(r)}
\deff \pi_r^*J$ in $B$. The map $u^{(r)}_j$ is a parameterization of $J^{(r)}$-holomorphic curves $\pi_r^{-1}\Gamma_j$. One can see that $\Vert u^{(r)}_j(z) - z^{\mu_j} a_j \Vert_{C^0(\Delta)} \le C\cdot
r^{\mu_j+\alpha}$ and $\Vert du^{(r)}_j(z) - \mu_j\cdot z^{\mu_j-1} a_j \cdot
dz \Vert_{C^0(\Delta)} \le C\cdot r^{\mu_j-1+\alpha}$. In particular, there exists $r_j>0$ such that for all $0<r<r_j$ the maps $u^{(r)}$ are transversal to all spheres $S^3_1$. On the other hand, $\Vert J^{(r)} - J\st \Vert_{C^1(B)}
= O(r^{\mu_j})$, and for any $\epsi>0$ we can choose sufficiently small $r_j>0$ such that $\Vert J^{(r)} - J\st \Vert_{C^1(B)} \le \epsi$ for all $0<r<r_j$. Thus, by replacing $B$, $J$ and all the maps $u_j$ by their rescaled counterpart, we may assume that the almost complex structure $J$ satisfies the estimate $$\Vert J - J\st \Vert_{C^1(B)} \le \epsi.
\eqno(A2.2.2)$$

[*Step 2. Reduction to the case of holomorphic cusp-points*]{}.

Recall that in [*Lemma 1.4.1*]{} a natural diffeomorphism between the space $\jj$ of orthogonal complex structures in $\rr^4$ and the unit sphere $S^2\deff \{\,(c_1, c_2, s)\in \rr^3 : c_1^2 + c_2^2
+ s^2 =1\, \}$ was established. Define the function $\Phi :\jj \to \rr^2$ by setting $\Phi(J)
=(c_1, c_2)$. The map $\Phi$ defines a diffeomorphism between the upper half-sphere in $S^2$ and the unit disk $\rr^2$ such that the north pole $(0,0,1)\in S^2$ corresponds to the center of the disk.

Define the function $f:B \to \rr^2$ setting $f(w)\deff \Phi(J(w))$. Due to $(A2.2.3)$, $f$ parameterizes the given almost complex structure $J$, $\Vert f \Vert_{C^1(B)} \le C\epsi$, and $f(0)=0$. Fix a cut-off function $\chi$ in $B$ such that $0\le\chi\le1$, $\chi\bigm|_{B(1/2)}\equiv 1$, $\supp \chi \comp B$, and $\Vert d\chi \Vert_{C^1(B)} \le 3$. For $0\le t \le1$ and $0<\sigma<1$ set $f_{\sigma, t}(w) \deff (1-t\chi(w/\sigma))\cdot f(w)$ and define $J_{\sigma, t}\deff \Phi^{-1}(f_{\sigma, t})$. One can easily see that $\Vert f_{\sigma, t} \Vert_{C^1(B)}  \le 4\cdot\Vert f \Vert_{C^1(B)}$, and consequently $\Vert J_{\sigma, t} - J\st \Vert_{C^1(B)} \le C_1\cdot
\Vert J - J\st \Vert_{C^1(B)}\le C_1\cdot \epsi$. Here the constant $C_1$, as well as the constants $C_2,\ldots, C_5$ below in the proof, are independent of $\epsi$, $t$ and $\sigma$. For fixed $\sigma$, the curve $J_{\sigma, t}$, $0\le t\le1$ is a homotopy between $J\equiv J_{\sigma, 0}$ and an almost complex structure $J_\sigma \deff J_{\sigma, 1}$ such that $J_{\sigma, t} \equiv J$ in $B\backslash B(\sigma)$ and $J_\sigma \equiv J\st$ in $B({\sigma\over2})$. Moreover, we have $$\Vert J_{\sigma, t} - J\st \Vert_{C^1(B)} +
\Vert \msmall{ dJ_{\sigma, t} \over dt} \Vert_{C^1(B)} \le C_2\cdot \epsi.
\eqno(A2.2.3)$$

Now we fix some $p>p'>2$ and set $\alpha\deff {2\over p'} -{2\over p}$. Due to (4.2.3) we can apply [*Lemma 3.3.1*]{} to the map $u_j$ with $\nu_j=\mu_j$, $v_j\equiv0$, and with the curve of $C^1$-smooth almost complex structures $J_{\sigma, t}$. As result we obtain a curve of maps $u_{j, \sigma, t} = u_j + z^{\mu_j}w_{j, \sigma, t}$ with $w_{j, \sigma, t}(0)
=0$ and $\Vert w_{j, \sigma, t} \Vert_{L^{1,p}(\Delta)} \le C_3\cdot \Vert u_j
\Vert_{L^{1,p}(B)}$. The condition $\supp \dot J_{\sigma, t} \subset B(\sigma)$ together with [*Corollary 3.1.3*]{} provide that $\supp (\dot J_{\sigma, t}
\scirc du_{j, \sigma, t}) \subset \Delta(\rho)$ with $\rho\sim\sigma^{1/\mu_j}$. Due to (3.3.8) and the Hölder inequality we obtain $$\left\Vert z^{-\mu_j}\cdot \dot J_{\sigma, t} (u_{j, \sigma, t})
\scirc du_{j, \sigma, t} \right\Vert_{L^{p'}_{\vphantom{1}}(\Delta)}
\le C_4 \cdot \sigma^{\alpha/\mu_j}\cdot \Vert u_j  \Vert_{L^{1,p}(\Delta)}^2
\cdot \Vert \dot J_{\sigma, t} \Vert_{C^1(B)},
\eqno(A2.2.4)$$ and consequently

$$\Vert w_{j, \sigma, 1} \Vert_{L^{1,p'}_{\vphantom{1}}(\Delta)}
\le C_5 \cdot \sigma^{\alpha/\mu_j}\cdot \Vert u_j  \Vert_{L^{1,p}(\Delta).}^2
\eqno(A2.2.5)$$ [*Lemma 3.2.3*]{} provides that for $\sigma$ small enough the $J_\sigma$-holomorphic maps $u_{j, \sigma} \deff u_{j, \sigma, 1}$ are transversal to all spheres $S^3_r$, $0<r<{1\over2}$, have no self-intersection points in $B({1\over2}) \backslash B({1\over4})$, and the Bennequin index of $u_{j, \sigma}(\Delta)\cap S^3_r$, ${1\over4}<r<{1\over2}$, coincides with the one of $\Gamma_j\cap S^3_r$, $0<r<1$. Moreover, we may match $u_{j, \sigma}$ to the rest of $M$, changing in an appropriate way the almost complex structure $J$, see [*Lemma A2.3.1*]{}.

As a result, we conclude that for appropriate small enough neighborhoods $U_1
\comp U$ of any cusp-point $p\in M$ there exist a perturbed almost complex structure $\tilde J$ and $\tilde J$-holomorphic curve $\widetilde M$, which is parameterized by $\tilde u:
\bigsqcup_{j=1}^d S_j \longrightarrow X$ and has the following properties:

$\tilde J$ and $\widetilde M$ coincide on $X\backslash U$ with $J$ and $M$ correspondingly;

$U_1$ is a ball centered in $p$ and $\widetilde M \cap U_1$ is obtained from $M\cap U_1$ by perturbing its components in the above described way;

the (possibly reducible) curve $\tilde \gamma \deff \d U_1\cap \wt M$ is isotopic to $\gamma \deff \d U_1\cap M$, in particular all the corresponding components $\tilde \gamma_j$ of $\tilde \gamma$ and $\gamma_j$ of $\gamma$ have the same Bennequin index, and the linking number $l(\tilde \gamma_i,
\tilde \gamma_j)$ is equal to $l(\gamma_i,\gamma_j)$;

$\tilde u$ is homotopic to $u$;

the cusp-points of $\widetilde M$ coincide with the ones of $M$ and $\tilde\delta + \tilde\varkappa= \delta +\varkappa$,

$\tilde J$ is integrable in a neighborhood.

The last equality of follows from due to [*Lemma 4.1.2*]{}. Thus the formulas (4.2.1) for $M$ and $\widetilde M$ are equivalent.

[*Step 3. Final reduction to the case of an immersed curve*]{}.

This step is rather obvious and uses the following fact, shown in \[Bn\]. Let $B$ be the unit ball in $\cc^2$ and let $\Gamma_0$ be an irreducible holomorphic curve in $B$, which is transversal to $\d B$ and is defined as a zero-divisor of a holomorphic function $f$. Then for any sufficiently small nonzero $\epsi\in\cc$ the curve $\Gamma_\epsi$, defined as the zero-divisor of the function $f+\epsi$, are smooth and of the same genus $g$. Moreover, all $\Gamma_\epsi$ are transversal to $\d B$, and the Bennequin index of $\gamma_\epsi \deff \Gamma_\epsi \cap \d B$ equals $2g-1$. In particular, the conductor of a single cusp-point of a holomorphic curve in $B$ can be defined as a genus of general small perturbation to a smooth curve.

In general, let $M$ be a $J$ - complex curve in $X$, such that $J$ is integrable in a neighborhood of every cusp-point of $M$. One can now see that we can perturb $J$ and $M$ to an almost complex structure $\tilde J$ and a $\tilde J$-holomorphic curve $\widetilde M$, satisfying the conditions [*i)*]{}–[*iv)*]{} from above, and the desired condition

[*v$'$)*]{} $\widetilde M$ has no cusp-points, $\sum\tilde g_j =\sum g_j +
\varkappa$, $\tilde \delta =\delta$.

An important corollary of the proof of [*Theorem 3*]{} is the estimate from below of the conductor number of a cusp point.

Corollary A2.2.2. *Let $J$ be an almost complex structure in the unit ball $B\subset \cc^2$ with $J(0)=J\st$, and let $u:\Delta \to B$ be a $J$ - complex curve with the cusp-point $u(0)=0$. Then $\varkappa_0$ is an integer, $\varkappa_0 \ge \ord_0du$, or equivalently, for all sufficiently small $r>0$ the Bennequin index of $\gamma_r\deff u(\Delta) \cap S^3_r$ is odd and satisfies the inequality $$b(\gamma_r)\ge 2\cdot \ord_0du - 1. \eqno(A2.2.6)$$*

**Proof. Rescaling $u$ as in the [*Step 1*]{} of the proof of [*Theorem 3*]{} we may assume that $\Gamma\deff u(\Delta)$ has no nodes and cusps, excepting $0\in \Delta$ , and is transversal to all spheres $S^3_r$, $0<r<1$, so that the Bennequin index $b(\gamma_r)$ is the same and equals $2\varkappa_0 -1$. We may also assume that for the rescaled almost complex structure $J$ the estimate $\Vert J - J\st \Vert_{C^1(B)} \le \epsi$ with the appropriate $\epsi$ is fulfilled. Applying [*Step 2*]{} with a sufficiently small $\sigma$, we can deform $u$ into a $\tilde J$ holomorphic map $\tilde u$ which has the following properties:**

$\widetilde \Gamma \deff \tilde u(\Delta)$ is transversal to all spheres $S^3_r$, $0<r<1$;

$\widetilde \Gamma$ has no self-intersection points in $B\backslash
B({1\over2})$, and the Bennequin index of $\tilde\gamma_r \deff \widetilde
\Gamma\cap S^3_r$, ${1\over2}<r<1$, coincides with the one of $\gamma_r$;

$\tilde J$ coincides with $J$ in $B\backslash B({1\over2})$, is integrable in the neighborhood of $0\in B$, and $\ord_0 d\tilde u =
\ord_0 du $.

Now the corollary follows from [*Lemmas A2.1.2*]{} and the fact that for [*integrable*]{} complex structures the same statement is true, see \[Bn\].

Another modification of the proof of [*Theorem 3*]{} leads to the following consequence.

Lemma A2.2.3 *Let $J$ be a $C^1(X)$-almost complex structure on $X$ and $M\subset X$ a compact $J$ holomorphic curve parameterized by $u: S \to X$. Then*

$u$ can be $L^{2,p}$-approximate by $J_n$-holomorphic immersions $u_n: S \to X$ with $J_n \longrightarrow J$ in $C^1(X)$.

there exists a $C^1(X)$-approximation $J_n$ of $J$ and a sequence of $J_n$-holomorphic imbedded curves $M_n$ which converge to $M$ in the Gromov topology.

Proof. We do not need this result for the purpose of this paper; therefore, we give only a sketch of the proof.

In the first step, one applies the rescaling procedures in order to find appropriate small neighborhoods of the cusp-points of $M$.

In the second step, one applies [*Lemma 3.1.3*]{} to the chosen neighborhoods, taking $\nu=1$, $v$ sufficiently small, and $J$ unchanged. After matching procedure we obtain the statement [*i)*]{}. To obtain the statement , one must first deform all the nodes of $M$ into simple transversal ones and then find an appropriate small neighborhood $U$ of every node. In some complex coordinates $(w_1, w_2)$ in $U$, the curve $M$ is defined by the equation $w_1\cdot w_2=0$. It remains to replace a node $M\cap U$ by a “small handle” $M_\epsi\deff \bigl\{\,
(w_1, w_2) \in U  :  w_1\cdot w_2=\epsi\,\}$ with $\epsi$ sufficiently small and to use the matching procedure once more.

*A2.3. A matching Structures Lemma.*

Let $B(r)$ be a ball of radius $r$ in $\rr^4$ centered at zero, and $J$ a $C^1$-smooth almost-complex structure on $B(2)$, $J(0)=J\st$. Further, let $M = u(\Delta )$ be a closed primitive $J$ - complex disk in $B(2)$ such that $u(0)=0$ and $M$ transversally meet $S^3_r$ for $r\ge 1/2$. Here $S^3_r = \d B(r) $ and transversality are understood with respect to both $TS^3_r$ and $F_r$, see [*paragraph 4.1.*]{}

By $B(r_1,r_2)$ we shall denote the spherical shell $\{ x\in \rr^4:r_1<\Vert
x\Vert <r_2 \}$. In the lemma below denote by $D_{\delta }$ the pre-image of $B(1+\delta )$ by $u$.

Lemma A2.3.1. *For any positive $\delta >0$ there exists an $\epsi >0$ such that if an almost complex structure $\tilde J$ in $B(1+\delta )$ and a closed $\tilde J$-holomorphic curve $\tilde u : D_{\delta }\to B(1+\delta )$ satisfy $\Vert \tilde J - J\Vert_{C^1(\bar B(1+\delta ))} < \epsi $ and $\Vert \tilde u - u\Vert_{L^{1,p}(\bar D_{\delta })} < \epsi $, then there exists an almost-complex structure $J_1$ in $B(2)$ and $J_1$-holomorphic disk $M_1$ in $B(2)$ such that:*

a\) $J_1\ogran_{B(1-\delta )} = \tilde J\ogran_{B(1-\delta )}$ and $J_1\ogran_{B(1+\delta ,2)} = J\ogran_{B(1+\delta ,2)}$.

b\) $M_1\ogran_{B(1-\delta )} = \tilde u(D_{\delta })\cap B(1-\delta )$ and $M_1\cap B(1+\delta ,2) = M\cap B(1+\delta ,2)$.

Proof.

We have chosen the parameterization of $M$ to be primitive. Thus, $u$ is an imbedding on $D_{-\delta ,\delta } = u^{-1}(B_{1-\delta ,1+\delta })$. Let us identify a neighborhood $V$ of $u(D_{-\delta ,\delta })$ in $B_{1-
\delta ,1+\delta }$ with the neighborhood of the zero-section in the normal bundle $N$ to $u(D_{-\delta ,\delta })$. Now $\tilde u\ogran_{D_{-\delta ,\delta
}}$ can be viewed as a section of $N$ over $u(D_{-\delta ,\delta })$ which is small [*i.e.,*]{} contained in $V$. Using an appropriate smooth function $\phi $ on $D_{-\delta ,\delta }$ (or equivalently on $u(D_{-\delta ,\delta} )$), $\phi
\ogran_{B(1-\delta )\cap D_{\delta }}\equiv 1$, $\phi\ogran_{\partial D_{\delta
}
}\equiv 0$, $0\le \phi \le 1$ we can glue $\tilde u$ and $u$ to obtain a *symplectic surface $M_1$ which satisfies (b).*

Patching $J$ and $\tilde J$ and simultaneously making $M_1$ complex can be done in an obvious way.

\#1

[Chapter II. Compactness Theorem.]{}

The goal of this chapter is to give a proof of the Gromov Compactness Theorem for continuously varying almost complex structures and for the sequences of complex curves parameterized by some fixed real surface.

More precisely, we consider a sequence $J_n$ of continuous (of class $C^0$) almost complex structures on a manifold $X$ which converge uniformly to some $J_\infty$, again of class $C^0$. Further, let $(C_n,j_n)$ be a sequence of Riemann surfaces with boundaries of fixed topological type. This means that each $(C_n,\d C_n)$ can be parameterized by the same real surface $(\Sigma, \d\Sigma ) $ (see §2 for details). Denote by $\delta_n:\Sigma
\to C_n$ some parameterizations. However, the complex structures $j_n$ on $C_n$ may vary in an arbitrary way. Finally, let some sequence of $(j_n,J_n)$-holomorphic maps $u_n:C_n\to X$ be given.

Theorem 2.1.

*If the areas of $u_n(C_n)$ are uniformly bounded (with respect to some fixed Riemannian metric on $X$) and the structures $j_n$ do not degenerate at the boundary (see [*Definition 1.7*]{}), then there exists a subsequence, denoted $(C_n, u_n)$, such that*

*1) *$(C_n,j_n)$ converge to some nodal curve $(C_\infty,j_\infty)$ in an appropriate completion of the moduli space of Riemann surfaces of given topological type, i.e., there exists a parameterization map $\sigma_\infty:
\Sigma \to C_\infty$ by the same real surface $\Sigma$;**

*2) *one can choose a new parameterizations $\sigma_n$ of $C_n$ in such a way that $\sigma_n^*j_{C_n}$ will converge to $\sigma_\infty^*j_{C_\infty}$ in the $C^\infty$-topology on compact subsets outside of the finite set of circles on $\Sigma$, which are pre-images of the nodal points of $C_\infty$ by $\sigma_\infty$, where $j_{C_n}$ denote the complex structures on $C_n$;**

[*3)*]{} the maps $u_n\scirc \sigma_n$ converge, in the $C^0$-topology on $\Sigma$ and in the $L^{1,p}_\loc$-topology (for all $p<\infty$) outside of the pre-images of the nodes of $C_\infty$ to a map $u_\infty \scirc
\sigma_\infty$ such that $u_\infty$ is a $(j_\infty, J_\infty) $-holomorphic map $C_\infty\to X$.

This description of convergence is precisely the one given by Gromov in \[G\]. Our statement is slightly more general in two directions. First, we consider not only the case of closed curves, but also the case where $C_n$ is open and of a “fixed topological type”. Second, we note that the Gromov compactness theorem is still valid for continuous and continuously varying almost complex structures. This could have an interesting application, because now one can consider $C^0$-perturbations of complex structures being assured that at least the compactness theorem still holds true. For the definitions involved and the formal statement see §4.1 and [*Theorem 4.1.1*]{}.

We also prove the compactness theorem for curves with boundaries on a totally real submanifold. This “boundary” result needs appropriate generalizations of all “inner” constructions and estimates. The related considerations are shown in Appendix III.

Another result of this paper, which we would like to mention in the introduction, is the improvement of the removable singularity theorem in two directions.

First we prove (see [*Corollary 5.2.1*]{}) the following generalization of the removability theorem for the point singularity.

Theorem 2.2. [*If the area of the image of $J$-holomorphic map $u:(\check\Delta, J\st)\to (X,J)$ from the punctured disk into a compact almost complex manifold “is not growing too fast", i.e., if $\area(u(R_k))\le \eps $ for all annuli $R_k \deff \{ z\in \cc :{1\over e^{k+1}}\le | z| \le {1 \over
e^k}\}$ with $k>\!>1$, then $u$ extends to the origin.* ]{}

The positive constant $\eps $ here depends on the Hermitian structure $(J,h)$ of $X$. This theorem, under the stronger assumption $\sum_k
\area(u(R_k)) \equiv \norm{du}^2_{L^2(\check\Delta)} < \infty$, was proved by Sacks and Uhlenbeck \[S-U\] for harmonic maps, and by Gromov \[G\] for $J$-holomorphic maps.

This fact (which is proved here for continuous $J$’s) is new even in the integrable case. In fact, it measures the “degree of non-hyperbolicity" (in the sense of Kobayashi) of $(X,J,h)$.

Another (see [*Corollary A3.3.5*]{}) is a generalization of Gromov’s result about removability of the boundary point singularity, see \[G\]. An improvement is that the statement remains valid also when one has [*different*]{} boundary conditions to the left and to the right of a singular point. Let us explain this in more detail.

Define the (punctured) half-disk by setting $\Delta^+ \deff \{ z\in \Delta :
\im(z)>0\}$ and $\check\Delta^+ \deff \Delta^+\bs \{0\}$. Define $I_- \deff
]{-1},0[ \subset \d\check\Delta^+$ and $I_+ \deff ]0,+1[ \subset \d\check
\Delta^+$. Let a $J$-holomorphic map $u:(\check\Delta^+, J\st)\to (X,J)$ be given, where $J$ is again continuous. Suppose further that $u(I_+)\subset
W_+$ and $u(I_-) \subset W_-$, where $W_+,W_-$ are totally real submanifolds of dimension $n= \half \dim_\rr X$ and intersect transversally.

Theorem 2.3. [*There is an $\eps^b>0$ such that if for all half-annuli $R^+_k\deff\{ z\in \Delta^+ :e^{-(k+1)} \le | z| \le e^{-k} \}$ one has $\area(u(R^+_k))\le \eps^b$, then $u$ extends to the origin $0\in
\Delta^+$ as an $L^{1,p}$-map for some $p>2$.* ]{}

As in the “inner” case, the necessary condition is weaker than the finiteness of energy. But unlike “inner" and smooth boundary cases, it is possible that the map $u$ in the last statement is $L^{1,p}$-regular in the neighborhood of the “corner point” $0\in \Delta^+$ only for some $p>2$. For example, the map $u(z) = z^\alpha$ with $0<\alpha <1$ satisfies the totally real boundary conditions $u(I_+) \subset \rr$, $u(I_-) \subset e^{\alpha\pi\isl}\rr$ and is $L^{1,p}$-regular only for $p< p^* \deff {2\over1-\alpha}
\cdot$ Note also that by the Sobolev imbedding $L^{1.p}\subset C^{1,\alpha }$ with $\alpha = 1-{2\over p}$, $u$ extends to zero at least continuously. Thus $u(0)\in W_+ \cap W_-$.

One can see such a point $x$ as a [*corner point*]{} for a corresponding complex curve. A typical example appears in symplectic geometry when one takes Lagrangian submanifolds as boundary conditions.

The organization of the chapter is the following. In Lecture 4 we present, for the convenience of the reader, the basic notions concerning the topology on the space of stable curves and complex structure on the Teichmüller space of Riemann surfaces with boundary. In Lecture 5 we give the necessary a priori estimates for the inner case, and the proof of [*Theorem 2.1*]{}, related to curves with free boundary. This includes the case of closed curves. In Appendix III we consider curves with totally real boundary conditions, obtain necessary a priori estimates at a “totally real boundary”, and prove the compactness theorem for such curves. In particular, we prove [*Theorem 2.3*]{} there.

[Lecture 4]{} [Space of Stable Curves]{}

[4.1. Stable Curves and Gromov Topology.]{}

Before stating the Gromov compactness theorem, we need to introduce an appropriate category of complex curves. Since the limit of a sequence smooth curves can be singular,i.e., a cusp-curve in Gromov’s terminology, we need to allow certain types of singularities of curves. On the other hand, it is desirable to have singularities as simple as possible.

A similar problem appears in looking for a “good” compactification of moduli spaces $\calm_{g,m}$ of abstract complex smooth closed curves of genus $g$ with $m$ marked points. The Deligne-Mumford compactification $\barr \calm
_{g, m}$, obtained by adding [*stable curves*]{}, gives a satisfactory solution to this problem and suggests a possible way of generalizing to other situations. In fact, the only singularity type one should allow are nodes, or nodal points. An appropriate notion for curves in a complex algebraic manifold $X$ was introduced by Kontsevich in \[K\]. Our definition of stable curves over $(X,J)$ is simply a translation of this notion to almost complex manifolds. The change of terminology from [*stable maps*]{} to [*stable curves over $(X,J)$*]{} is motivated by the fact that we want to consider our objects as curves rather than maps.

Recall that a [*standard node*]{} is the complex analytic set $\cala_0 \deff
\{ (z_1,z_2)\in \Delta^2 : z_1\cdot z_2 =0\}$. A point on a complex curve is called a [*nodal point*]{} if it has a neighborhood biholomorphic to the standard node.

Definition 4.1.1. [*A [*nodal curve*]{} $C$ is a complex analytic space of pure dimension 1 with only nodal points as singularities*]{}.

In other terminology, nodal curves are called [*prestable*]{}. We shall always suppose that $C$ is connected and has a “finite topology”, i.e., $C$ has finitely many irreducible components, finitely many nodal points, and that $C$ has a smooth boundary $\d C$ consisting of finitely many smooth circles $\gamma_i$, such that $\barr C \deff C \cup \d C$ is compact.

Definition 4.1.2.

*We say that a real connected oriented surface with boundary $(\Sigma, \d\Sigma)$ [*parameterizes*]{} a complex nodal curve $C$ if there is a continuous map $\sigma :\barr\Sigma \to \barr C$ such that*

if $a\in C$ is a nodal point, then $\gamma_a = \sigma\inv(a)$ is a smooth imbedded circle in $\Sigma \bs \d \Sigma $, and if $a\not= b$ then $\gamma_a \cap \gamma_b= \emptyset$;

$\sigma :\barr\Sigma \bs \bigcup_{i=1}^N\gamma_{a_i}\to \barr C \bs \{
a_1,\ldots ,a_N\} $ is a diffeomorphism, where $a_1,\ldots ,a_N$ are the nodes of $C$.

=12.5pt plus 1.5pt

Note that such a parameterization is not unique: if $g:\barr\Sigma \to
\barr\Sigma$ is any orientation preserving diffeomorphism, then $\sigma \scirc
g: \barr\Sigma \to \barr C$ is again a parameterization.

A parameterization of a nodal curve $C$ by a real surface can be considered as a method of “smoothing” of $C$. An alternative method of “smoothing” — the normalization — is also useful for our purposes.

Consider the normalization $\hat C$ of $C$. Mark on each component of this normalization the pre-images (under the normalization map $\pi_C: \hat C \to
C$) of nodal points of $C$. Let $\hat C_i$ be a component of $\hat C$. We can also obtain $\hat C_i$ by taking an appropriate irreducible component $C_i$, replacing nodes contained in $C_i$ by pairs of disks with marked points, and marking remaining nodal points. Since it is convenient to consider components in this form, we make the following

Definition 4.1.3. *A component $C'$ *of a nodal curve $C$ is a normalization of an irreducible component of $C$ with marked points selected as above.**

This definition allows us to introduce the Sobolev and Hölder spaces of functions and (continuous) maps of nodal curves. For example, a continuous map $u: C \to X$ is Sobolev $L_\loc ^{1,p}$-smooth if its restrictions on every component of $C$ is $L_\loc^{l,2}$-smooth as well. The most interesting case is, of course, the one of continuous $L_\loc ^{1,2}$-smooth maps. In this case the energy functional $\norm{du}^2_{L^2(C)}$ is defined. The definition of the energy $\norm{du}^2
_{L^2(C)}$ involves Riemannian metrics on $X$ and $C$ which are supposed to be fixed.

Let $C$ be a nodal curve and $(X,J)$ an almost complex manifold with continuous almost complex structure $J$.

Definition 4.1.4. [*A continuous map $u:C \to X$ is $J$-holomorphic if $u\in L^{1,2}_\loc(C,X)$ and $$du_x + J\scirc du_x\scirc j_C=0
\eqno(4.1.1)$$ for almost all $x\in C$. Here $j_C$ denotes the complex structure on $C$.* ]{}

Recall that the area of a $J$-holomorphic map is defined as $$\area(u(C)) \deff \norm{du}^2_{L^2(C)}.$$

See the end of §1.3. We shall show later that every $J$-holomorphic $u$ is, in fact, $L^{1,p}_\loc(C,X)$-smooth for all $p< \infty$, see [*Corollary 2.4.2*]{}. The following notion of stability was introduced by Kontsevich in \[K\].

Definition 4.1.5. [*A [*stable curve over $(X,J)$*]{} is a pair $(C,u)$, where $C$ is a nodal curve and $u:C\to X$ is a $J$-holomorphic map, satisfying the following condition. If $C'$ is a compact component of $C$, such that $u$ is constant on $C'$, then there exist finitely many biholomorphisms of $C'$ which preserve the marked points of $C$.* ]{}

Remark. One can see that the stability condition is nontrivial only in the following cases:

[*1)*]{} some component $C'$ is biholomorphic to $\cc\pp^1$ with 1 or 2 marked points; in this case $u$ should be non-constant on any such component $C'$;

[*2)*]{} some irreducible component $C'$ is $\cc\pp^1$ or a torus without nodal points.

Since we consider only connected nodal curves, case [*2)*]{} can happen only if $C$ is irreducible, so that $C'=C$. In this case $u$ must be non-constant on $C$.

Now we are going to describe the Gromov topology on the space of stable curves over $X$ introduced in \[G\]. Let a sequence $J_n$ of continuous almost complex structures on $X$ be given. Suppose that $\{J_n\}$ converges to $J_\infty$ in the $C^0$-topology. Furthermore, let $(C_n,
u_n)$ be a sequence of stable curves over $(X, J_n)$ such that all $C_n$ are parameterized by the same real surface $\Sigma$.

Definition 4.1.6.

*We say that $(C_n,u_n)$ [*converges to a stable $J_\infty$-holomorphic curve $(C_\infty,u_\infty)$ over $X$*]{} if the parameterizations $\sigma_n: \barr\Sigma \to \barr C_n$ and $\sigma_\infty:
\barr \Sigma \to \barr C_\infty$ can be chosen in such a way that the following hold:*

$u_n\scirc \sigma_n$ converges to $u_\infty\scirc \sigma_\infty$ in the $C^0( \Sigma, X)$-topology;

if $\{ a_k \}$ is the set of nodes of $C_\infty$ and $\{\gamma_k\}$ are the corresponding circles in $\Sigma$, then on any compact subset $K\comp
\Sigma \bs \cup_k \gamma_k$ the convergence $u_n\scirc \sigma_n \to u_\infty
\scirc \sigma_\infty$ is $L^{1,p}(K, X)$ for all $p< \infty$;

for any compact subset $K\comp \barr\Sigma \bs \cup_k\gamma_k$ there exists $n_0=n_0(K)$ such that $ \sigma_n(K) \subset C_n\setminus \{ nodes\} $ for all $n\ge n_0$ and the complex structures $\sigma_n^*j_{C_n}$ converge smoothly to $\sigma_0^*j_{C_0}$ on $K$;

the structures $\sigma_n^*j_{C_n}$ are constant in $n$ near the boundary $\d\Sigma$.

The reason for introducing the notion of a curve stable over $X$ is similar to the one for the Gromov topology. We are looking for a completion of the space of smooth imbedded complex curves which has “nice” properties, namely: 1) such a completion should contain the limit of some subsequence of every sequence of smooth curves, bounded in an appropriate sense; 2) the same should also hold for every sequence in the completed space; 3) such a limit should be unique. Gromov’s compactness theorem insures us that the space of curves stable over $X$ enjoys these nice properties.

Condition is trivial if $\Sigma $ is closed, but it is important when one considers the “free boundary case”, i.e., when $\Sigma$ (and thus all $C_n$) are not closed and no boundary condition is imposed. However, we would like to point out that in our approach the “free boundary case” is essentially involved in the proof of the compactness theorem also in the case of closed curves. On the other hand, in the case of curves with boundary on totally real submanifolds such a condition is unnecessary.

Recall that a complex annulus $A$ has a conformal radius $R>1$ if $A$ is biholomorphic to $A(1,R) \deff \{ z\in \cc \,:\, 1 <|z| < R \}$. An annulus $A$ is said to be [*adjacent*]{} to a circle $\gamma$, if $\gamma$ is one of its boundary components.

Definition 4.1.7. [*Let $C_n$ be a sequence of nodal curves, parameterized by the same real surface $\Sigma$. We say that complex structures on $C_n$ [*do not degenerate near the boundary*]{}, if there exists $R>1$ such that for any $n$ and any boundary circle $\gamma_{n, i}$ of $C_n$ there exists an annulus $A_{n,i} \subset C_n$ adjacent to $\gamma_{n, i}$ such that all $A_{n,i}$ are mutually disjoint, do not contain nodal points of $C_n$, and have the same conformal radius $R$.*]{}

Since the conformal radii of all the $A_{n, i}$ are the same, we can identify them with $A(1,R)$. This means that all changes of complex structures of $C_n$ take place away from the boundary. The condition is trivial if $C_n$ and $\Sigma$ are closed, $\d\Sigma = \d C_n = \emptyset$.

Remark. Changing our parameterizations $\sigma_n: \Sigma \to C_n$, we can suppose that for any $i$ the pre-image $\sigma_n\inv (A_{n,i} )$ is the same annulus $A_i$ independent of $n$.

Now we state our main result. Fix some Riemannian metric $h$ on $X$ and some $h$-complete set $A\subset X$.

Theorem 4.1.1.

*Let $\{(C_n,u_n)\}$ be a sequence of stable $J_n$-holomorphic curves over $X$ with parameterizations $\delta_n: \Sigma \to
C_n$. Suppose that*

[*a)*]{} $J_n$ are continuous almost complex structures on $X$, $h$-uniformly converging to $J_\infty$ on $A$ and $u_n(C_n)\subset A$ for all $n$;

[*b)*]{} there is a constant $M$ such that $\area [u_n (C_n)]\le M$ for all $n$;

[*c)*]{} complex structures on $C_n$ do not degenerate near the boundary.

Then there is a subsequence $(C_{n_k},u_{n_k})$ and parameterizations $\sigma
_{n_k}: \Sigma \to C_{n_k}$ such that $(C_{n_k}, u_{n_k}, \sigma_{n_k})$ converges to a stable $J_\infty$-holomorphic curve $(C_\infty, u_\infty,
\sigma_\infty)$ over $X$.

Moreover, if the structures $\delta_n^*j_{C_n}$ are constant on the fixed annuli $A_i$, each adjacent to a boundary circle $\gamma_i$ of $\Sigma$, then the new parameterizations $\sigma_{n_k}$ can be taken equal to $\delta_{n_k}$ on some subannuli $A'_i \subset A_i$, also adjacent to $\gamma_i$.

Remarks. 1. In the proof, we shall give a precise description of convergence with estimates in neighborhoods of the contracted circles $\gamma_i$. The convergence of curves with boundary on totally real submanifolds will be studied in §5.5. 2. In applications, one uses a generalized version of the Gromov compactness theorem for nodal curves with a marked point. This version is an immediate consequence of [*Theorem 1.1*]{} due to the following construction. Consider a nodal curve $C$ and let a $J$-holomorphic map $u: C \to X$. Let $\bold
x \deff \{x_1,\ldots,x_m\}$ be the set of marked points on $C$ which are supposed to be distinct from the nodal points of $C$. Define a new curve $C^+$ as the union of $C$ with disks $\Delta_1,\ldots, \Delta_m$ such that $C
\cap \Delta_i = \{x_i\}$ and any $x_i$ becomes a nodal point of $C^+$. Extend $f$ to a map $f^+: C^+ \to X$ by setting $f^+\ogran_{\Delta_i}$ to be constant and equal to $f(x_i)$. An appropriate definition of stability, used for triples $(C, \bold x, f)$, is equivalent to stability of $(C^+, f^+)$. Similarly, the Gromov convergence $(C_n, \bold x_n, f_n) \to (C_\infty, \bold x_\infty, f_\infty)$ is equivalent to the Gromov convergence $(C^+_n, f^+_n) \to (C^+ _\infty, f^+_\infty)$. Thus the Gromov compactness for curves with marked points reduces to the case considered in our paper. However, we shall consider curves with marked points as well.

plus 30pt 

[4.2. Fenchel-Nielsen Coordinates on the Space of Nodal Curves.]{}

In the rest of this section we shall describe topology and conformal geometry of nodal curves and compute the set of moduli parameterizing deformations of a complex structure. As a basic reference we use the book of Abikoff \[Ab\].

Let $C$ be a complex nodal curve parameterized by $\Sigma$.

Definition 4.2.1.

*A component $C'$ of $C$ is called [*nonstable*]{} if one of the following two cases occurs:*

[ 1)]{} $C'$ is $\cc\pp^1$ and has one or two marked points;

[ 2)]{} $C'$ is $\cc\pp^1$ or a torus and has no marked points.

This notion of stability of abstract closed curves is due to Deligne-Mumford, see \[D-M\]. It was generalized by Kontsevich \[K\] for the case of maps $f:C
%%%%%%%%%%%%%%%%%%Question%%%%%%%%%%%%%%%%%%%
\to X$, i.e., for curves over $X$ in our terminology. As was already noted, the last case can happen only if $C=C'$. Strictly speaking, this case should be considered separately. However since such considerations require only obvious changes, we just skip them and suppose that case [*2)*]{} does not occur.

Our first aim is to analyze the behavior of complex structures in the sequence $(C_n,u_n)$ of $J_n$-holomorphic curves stable over $X$ with uniformly bounded areas, which are parameterized by the same real surface $\Sigma$. At the moment, the uniform bound on the area of $u_n(C_n)$ is needed only to show that the number of components of $C_n$ is bounded. Passing to a subsequence, we can assume that all $C_n$ are homeomorphic. This reduces the problem to a description of complex structures on a fixed nodal curve $C$.

To obtain such a description, it is useful to cut the curve into pieces where the behavior of a complex structure is easy to understand. Such a procedure is a [*partition into pants*]{} which is well- known in the theory of moduli spaces of complex structures on curves, see, e.g., \[Ab\], p.93. Here we shall make use of a related but slightly different procedure. Namely, we shall choose a special covering of $\Sigma$ instead of its partition. Further, as blocks for our construction we shall use not only pants, but also disks and annuli. The reasons are that, first, the considered curves can have unstable components and, second, it is convenient to use annuli for a description of the deformation of the complex structure on curves. We start with

Definition 4.2.2. *An annulus $A$ *on a real surface or on a complex curve is a domain which is diffeomorphic (resp. biholomorphic) to the standard annulus $A(r,R) \deff \{ z\in \cc \;:\; r<|z|<R \}$ such that its boundary consists of smoothly imbedded circles. *Pants *(also called a *pair of pants*) on a real surface or on a complex curve is a domain which is diffeomorphic to a disk with 2 holes.******

The boundary of pants consists of three components, each of them being either a smoothly imbedded circle or a point. This point can be considered as a puncture of pants or as a marked point. An annulus or pants is [*adjacent to a circle $\gamma$*]{} if $\gamma$ is one of its boundary components.

Let $C$ be a nodal curve parameterized by a real surface $\Sigma$. We shall associate with every such curve $C$ a certain graph $\Gamma_C$ which determines $C$ topologically in a unique way. In fact, $\Gamma_C$ will also determine a decomposition of some components of $C$ into pants.

By definition, a compact component $C'$ is stable if it contains only a finite number of automorphisms preserving marking points. In this case $C'
\bs\mapo$ possesses a unique so-called intrinsic metric.

Definition 4.2.3.

*The [*intrinsic metric*]{} for a smooth curve $C$ with marked points $\{ x_i \}$ and with boundary $\d C$ is a metric $g$ on $C\bs \mapo$ satisfying the following properties:*

$g$ induces the given complex structure $j_C$;

the Gauss curvature of $g$ is constantly -1;

$g$ is complete in a neighborhood of every marked point $x_i$;

every boundary circle $\gamma$ of $C$ is geodesic $g$.

Note that such a metric, if it exists, is unique, see .

Now consider a component $C'$ of $C$ adjacent to some boundary circle of $C$. Then $C' \bs \mapo$ is one of the following:

a) a disk $\Delta$, or

b) an annulus $A$, or

c) a punctured disk $\check\Delta$, or else d) $C'\bs \mapo$ admits the intrinsic metric.

Note that if a component $C'$ is a disk or an annulus (both without marked points), then $C'$ is the whole curve $C$. We shall consider cases a) and b) later. Now we assume, for simplicity, that cases a) and b) do not occur.

Definition 4.2.4 *A component $C'$ of a nodal curve $C$ is called [*non-exceptional*]{} $C' \bs \mapo$ admits the intrinsic metric.*

In particular, nonstable components are exceptional compact ones, and exceptional non-compact components are those of types a)–c) above.

Take some non-exceptional component $C'$ of $C$. There is a so- called maximal partition of $C'\bs \mapo$ into pants $\{ C_1,\ldots C_n\}$ such that all boundary components of these pants are either simple geodesic circles in intrinsic metric or marked points, see \[Ab\]. Let us fix such a partition and mark the obtained geodesic circles on $C'$.

Now let $\sigma: \Sigma \to C$ be some parameterization of $C$. This defines the set $\bfgamma'$ of the circles on $\Sigma$ which correspond to the nodes of $C$. Let $\bfgamma''$ be the set of $\sigma $-pre-images of the geodesics chosen above. Then $\bfgamma\deff \bfgamma' \sqcup \bfgamma''$ forms a system of disjoint “marked” circles on $\Sigma$, which encodes the topological structure of $C$. Now the graph $\Gamma_C$ in question can be constructed as follows.

Define the set $V_C$ of vertices of $\Gamma_C$ to be the set $\{ S_j \}$ of connected components of $\Sigma \bs \cup_{\gamma \in \bfgamma} \gamma =
\sqcup_j S_j$. Any $\gamma \in \bfgamma$ lies between 2 components, say $S_j$ and $S_k$, and we draw an edge connecting the corresponding 2 vertices. Further, any boundary circle $\gamma$ of $\Sigma$ has the uniquely defined component $S_j$ adjacent to $\gamma$. For any such $\gamma$ we draw a [*tail*]{}, i.e., an edge with one end free, attached to vertex $S_j$. Finally, we mark all edges which correspond to the circles $\bfgamma'$, i.e., those coming from the nodes.

Having the graph $\Gamma$, which characterizes uniquely the topological structure of $C$, we are now going to describe the set of parameters defining (uniquely) the complex structure of the curves $C$. This is equivalent to determining the complex structure and marked points on all components of $C$. If such a component $C'$ is a sphere with 1 or 2 marked points or a disk with 1 marked point, then its structure is defined by its topology uniquely up to diffeomorphism. Otherwise, the component $C'$ is non-exceptional. In this case the complex structure and the marked points can be restored by the so- called [*Fenchel-Nielsen coordinates*]{} on the Teichmüller space $\ttt_{g,m,b}$. Recall that the space $\ttt_{g,m,b}$ parameterizes the complex structures on a Riemann surface $\Sigma$ of genus $g$ with $m$ punctures (marked points) and with a boundary consisting of $b$ circles, see \[Ab\].

Let $C$ be a smooth complex curve with marked points of non-exceptional type, so that $C$ admits the intrinsic metric. Fix some parameterization $\sigma:
\Sigma \to C$. Consider the pre-images of the marked points on $C$ as marked points on $\Sigma$ or, equivalently, as punctures of $\Sigma$. Let $C \bs
\mapo = \cup_j C_j$ be a decomposition of $C$ into pants and $\Sigma\bs
\mapo = \cup_j S_j$ the induced decomposition of $\Sigma$.

Let $\{\gamma_i\}$ be the set of boundary circles of $\Sigma$. The boundary of every pants $S_j$ has three components, each of them being either a marked point of $\Sigma$ or a circle. In the last case this circle is either a boundary component of $\Sigma$ or a boundary component of another pants, say $S_k$. In this situation we denote by $\gamma _{jk}$ the circle lying between the pants $S_j$ and $S_k$. Fix the orientation on $\gamma _{jk}$, induced from $S_j$ if $j<k$ and from $S_k$ if $k<j$. For any such circle $\gamma
_{jk}$, fix a boundary component of $S_j$ different from $\gamma_{jk}$ and denote it by $\d_k S_j$. In the same way fix a boundary component $\d_j S_k$. Make similar notations on $C$ using primes to distinguish the circles on $C$ from those on $\Sigma$, i.e., set $\gamma'_i \deff \sigma(\gamma_i)$ and $\gamma'_{jk} \deff \sigma(\gamma_{jk})$.

By our construction, $\gamma'_{jk} = \sigma( \gamma_{jk})$ is a geodesic the intrinsic metric in $C$.

Using the same procedure in $C_k$, we obtain another point $x^*_{k,j} \in
\gamma' _{jk}$. Denote by $\ell_{jk}$ (resp. by $\ell_i$) the intrinsic length of $\gamma'_{jk}$ (resp. of $\gamma'_i \deff \sigma( \gamma_i)$) in $C$. For $j<k$ define $\lambda_{jk}$ as the intrinsic length of the arc on $\gamma' _{jk}$, which starts at $x^* _{j,k}$ and goes to $x^* _{k,j}$ in the direction determined by the orientation of $\gamma _{jk}$. Set $\vartheta
_{jk} \deff {2\pi \lambda _{jk} \over \ell_{jk} }$. We shall consider $\vartheta _{jk}$ as a function of the complex structure $j_C$ on $C$ with values in $S^1 \cong \rr/ 2\pi \zz$.

The parameters $\bfell \deff (\ell_i, \ell_{jk} )$ and $\bfvartheta \deff(\vartheta_{jk})$ are called [*Fenchel-Nielsen coordinates*]{} of the complex structure $j_C$. The reason is that these parameters determine up to isomorphism the complex structure $j_C$ on the smooth complex curve with marked points parameterized by a real surface $\Sigma$. In other words, $(\bfell, \bfvartheta)$ can be considered as coordinates on $\ttt_{g,m,b}$. More precisely, one has the following

Proposition 4.2.2. *Let $\Sigma$ be a real surface of genus $g$ with $m$ marked points and with the boundary consisting of $b$ circles, so that $2g+m +b\ge3$. Let $\Sigma \bs \mapo = \cup_j S_j$ be its decomposition into pants. Then*

[*i)*]{} for any given tuples $\bfell= (\ell_i, \ell_{jk})$ and $\bfvartheta = (\vartheta_{jk})$ with $\ell_i, \ell_{jk} >0$ and $\vartheta_{jk} \in S^1$ there exists a complex structure $j_C$ on $\Sigma$ such that boundary circles of all $S_j$ are geodesic the intrinsic metric on $\Sigma\bs \mapo$ defined by $j_C$ and the given $(\bfell, \bfvartheta)$ are Fenchel-Nielsen coordinates of $j$; moreover, such a structure $j_C$ is unique up to a diffeomorphism preserving the pants $S_j$ and the marked points;

[*ii)*]{} let $C$ be a smooth complex curve with parameterization $\sigma:
\Sigma \to C$ which has $m$ marked points; then there exists a parameterization $\sigma_1: \Sigma \to C$ isotopic to $\sigma$, which maps boundary components and marked points of $\Sigma$ onto the ones of $C$ in prescribed order such that the boundary circles of $\sigma(S_j)$ are geodesic the intrinsic metric on $C\bs \mapo$.

Proof. See \[Ab\].

[4.3. Complex Structure on the Space $\ttt_\Gamma$.]{}

Let $\Sigma$ be a real surface of genus $g$ with $m$ marked points and with the boundary consisting of $b$ circles. Assume that $2g+m +b\ge3$. Then there exists a decomposition of $\Sigma \bs \mapo$ into pants, which is in general not unique. The topological type of such a decomposition can be encoded in graph $\Gamma$, associated with the decomposition. It is constructed in a similar way to that above, but this time we must draw a tail for every marked point, and then mark all those tails on the graph.

Let such a graph $\Gamma$ be fixed. We call two complex structures $J_1$ and $J_2$ on $\Sigma$ isomorphic if there exists a biholomorphism $\phi: (\Sigma,
J_1) \cong (\Sigma, J_1)$ preserving the marked points of $\Sigma$ and the decomposition of $\Sigma$ into pants given by graph $\Gamma$. Denote by $\ttt' _\Gamma$ the space of isomorphism classes of complex structures on $\Sigma$. By [*Proposition 1.2*]{}, Fenchel-Nielsen coordinates identify $\ttt'_\Gamma$ with the real manifold $\rr_+^{3g-3+m+2b} \times (S^1)^{3g-3+m+b}$.

It is desirable to equip $\ttt'_\Gamma$ with some natural complex structure. In so doing, the main difficulty is that the real dimension of $\ttt'_\Gamma$ can be odd. A possible explanation of this fact is that not all relevant information (parameters) about a complex structure has been taken into consideration. Note that for any “inner circle” $\gamma_{jk}$ which appears after the decomposition into pants, we have obtained a pair of coordinates, mainly the length $\ell_{jk}$ and the angle $\vartheta_{jk}$. On the other hand, for any boundary circle $\gamma_i$ of $\Sigma$ we have only the length $\ell_i$. An obvious way to produce additional angle coordinates is to introduce an additional marking of every boundary circle.

Definition 4.3.1. *A real surface $\Sigma$ or a nodal complex curve $C$ is said to have a [*marked boundary*]{} if on every boundary circle of $\Sigma$ (resp. $C$) a point is fixed.*

Remark. Later in §5 we shall consider complex curves with several marked points on boundary circles. But now we shall assume that on every boundary circle exactly one point is marked.

“Missed” angle coordinates $\vartheta_i$ can be now introduced similarly to $\vartheta_{jk}$. For a boundary circle $\gamma_i$ we consider the adjacent pants $S_j$. Fix a boundary component $\d_i S_j$ different from $\gamma_i$. Let $J$ be a complex structure on $\Sigma$ such that the boundary circles of all pants $S_k$ are geodesic the intrinsic metric defined by $J$. Using constructions from above, find a geodesic (resp. a ray) $\alpha_i$ starting at point $x^*_i \in \gamma_i$ and ending at the boundary circle $\d_i S_j$ (resp. approaching marked point $\d_i S_j$ of $\Sigma$). Take the marked boundary point $\zeta_i$ on $\gamma_i$ and consider the length $\lambda_i$ of the geodesic ark on $\gamma_i$, starting at $x^*_i$ and going to $\zeta_i$ in the direction defined by the orientation of $\gamma_i$. Define $\vartheta_i
\deff {2\pi \lambda_i \over \ell_i }$, $\vartheta_i \in S^1\cong \rr/ 2\pi\zz$. We include the coordinates $\vartheta_i$ in the system of angle coordinates $\bfvartheta$. Denote by $\ttt_\Gamma$ the set of isomorphism classes of complex structures on $\Sigma$ with marked boundary and with a given decomposition into pants.

Let $C$ be a smooth complex curve with marked points and a marked boundary, $C
\bs \mapo = \cup_j C_j$ its decomposition into pants, $\sigma: \Sigma \to C$ a parameterization, and $\Sigma \bs \mapo = \cup_j S_j$ the induced decomposition of $\Sigma$. To define a complex structure on $\ttt_\Gamma$, we introduce special local holomorphic coordinates in a neighborhood of boundary of pants on $C$. Consider some pants $S_j$ and its boundary circle $\gamma^*$. It can be a boundary circle of $\Sigma$, $\gamma_i$ in our previous notation, or a circle $\gamma_{jk}$ separating $S_j$ from another pants $S_k$. Let $\ell^*$ be the intrinsic length of $\gamma^*$. Fix some small $a>0$ and consider the annulus $A$ consisting of those $x\in S_j$ for which the intrinsic distance $\dist(x,\gamma^*) <a$. The universal cover $\ti A$ can be imbedded into the hyperbolic plane $\hh$ as an infinite strip $\Theta$ of constant width $a$ such that one of its borders is a geodesic line $L$. The action of a generator of $\pi_1(A) \cong \zz$ on $\ti A$ is defined by the shift of $\Theta$ along $L$ by distance $\ell^*$.

Now consider the annulus $A' \deff [0, {\pi^2\over \ell^*}] \times S^1$ with coordinates $\rho, \theta$, $0\le\rho < {\pi^2\over \ell^*}$, $0\le \theta \le
2\pi$ and with the metric $({\ell^* \over 2\pi} / \cos {\ell^* \rho\over 2\pi}
)^2 (d\rho^2 + d\theta^2)$. A direct computation shows that this metric is of constant curvature $-1$ and that the boundary circle $\d_0A_1 \deff S^1 \times 
\{0\}$ is geodesic of length $\ell^*$, whereas $A'$ is complete in a neighborhood of the other boundary circle. Consequently, the universal cover $\ti A{}'$ of $A'$ can be imbedded in the hyperbolic plane $\hh$ as a hyperbolic half-plane $\hh^+_L$ with a boundary line $L$ such that $\Theta \subset \hh^+_L$. Moreover, the action of $\pi_1(A') \cong \zz$ on $\ti A{}' \cong \hh^+_L$ is the same as for $\ti A \cong \Theta$. This shows that there exists an [*isometric*]{} imbedding of $A$ into $A'$ which maps $\gamma^*$ onto $\d_0 A'$. Moreover, such an imbedding is unique up to rotations in the coordinate $\theta$. This leads us to the following

Proposition 4.3.1. *Let $C_j$ be pants with a complex structure and $\gamma^*$ its boundary circle of the intrinsic length $\ell^*$. Let $x^*$ be a point on $\gamma$. Then some collar annulus $A$ of $\gamma^*$ possesses the uniquely defined conformal coordinates $\theta\in S^1\cong \rr/
2\pi\zz$ and $\rho$ such that the intrinsic metric has the form $({\ell^*
\over 2\pi} / \cos {\ell^* \rho\over 2\pi})^2 (d\rho^2+ d\theta^2)$, $\rho|_{
\gamma^*} \equiv0$, $\theta(x^*) =0$ and the orientation on $S_j$ is given by $d\theta\wedge d\rho$.*

We shall represent $\rho$ and $\theta$ also in the complex form $\zeta \deff
e^{-\rho + \isl \theta}$ and call $\zeta$ the [*intrinsic coordinate*]{} of the pants $C_j$ at $\gamma^*$. An important corollary of the description of the intrinsic metric in a neighborhood of the boundary circle is the following statement about non-degenerating complex structures in pants, see [*Definition 1.7*]{}.

Lemma 4.3.2. *Let $C$ be a smooth complex curve with marked points admitting the intrinsic metric and let $\gamma^*$ be a boundary circle of $C$ of length $\ell^*$.*

If there exists an annulus $A \subset C$ of conformal radius $R$ ($A
\cong \{\,z \in \cc \;:\; 1<|z| < R \;\})$, adjacent to $\gamma^*$ and containing no marked points, then $\log R \le {\pi^2 \over \ell^*}$.

There exists a universal constant $a^*$ such that the condition $\ell^*
\le 1$ implies that there exists an annulus $A \subset C$ of conformal radius $R$ with $\log R \ge { \pi^2\over \ell^*} -{2\pi\over a^*}$, which is adjacent to $\gamma^*$, has area $a^*$ and contains no marked points of $C$.

Let $\gamma\subset C$ be a simple geodesic circle of the length $\ell$ and $A \subset C\bs\mapo$ annulus of conformal radius $R$ homotopy equivalent to $\gamma$. Then $\log R \le {2\pi^2 \over \ell}$.

Proof. Let $\Omega$ be the universal cover of $C\bs \mapo$ equipped with the intrinsic metric lifted from $C$. Then $\Omega$ can be isometrically imbedded into the hyperbolic plane $\hh$ as a domain bounded by geodesic lines such that each of these lines covers some boundary circle $\gamma_i$. Take some (not unique!) line $L$ covering the circle $\gamma^*$ and fix a hyperbolic half-plane $\hh^+_L$ with a boundary line $L$, so that $\Omega \subset
\hh^+_L$.

Now consider the universal cover $\ti A$ of the annulus $A$ and provide it with the metric induced from $C$. Then we can isometrically imbed $\ti A$ in $\hh^+_L$ in such a way that the line covering $\gamma^* \subset \d A$ will be mapped onto $L$. The action of a generator of $\pi_1(A) \cong \zz$ on $\ti A$ is defined by the shift of $\hh$ along $L$ onto a distance $\ell^*$. Consequently, $A$ can be isometrically imbedded into $\hh^+_L/ \pi_1(A)$, which is the annulus $A'= [0, {\pi^2\over \ell^*}] \times S^1$ with coordinates $\rho, \theta$, $0\le\rho < {\pi^2\over \ell^*}$, $0\le \theta \le 2\pi$ and with a metric $({\ell^* \over 2\pi} / \cos {\ell^* \rho\over 2\pi})^2 
(d\rho^2 +d\theta^2)$. Note that the conformal radius of $A'$ is $e^{\pi^2/\ell^*}$. The monotonicity of the conformal radius of annuli (see , Ch.II, §1.3) yields the inequality $R\le e^{\pi^2/\ell^*}$ which is equivalent to first assertion of the lemma.

Part of the lemma can be proved by same argument. More precisely, under the hypothesis of part we imbed the annulus $A$ into the annulus $A''= ]-{\pi^2\over \ell}, {\pi^2\over \ell}[ \times S^1$ with coordinates $\rho, \theta$, $- {\pi^2\over \ell} < \rho < {\pi^2\over \ell}$, $0\le \theta \le 2\pi$ and with a metric $({\ell \over 2\pi} / \cos {\ell \rho \over 2\pi})^2 (d\rho^2 +d\theta^2)$. The conformal radius of $A$ is now estimated by the conformal radius of $A''$, which is equal to $e^{2\pi^2\over \ell}$.

The second part of our lemma follows from results of Ch.II, §3.3 of \[Ab\]. [*Lemma 2*]{} says that there exists a universal constant $a^*$ with the following property: If $\ell^* \le 1$, then there exists a collar neighborhood $A$ of constant width $\rho^*$ and of area $a^*$, which is an annulus imbedded in $C$ and contains no marked points of $C$. In particular, we can extend the intrinsic coordinates $\rho$ and $\theta$ in $A$. Using these coordinates, we present $A$ in the form $\{ (\rho,\theta): 0\le \rho \le \rho^*\}$ and compute the area, $$a^*=\area A = 2\pi \int_{\rho=0}^{\rho^*} \left(\msmall{\ell^*/2\pi \over
\cos(\ell^*\rho/2\pi) }\right)^2 d\rho =
\ell^* \tan\left(\msmall{\ell^*\rho^* \over 2\pi}\right).$$ Consequently, $\tan\left({\pi\over2}-{\ell^*\rho^* \over 2\pi}\right) =
\cotan\left({\ell^*\rho^* \over 2\pi}\right) = {\ell^*\over a^*}$. This implies ${\pi\over2}-{\ell^*\rho^* \over 2\pi} \le {\ell^*\over a^*}$, which is equivalent to $\rho^* \ge { \pi^2\over \ell^*} -{2\pi\over a^*}$. To finish the proof we note that the conformal radius $R$ of $A$ is equal to $e^{\rho^*}$.

Let $C$ be a smooth complex curve with marked points, $C_j$ a piece of a decomposition of $C\bs \mapo$ into pants and $\gamma^*$ its boundary circle. Then as a “base point” $x^*=\{\theta=0=\rho\}$ for the definition of the intrinsic coordinate we shall use the point $x^*_{j,k}$ if $\sigma( \gamma^*
)$ is the geodesic separating $C_j$ from another pants $C_k$, or respectively, the point $x^*_i$ if $\gamma^*$ is a boundary circle of $C$. We denote these coordinates $\zeta_{j,k}= e^{-\rho_{j,k} + \isl \theta_{j,k}}$ and $\zeta_i=
e^{-\rho_i + \isl \theta_i}$. Note that $\vartheta_i$ is exactly the $\theta$-coordinate of the marked boundary point $x_i\in \gamma_i$ with respect to $x^*_i$, and $\vartheta_{jk}$ is the $\theta$-coordinate of $x^*_{k,j}$ with respect to $x^*_{j,k}$.

Note also that any intrinsic coordinate of a pair $(\zeta_{j,k}, \zeta_{k,j}
)$ extends canonically from one collar neighborhood of $\gamma_{jk}$ to another side in such a way that the formula for the intrinsic metric remains valid. This extension possesses the property $\zeta_{j,k}\cdot \zeta
_{k,j} \equiv e^{\isl \vartheta_{jk}}$, where $\vartheta_{jk}$ is a constant function. We can view this relation as the transition function from $\zeta_{j,k}$ to $\zeta_{k,j}$.

A similar construction is possible in the case of a boundary circle $\gamma_i$. Namely, allowing $\rho_i$ to change also in the interval $(-{\pi^2 \over
\ell_i}, 0]$ and maintaining the formula $({\ell_i \over 2\pi} / \cos {\ell_i
\rho_i\over 2\pi})^2 (d\rho_i^2+ d\theta_i^2)$ for the metric, we can glue to $\Sigma$ an annulus $(-{\pi^2\over \ell_i}, 0] \times S^1$ and extend the coordinate $\zeta_i = e^{-\rho_i + \isl \theta_i}$ there.

Making such a construction with every boundary circle $\gamma_i$, we obtain a complex curve $C^{(N)}$ with the following properties. $C$ is relatively compact in $C^{(N)}$ and the intrinsic metric of $C$ extends to a complete Riemannian metric on $C^{(N)}$ with constant curvature $-1$. Such an extension and the metric are unique. $C^{(N)}$ is called the [*Nielsen extension of $C$*]{}, see \[Ab\]. Note that the complex coordinate $\zeta_i$ can be extended further to the unit disk $\{|\zeta_i|<1\}$.

Using the introduced complex coordinates $\zeta_i$ and $\zeta_{j,k}$, we define a deformation family of complex structures on the curve $C$ with marked boundary. Let $\lambda_i$ and $\lambda_{jk}$ be complex parameters changing in small neighborhoods of $e^{\isl\theta_i}$ and $e^{\isl \theta
_{jk}}$, respectively. Having these data $\bflambda =(\lambda_i, \lambda_{jk}
)$, construct a complex curve $C_\bflambda$ in the following way. Take the pants $\{C_j\}$ of the given decomposition of $C$ and extend all the complex coordinates $\zeta_i$ and $\zeta_{jk}$ outside the pants. Glue the pairs of coordinates $(\zeta_{j,k},\zeta_{k,j})$ with new transition relations $\zeta_{j,k}\cdot \zeta_{k,j}= \lambda_{jk}$ (constant functions). Move original boundary circles $\gamma_i= \{|\zeta_i|=1\}$ of $C$ to new positions defined by the equations $|\zeta_i|= |\lambda_i|$ and mark the points $\zeta_i = \lambda_i$ on them.

Theorem 4.3.3. *The natural map $F:\bflambda \to(\bfell,\bfvartheta)$ is non-degenerated. In particular, $\bflambda$ can be considered as the set of local complex coordinates on $\ttt_\Gamma$ and $\calc\deff \{ C_\bflambda
\}$ as a (local) universal holomorphic family of curves over $\ttt_\Gamma$.*

Proof. Write the functions $\bflambda = (\lambda_i, \lambda_{jk})$ in the form $\lambda_i= e^{-r_i + \isl\phi_i}$, $\lambda_{jk}= e^{-r_{jk} +
\isl\phi_{jk}}$. From the definition of the map $$\textstyle
F: (e^{-r_i + \isl\phi_i}, e^{-r_{jk} + \isl\phi_{jk}})
\mapsto (\ell_i, \ell_{jk}; \vartheta_i, \vartheta_{jk})$$ it is easy to see that ${\d (\vartheta_i, \vartheta_{jk}) \over \d(\phi_i, \phi_{jk})}$ is the identity matrix, whereas ${\d (\ell_i, \ell_{jk}) \over \d(\phi_i, \phi_{jk}
)}$ is equal to 0. So it remains to show that the matrix ${\d (\ell_i,
\ell_{jk}) \over \d(r_i, r_{jk})}$ is non-degenerate.

Consider a special case where $C$ is pants with the boundary circles $\gamma_i$ (at least one) and, possibly, with marked points $x_j$. We shall consider such points as punctures of $C$. Let $J$ denote the complex structure on $C$ and let $\mu_0$ be the intrinsic metric. Extend the coordinates $\zeta_i$ and the metric $\mu_0$ outside of $\gamma_i$ to some bigger complex curve $\wt C$ with $C \comp \wt C$.

Fix real numbers $v_i$ and consider the domains $C_t$ in $\wt C$ defined in local coordinates $\zeta_i= e^{-\rho_i + \isl\theta_i}$ by the inequalities $\rho_i \ge v_it$. This defines a family of deformations of $C= C_0$ parameterized by a real parameter $t$, corresponding to a real curve in the parameter space $\{ \bflambda \}$ given by $\lambda_i(t) = e^{v_it}$. Note that the deformation is made in such a way that the original complex structure $J$ and local holomorphic coordinates are preserved. Thus we can use them as an “invariable basis” in our calculations.

Let $\mu_t$ be the intrinsic metric of $C_t$. Without loss of generality we may assume that $\mu_t$ extends to $\wt C$ as a metric with constant curvature $-1$, which induces the original complex structure $J$ on $\wt C$. Since the intrinsic metric depends smoothly on the operator $J$ of a complex structure, $\mu_t$ are smooth in $t$. In a local holomorphic coordinate $z=x
+\isl y$ we can present $\mu_t$ in the form $e^{2\psi(t,z)}(dx^2 + dy^2)$. The condition ${\sf Curv}(\mu_t) \equiv -1$ is equivalent to the differential equation $$\d^2_{xx}\psi(t,\cdot) + \d^2_{yy} \psi(t, \cdot)= e^{2\psi(t, \cdot)},$$ where $\d_x$ denotes the partial derivation ${\d \over \d x}$ and so on. Differentiating it in $t$, we obtain $e^{-2\psi(t,\cdot)}(\d^2_{xx}+ \d^2_{yy})
\dot \psi(t,\cdot) = \dot\psi(t,\cdot)$, where $\dot\psi(t,\cdot)$ denotes the derivative of $\psi(t,\cdot)$ in $t$.

Note that $\d_t\mu_t = 2 \dot \psi(t,\cdot)\cdot \mu_t $, so $\dot \psi_t
(\cdot)= \dot\psi(t, \cdot)$ is independent of the choice of a local holomorphic coordinate $z= x+ \isl y$ and is defined globally. The equation $e^{-2\psi_z(t,\cdot)} (\d^2_{xx}+ \d^2_{yy})\dot\psi_z(t,\cdot) = \dot\psi_z
(t,\cdot)$ can be rewritten in the form $\Delta_t \dot\psi_t = 2 \dot \psi_t
$, with $\Delta_t$ denoting the Laplace operator for the metric $\mu_t$.

The condition that the circle $\gamma_i(t) \deff \{ \rho_i = v_it\}$ is $\mu_t $-geodesic means that the covariant derivative $\nabla_{\theta_i}
(\d_{\theta_i})$ of the vector field $\d_{\theta_i}$, the tangent vector field to $\gamma_i(t)$, must be parallel to $\d_{\theta_i}$ along $\gamma_i(t)$. Expressing this relation in local coordinates $\rho_i$ and $\theta_i$, we obtain $\d_{\rho_i}\psi_i(t; v_it, \theta_i)=0$, where $\mu_t =e^{2 \psi_i (t;
\rho_i, \theta_i)}(\d\rho_i^2 + \d\theta_i^2)$ is a local representation of the metric $\mu_t$. Deriving in $t$, we obtain $\d_{\rho_i}\dot \psi_i(t; v_it,
\theta_i) + v_i \d^2_{\rho_i\rho_i}\psi_i(t; v_it, \theta_i) =0$.

In the case where $t=0$ we have $\psi_i(0; 0, \theta_i) \equiv \log{\ell_i \over
2\pi}$, a constant. Hence $\d^2_{\theta_i\theta_i}\psi_i(0; 0, \theta_i) =
e^{2 \psi_i(0; 0, \theta_i)} $ and $\d_{\rho_i}\dot \psi_i(0; 0, \theta_i) =
- v_i e^{2 \psi_i(0; 0, \theta_i)}= - v_i \left({\ell_i \over2\pi} \right)^2
$. On the other hand, $\d_{\rho_i} = - {\ell_i \over 2\pi} \d_\nu$ on $\gamma_i(0) = \gamma_i$, where $\nu$ denotes the unit outer normal field to $C_0=C$. Consider the integral $\int_C |d\dot\psi {}_0|^2 + 2 \dot\psi{}_0^2
d\mu_0$. Integrating by parts, we get $$\eqalignno{
&\int_C |d\dot\psi{}_0|^2 + 2 \dot\psi{}_0^2 d\mu_0 =
\int_C \dot\psi_0(2 \dot\psi_0 - \Delta_0 \dot\psi_0) d\mu_0
+ \int_{\d C} \dot\psi_0 \d_\nu\dot\psi_0 dl=
\cr
\noalign{\vskip0pt\allowbreak}
=&
\sum_i \int_{\gamma_i} \dot\psi_0 \d_\nu\dot\psi_0
\msmall{\ell_i \over 2\pi}
d\theta_i =
\sum_i - \int_{\gamma_i}  \dot\psi_0 \d_{\rho_i}\dot\psi_0
d\theta_i =
\sum_i \int_{\gamma_i}  \dot\psi_0 v_i \left(\msmall
{\ell_i \over 2\pi}\right)^2 d\theta_i =
\cr
\noalign{\vskip0pt\allowbreak}
=&
\sum_i \msmall{v_i \ell_i \over 2\pi} \int_{\gamma_i}  \dot\psi_0
e^{\psi_i(0; 0, \theta_i)} d\theta_i =
\sum_i \msmall{v_i \ell_i \over 2\pi} \int_{\gamma_i}  (\dot\psi_0 +
v_i \d_{\rho_i} \psi_i(0; 0, \theta_i))e^{\psi_i(0; 0, \theta_i)} d\theta_i =
\cr
\noalign{\vskip0pt\allowbreak}
=&
\sum_i \msmall{v_i \ell_i \over 2\pi}
\left.\msmall{\d \over \d t}\right|_{t=0} \int_{\gamma_i}
e^{\psi_i(t; v_it, \theta_i)} d\theta_i =
\sum_i \msmall{v_i \ell_i \over 2\pi}
\left.\msmall{\d \over \d t}\right|_{t=0}\ell_i(t) =
\sum_i \msmall{ \ell_i \over 2\pi} v_i \dot\ell_i.
}$$ Here $\dot\ell_i$ denotes the derivative of the length parameter $\ell_i$ for the curve $C_t$ at $t=0$, so that $(\dot\ell_1,\dot\ell_2,\dot\ell_3)= dF (v_1, v_2, v_3)$. The obtained relation shows that the Jacobi matrix $dF= {\d(\ell_i, \ell_2, \ell_3) \over \d(r_1, r_2, r_3)}$ is non-degenerate. Otherwise there would exist a nonzero vector $(v_1, v_2, v_3)$ such that for the deformation constructed above we get $\dot\ell_i=0$. But then $\dot\psi_0
\equiv 0$, which is a contradiction.

Now consider a general situation. Let $\Sigma$ be a real surface with a marked boundary, $C$ a smooth curve with marked points, $\sigma:\Sigma \to C$ a parameterization, and $C \bs \mapo = \cup_j C_j$ a decomposition into pants with a given graph $\Gamma$. Let $\{\gamma_i\}$ be the set of boundary circles and $\{ \gamma_{jk} \}$ the set of circles lying between the pants $C_j$ and $C_k$, respectively. Consider these pants separately. Then for any circle $\gamma
_{jk} = \gamma_{kj}$ we obtain 2 distinguished ones, $\gamma_{j,k}$ considered as a boundary circle of $C_j$, and $\gamma_{k,j}$ considered as a boundary circle of $C_k$. Take real numbers $\bfv\deff (v_i, v_{j,k}, v_{k,j})$, where $v_i$ is associated with the circle $\gamma_i$, $v_{j,k}$ with $\gamma_{j,k}$, and $v_{k,j}$ with $\gamma_{k,j}$, respectively. Let $C_j(t\bfv)$ denote the pants obtained from $C_j$ by the above construction using the corresponding parameters $v_i$ and $v_{j,k}$. For $\bfv$ lying in a small ball $B = \{ |\bfv| < \eps \}$ all such families $C_j(t\bfv)$ can be extended for all $t \in [-1,1]$. Thus over $B$ we obtain a collection of deformation families $C_j(\bfv)$ of complex structure on pants $C_j$.

Let $\ell_i(\bfv)$, $\ell_{j,k}(\bfv)$, and $\ell_{k,j}(\bfv)$ denote the lengths of circles $\gamma_i$, $\gamma_{j,k}$, and $\gamma_{k,j}$ the obtained intrinsic metrics $\mu_j(\bfv)$ on $C_j(\bfv)$. Denote by $\dot\ell_i$ a linear functional $\d_t|_{t=0} \ell_i(t\bfv)$, and define $\dot\ell_{j, k}$ similarly. The explicit formula for an intrinsic metric near a boundary circle shows that $C_j(\bfv)$ can be glued to $C_k(\bfv)$ along $\gamma_{jk}$ exactly when $\ell_{j,k}(\bfv) = \ell_{k,j}(\bfv)$. Since the Jacobian $\d\bfell(\bfv) \over \d \bfv$ is non-degenerate, the conditions $\ell_{j,k}(\bfv) = \ell_{k,j}(\bfv)$ define a submanifold $V \subset B$ whose tangent space $T_0V$ is given by the relations $\dot\ell_{j, k}= \dot\ell
_{k,j}$. Note that this defines a deformation family of complex structures on $C$ over the base $V$ such that the map $\bfv \in V \mapsto \bfell(\bfv)$ is a diffeomorphism.

We state that the set $(v_i, v_{j,k} + v_{k,j})$ is a system of coordinates on $V$ in the neighborhood of $0\in V$. To prove this it is sufficient to show that the linear map $\bfv=(v_i, v_{j,k}, v_{k,j}) \in T_0V \mapsto (v_i,
v_{j,k} + v_{k,j})$ is non-degenerate. If it is not true, then there would exist a nontrivial $\bfv=(v_i, v_{j,k}, v_{k,j})\in T_0V$ with $v_i=0$ and $v_{j,k} + v_{k,j}=0$. Let $\dot\ell_i=\dot\ell_i(\bfv)$, $\dot\ell
_{j,k}= \dot\ell_{j,k}(\bfv)$ and $\dot \ell_{k,j} =\dot \ell_{k,j} (\bfv)$ be the corresponding derivatives of length. Then $\dot\ell_{j,k} = \dot
\ell_{k,j}$ and $$0< \sum_i \msmall{ \ell_i \over 2\pi} v_i \dot\ell_i +
\sum_{j<k} \msmall{ \ell_{jk} \over 2\pi} v_{j,k} \dot\ell_{j,k} +
\sum_{j<k} \msmall{ \ell_{jk} \over 2\pi} v_{k,j} \dot\ell_{k,j}=
\sum_{j<k} \msmall{ \ell_{jk} \over 2\pi} (v_{j,k}+ v_{k,j}) \dot\ell_{j,k}
=0.$$ The obtained contradiction leads us to the following conclusion. The functions $v_i$ and $v_{j,k}+ v_{k,j}$ define a coordinate system on $V$ equivalent to $\bfell=(\ell_i, \ell_{jk})$.

Let us return to the holomorphic deformation family of complex structures on $C$, defined by complex parameters $\lambda_i= e^{-r_i + \isl\phi_i}$ and $\lambda_{jk}= e^{-r_{jk} + \isl\phi_{jk}}$. It is easy to see that the Jacobian $\d(v_i, v_{j,k}+ v_{k,j}) \over \d( r_i , r_{jk})$ at the point $(r_i , r_{jk})=0$ is the identity matrix. This fact proves the statement of the lemma.

Remark. At this point we give a possible reason why the complex (holomorphic) structure introduced by the complex coordinates $\bflambda$ can be regarded as natural. Let $C$ be a complex curve with marked points and a nonempty marked boundary. In the case where $C$ is a disk or an annulus, assume additionally that at least one inner point of $C$ is marked. Then in a neighborhood of every boundary circle $\gamma_i$ of $C$ we can construct the intrinsic coordinate $\zeta_i$. Take 2 copies $C^+$ and $C^-$ of $C$ and denote by $\tau$ the natural holomorphic map $\tau: C^\pm
\to C^\mp$ interchanging the copies. Denote by $\zeta_i^\pm$ the local intrinsic coordinate on $C^\pm$ at boundary circles $\gamma_i^\pm$, both corresponding to $\gamma_i$. Now we can glue $C^+$ and $C^-$ together along every pair of circles $(\gamma_i^+, \gamma_i^-)$ by setting $\zeta_i^+ \cdot
\zeta_i^- =1$ as transition relations. We obtain a closed complex curve $C^d$ which admits a natural holomorphic involution $\tau: C^d \to C^d$. For the constructed family $\{C_\bflambda \}$, the corresponding family $\{C^d
_\bflambda \}$ will be holomorphic. In fact, the statement of [*Theorem 2.3*]{} means that $\{ C^d _\bflambda \}$ is a minimal complete family of deformation of $C^d$ in the class of curves with holomorphic involution. This construction of doubling should not be confused with another construction of the [*Schottky double*]{} $C^{Sch}$ of $C$ which provides an [*antiholomorphic*]{} involution $\tau^{Sch}: C^{Sch} \to C^{Sch}$. We shall use the Schottky double $C^{Sch}$ in [*Section 5*]{}, we will consider curves with totally real boundary conditions.

[4.4. Invariant Description of the Holomorphic Structure on $\ttt_\Gamma$.]{}

The construction of (holomorphic) double $C^d$ shows how to give an invariant description of a holomorphic structure on $\ttt_\Gamma$. Let $C$ be a smooth complex curve with marked points and marked boundary, and $x\in \ttt_\Gamma$ the corresponding point on the moduli space. Denote by $D$ the divisor of marked points. If the curve $C$ is not closed and $C^d$ is its double with holomorphic involution $\tau$, we denote by $D^d \deff D + \tau(D)$ the double of $D$.

Lemma 4.4.1. *If $C$ is closed, then the tangent space $T_x \ttt
_\Gamma$ is naturally isomorphic to $\sfh^1(C, \calo(TC)\otimes \calo(-D) )$.*

If $C$ is not closed, then the space $T_x\ttt_\Gamma$ is naturally isomorphic to the space $\sfh^1(C^d, \calo(TC^d)\otimes \calo(-D^d))^\tau$ of $\tau$-invariant elements in $\sfh^1(C^d, \calo(TC^d)\otimes \calo(-D^d))$.

In both cases the complex structure on $T_x\ttt_\Gamma$ induced by local complex coordinates $\bflambda$ coincides with those from $\sfh^1(C,
\calo(TC) \otimes \calo(-D))$ (resp. $\sfh^1(C^d, \calo(TC^d) \otimes
\calo(-D^d) )^{(\tau)}$). In particular, this defines a global complex structure on space $\ttt_\Gamma$.

Proof. The part concerning closed curves is well-known. In fact, the natural isomorphism $\psi: T_x\ttt_\Gamma \to \sfh^1(C, \calo(TC)\otimes
\calo(-D))$ is a Kodaira-Spencer map. Its description is very simple in the introduced local coordinates $\zeta_{j,k}$ on $C$ and $\bflambda =( \lambda
_{jk})$ on $\ttt_\Gamma$. Let $C\bs \mapo = \cup C_j$ be the decomposition of $C$ into pants with the graph $\Gamma$. For every pants $C_j$ choose an open set $\ti C_j$, containing a closure $\barr C_j= C_j \cup \d C_j$. Without loss of generality we may assume that $\ti C_j$ are chosen not too big, so that the covering $\calu\deff \{ \ti C_j \}$ is acyclic for the sheaf $\calo(TC)$ and the local coordinates $\zeta_{j,k}$ are well-defined in the intersections $\ti C_j \cap \ti C_k$. Then vector $v \in T_x\ttt_\Gamma$ with local representation $v= \sum_{j<k} v_{jk} {\d \over \d \lambda_{jk}}$ is mapped by the Kodaira-Spencer map $\psi$ to the Čech 1-cohomology class $$\psi(v) \in \sfh^1(C, \calo(TC)\otimes \calo(-D)) \cong
\check \sfh^1(\calu, \calo(TC) \otimes \calo(-D)),$$ represented by the 1-cocycle $$\left(v_{jk} \zeta_{j,k} \msmall{\d \over \d \zeta_{j,k}} \right)
\in \prod_{j<k} \Gamma(\ti C_j \cap \ti C_k,  \calo(TC)\otimes \calo(-D)).$$ For more details see \[D-G\].

Using this description of the Kodaira-Spencer map for closed curves with marked points, it is easy to handle the case of curves with boundary. Let $C$ be a non-compact curve with marked points and with decomposition $C \bs \mapo
= \cup_j C_j$. Take its double $C^d$ with the involution $\tau$. Then the decomposition of $C$ induces a $\tau$-invariant decomposition $C^d =\bigcap_j(C_j  \cap \tau C_j)$. The corresponding covering $\calu^d$ of $C^d$ can also be chosen to be $\tau$-invariant.

The local coordinates $\zeta_i$, corresponding to boundary circles $\gamma_i$ of $C$, can now be extended to a two-sided neighborhood of $\gamma_i$ in $C^d$. The coordinates $\zeta_{j,k}$, corresponding to inner circles $\gamma_{jk}$, induce local complex coordinates $\zeta^\tau_{j,k}\deff
\zeta_{j,k} \scirc \tau$ in $\tau( \ti C_j \cap \ti C_k)$.

Any deformation of the complex structure on $C$ induces a deformation of the complex structure on $C^d$. This defines a map $\phi: \ttt_\Gamma \to
\ttt_{\Gamma^d}$, with $\Gamma^d$ denoting the graph corresponding to the $\tau$-invariant decomposition of $C^d$ into pants. Using introduced coordinates, we present a tangent vector $v \in T_x \ttt_\Gamma$ in the form $$v= \sum_i v_i \msmall{\d \over \d \lambda_i} +
\sum_{j<k} v_{jk} \msmall{\d \over \d \lambda_{jk}}.$$ Then the composition of the Kodaira-Spencer map $\psi^d$ of $C^d$ with the differential of $\phi: \ttt_\Gamma \to \ttt_{\Gamma^d}$ maps $v$ to $$\psi^d \scirc d\phi(v) \in \sfh^1(C^d, \calo(TC^d)\otimes \calo(-D^d)) \cong
\check \sfh^1(\calu^d, \calo(TC^d) \otimes \calo(-D^d)),$$ represented by the Čech 1-cocycle $$\check v \deff \left(v_i \zeta_i \msmall{\d \over \d \zeta_i},
v_{jk} \zeta_{j,k} \msmall{\d \over \d \zeta_{j,k}},
v_{jk} \zeta^\tau_{j,k} \msmall{\d \over \d \zeta^\tau_{j,k}} \right)
\in \prod_i \Gamma(\ti C(i) \cap \tau C(i),  \calo(TC^d)\otimes \calo(-D^d))
\times$$ $$\prod_{j<k} \Gamma(\ti C_j \cap \ti C_k,  \calo(TC^d)\otimes \calo(-D^d))
\times \prod_{j<k} \Gamma(\tau(\ti C_j \cap \ti C_k),
\calo(TC^d)\otimes \calo(-D^d)),$$ where $C(i)$ denotes the pants of $C$ adjacent to circle $\gamma_i$. It is obvious that if all $v_i$ vanish, then this Čech 1-cocycle is $\tau
$-invariant. On the other hand, the relation $\zeta_i \cdot (\zeta_i \scirc
\tau) \equiv \lambda_i= \const$ implies that $\tau_*(\zeta_i {\d\over \d
\zeta_i}) = -\zeta_i {\d \over \d \zeta_i}$. The additional change of the sign of the corresponding part of cocycle $\check v$ comes from the fact that $\tau$ interchange $C(i)$ with $\tau(C(i))$. This shows that $\check v$ is $\tau
$-invariant and the statement of the lemma follows.

Now we study the connection between the geometry of $\ttt_\Gamma$ and the degeneration of complex structures on a real surface $\Sigma$ with marked points and marked boundary. Let $\Sigma \bs \mapo= \cup_j S_j$ be a decomposition into pants with graph $\Gamma$. The Fenchel-Nielsen coordinates on $\ttt_\Gamma$ define a map $(\bflambda, \bfvartheta): \ttt_\Gamma \to (\rr
\times S^1)^{3g-3+ m+2b}$, which is a diffeomorphism by [*Proposition 1.2*]{}. So, if $\{ j_n\}$ is a sequence of complex structures on $\Sigma$, its degeneration means that the sequence of Fenchel-Nielsen coordinates of $\{
j_n\}$ is not bounded in $(\rr \times S^1)^{3g-3+ m+2b}$.

One can see that, in fact, we have two types of the degeneration. The first one occurs when the maximum of the length coordinates $\ell_i$ and $\ell_{jk}$ of $j_n$ increases infinitely, and the second one is present when a minimum of the length coordinates of $j_n$ vanishes. It should be pointed out that for an appropriate sequence one can have both types of degeneration.

Note that by [*Proposition 1.2*]{} the Fenchel-Nielsen coordinates of a complex structure $j$ on $\Sigma$ are defined by a choice of a topological type of decomposition of $\Sigma$ into pants, encoded in the graph $\Gamma$. Thus the introduced notion of degeneration also depends on the choice of $\Gamma$. Possibly, the best choice of such decomposition is established by the following statement, proved in \[Ab\], Ch.II, §3.3.

Proposition 4.4.2. *Let $C$ be a complex curve with parameterization $\sigma: \Sigma \to C$. Then*

[*a)*]{} there exists a universal constant $l^*>0$ such that any two geodesic circles $\gamma'$ and $\gamma''$ on $C$ satisfying $\ell( \gamma') < l^*$ and $\ell(\gamma'') < l^*$ are either disjoint or they coincide;

[*b)*]{} there exists pants decomposition $C \bs \mapo= \cup_j S_j$ such that the lengths of inner boundary circles $\gamma_{jk} = \barr S_j \cap \barr
S_k$ are bounded from above by a constant $L$ which depends only on the topology of $\Sigma$ and the maximum $M$ of the lengths of the boundary circles of $C$; moreover, any simple geodesic circle $\gamma$ on $C$ with $\ell(\gamma) <l^*$ occurs as a boundary circle of some $C_j$.

Corollary 4.4.3. *Let $C_n$ be a sequence of nodal curves parameterized by a real surface $\Sigma$ with uniformly bounded number of components. Suppose that the complex structures of $C_n$ do not degenerate near the boundary. Then, passing to a subsequence, one can find a decomposition $\Sigma =\cup_j
S_j$ and new parameterizations $\sigma'_n : \Sigma \to C_n$, such that*

the decomposition $\Sigma =\cup_j S_j$ induces a decomposition of every non-exceptional component of $C_n$ into pants whose boundary circles are geodesics;

the intrinsic length of these geodesics are bounded uniformly in $n$.

Proof. By [*Lemma 4.3.2*]{}, the intrinsic lengths of boundary circles of non-exceptional components of $C_n$ are bounded uniformly in $n$. Find a decomposition into pants of every non-exceptional component of $C_n$ satisfying the conditions of part [*b)*]{} of [*Proposition 4.4.2*]{}. Let $\Gamma_n$ denote the obtained graph of the decomposition of $C_n$. Since the number of components of $C_n$ is uniformly bounded, we obtain a subsequence $C_{n_k}$ with the same graph $\Gamma$ for all $n_k$.

It follows from the proof in \[Ab\] that the constant $L$ from part [*b)*]{} of [*Proposition 4.4.2*]{} depends continuously on the maximum $M$ of the lengths of the boundary circles of $C_n$. This implies condition [*i$\!$i)*]{}. Applying [*Proposition 4.2.2*]{}, we complete the proof.

[4.5. Example: Degeneration to a Half-cubic Parabola on the Language of Stable Curves.]{}

[*Example.*]{} Consider a sequence of curves $T_n$ in $\cc\pp^2$ given by $$T_n= \{\; y^2z-x^3=n^{-6}z^3\;\} \subset \cc\pp^2$$ in homogeneous coordinates. In the affine chart $U_0=\{ z\not= 0\}$ the curves are given by the equation $T_n=\{ y^2-x^3=n^{-6}\} $. In any reasonable sense these curves should converge to a half-cubic parabola $T_\infty:= \{ y^2=x^3\} $. We shall now explain in great detail how the $T_n$ converge to $T_\infty$ in Gromov topology.

Each curve $T_n$ is the image of the curve $$T =  \{ y_1^2z_1-x_1^3=z_1^3\}\subset \cc\pp^2$$ w.r.t. an algebraic map $f_n: [x_1, y_1,z_1] \in C
\mapsto [x:y:z]:= [n^{-2} x_1: n^{-3} y_1: z_1]$.

We shall denote by $[x_1:y_1:z_1]$ the coordinates in the pre-image and by $[x:y:z]$ in the range.

[*Convergence of graphs.*]{} Consider the graphs $\Gamma_n$ of these mappings as subsets of $\cc\pp^2\times \cc\pp^2$. Then the $\Gamma_n$ converge to a [*reducible*]{} curve $\Gamma_\infty\in
\cc\pp^2\times \cc\pp^2$, which consists of the graph of a [*constant*]{} mapping $[x_1: y_1:z_1] \in T \mapsto [0:0:1]$, which is a “horizontal component”, plus a “vertical component” over $[x_1:y_1:z_1] =[0:1:0]$, which is a limit curve $\{y^2 = x^3\}$ of the sequence $T_n$.

The picture of the convergence is the following. Starting from $\Gamma_1$, the curves $\Gamma_n$ transform continuously (we can define $\Gamma_t$ for any real $t$), so that for $n>\!>1$ curve $\Gamma_n$ consists of an “almost horizontal” torus and an “almost vertical” sphere connected by a thin “neck”. As $n\longrightarrow \infty$ this neck is shrinking to a point $P:=([0:1:0], [0:0:1]) \in \pp^2 \times
\pp^2$ which is the only singular point of $\Gamma_\infty$ where its components intersect.

[*Explanation of the picture:*]{} $i_1,i_2,i_\infty$ are imbeddings of $C_n$’s into $\pp^2 \times \pp^2$ as the graphs $\Gamma_n$’s. The map $\pr:\pp^2 \times \pp^2 \to \pp^2$ is the natural projection onto the range $\pp^2$.

Thus, the picture of convergency on the language of parameterized curves should be the following: the surface parameterizing all $T_n$ (and also $T_\infty
=\{ y^2=x^3\} $) should be a torus $\Sigma $. Some fixed circle $\gamma $ on $\Sigma $ around some fixed point $p\in \Sigma $ should be mapped by parameterizations $\sigma_n :\Sigma \to C_n$ to “smaller circles". Finally the parameterization $\sigma_\infty :\Sigma \to C_\infty$ contracts $\gamma $ to a point. Thus $C_\infty$ should be a nodal curve which consists of a smooth torus and a smooth sphere intersecting at one point. The limit map $u_\infty$ should map the torus to the point $[0:0:1]$ and the sphere onto the limit curve $T_\infty$.

Now we must determine curves $C_n$, parameterizations $\sigma_n$ and mappings $u_n$ explicitly. Note that one cannot take as $C_n$ simply the graph $\Gamma_n$, because $\Gamma_\infty$ has a point $P=([0:1:0], [0:0:1]) \in \pp^2 \times \pp^2$ as a cusp and a node at the same time. However, by our definition of a nodal curve, $C_\infty$ should have at most nodes. This is not a big problem, because one should “holomorphically” parameterize $\Gamma_\infty$ by a nodal curve and at the same time one also needs to parameterize “coherently" $C_n$’s.

Let us do just that.

[*Explicit Parameterization.*]{}

Now we give a description of the convergence in terms of stable curves (terminology of \[KM\]; an alternative terminology speaks of “stable maps”). Consider homogeneous equations $$y_1^2z_1 -x_1^3=z_1^3
\hbox{\qquad and \qquad}
{x\over n^{-2}x_1}= {y \over n^{-3}y_1}= {z\over z_1},
\eqno(4.5.1)$$ defining $\Gamma_n$ in $\pp^2 \times \pp^2$. Let $Q=[x_1:y_1:z_1]$ be a point on $T$ with $z_1\not =0$. Then obviously $f_n(Q)= [n^{-2} x_1: n^{-3} y_1: 
z_1]$ converge to the point $[0:0:1]$ in the range $\pp^2$. Moreover, for a sufficiently small neighborhood $U$ of $Q$ in $T$ we obviously have a uniform convergence of the restriction $f_n|_U$ to a constant map $f_\infty:
 U \to \pp^2$, $f_\infty|_U  \equiv[0:0:1]$.

The set of points $[x_1:y_1:z_1]$ on $C$ with $z_1=0$ consists of one point $[0:1:0]$. This means that for any compact $K \Subset C \bs [0:1:0]$ the sequence of restricted maps $f_n|_K : \to \pp^2$ converge uniformly to the constant map $f_\infty: K \to \pp^2$, $u_\infty|_K \equiv[0:0:1]$. In terms of graphs $\Gamma_n$ this means that $\Gamma_n \cap(K\times \pp^2)$ converge to $\Gamma_\infty \cap(K \times \pp^2)$, which lies in the “horizontal” part $T \times [0:0:1]$ of $\Gamma_\infty$.

Consider the behavior of $f_n$ in a neighborhood $V\subset T$ of the point $[0:1:0]$. Setting $y_1=1$ and considering $x_1, z_1$ as an affine coordinate on the first $\pp^2$ we obtain the relation $$z_1= x_1^3 + z_1^3.$$ One can easily see that $x_1$ can be chosen as a local holomorphic coordinate in a neighborhood of $[0:1:0]$, and that there exists a holomorphic function $\phi(x_1)$ in the disk $\Delta(a)=\{ |x_1| <a\}$ such that $\phi(0)=1$ and $V=\{ z_1=x_1^3\phi(x_1)\} $.

Fix $a>{1\over b}>0$ and take $R>{1\over b}$. Denote by $\pr:V\to \Delta (a)$ the natural projection, see [*Fig. 7*]{}. For every $n$ sufficiently big, cover $\pr(V)=\Delta(a)$ by sets $V_n$, where $V_n:=\pr^{-1}\Delta({R\over n})$ and $V_n':=\pr^{-1}(A_n)$ with $A_n = \{ {1\over bn} <|x_1| <a\} $. First we consider the behavior of appropriately rescaled maps $f_n\circ \pr^{-1}$ in $V_n$. For a complex coordinate $\xi \in \cc$ we consider maps $v_n(\xi): \Delta(R) \to \pp^2$ with $v_n(\xi):=
f_n\circ \pr^{-1}({\xi\over n})$ and the domain of definition $\Delta(R):=\{|\xi|<R\}$. This means that we rescale $f_n\circ \pr^{-1}$ in $\Delta({R\over n})=$ by setting $nx_1 =\xi$. Then $$\textstyle
v_n(\xi)=f_n\circ \pr^{-1}\left(\left[{\xi\over n}:1: {\xi^3\over n^3}
\phi\left({\xi\over n}\right) \right]\right)=
\left[{\xi\over n^3}:{1\over n^3}: {\xi^3\over n^3}
\phi\left({\xi\over n}\right) \right]=
\left[\xi :1 :\xi^3\phi\left({\xi\over n}\right) \right].$$ Thus, images $v_n(\Delta(R))=f_n(\Delta ({R\over n}))$ lie in the affine chart $y\not=0$ in the range $\pp^2$, and in the affine coordinates $x,z$ ($y=1$) we obtain a representation $v_n(\xi)= (\xi, \xi^3 \phi({\xi\over n}))$. We easily see that $\phi({\xi\over n})$ converge on compacts $\Delta(R)$ to the constant function 1. This implies that maps $v_n$ converge on $\Delta(R)$ to the map $v_\infty: \cc \to \pp^2$, $v_\infty(\xi)= [\xi:1:\xi^3]$. This map is a parameterization of the “vertical part” of $\Gamma_\infty$.

Finally, we want to describe the behavior of the “neck” of $\Gamma_n$ when $n\lrar \infty$. Again, take $x_1$ to be a local holomorphic coordinate on $T$ in a neighborhood of $[0:1:0]$ setting $y_1=1$. Consider imbeddings $h_n: A_n \to \Delta^2(a,b)= \{ |x_1| < a\} \times \{|t|<b\}$ with $h_n(x_1)
=(x_1, {1\over n x_1})$. Denote by $\cala_n$ the images $h_n(A_n)$. Then $\cala_n = \{ (x_1,t) \in \Delta^2(a,b):
x_1{\cdot}t= {1\over n}\}$. As $n\lrar \infty$, the annuli $\cala_n$ converge to a subset $\cala_\infty \subset \Delta^2(a,b)$, which is the union of two disks $\Delta(a) \times \{0\}\cup \{0\} \times
\Delta(b)$.

Note that, along $\cala_n \subset \Delta^2(a,b)$ with coordinates $x_1,t$, we can represent $f_n\circ \pr^{-1}\circ h_n^{-1}$ in the form $$f_n\circ \pr^{-1}\circ h_n^{-1}:(x_1, t) \in A_n \mapsto \left[{x_1 \over n^2}
: {1\over n^3}: x_1^3
\phi(x_1) \right] =[ nx_1 t^3: t^3: n^3x_1^3 t^3\phi(x_1)]=$$ $$=[t^2:t^3: \phi(x_1)],$$ where we use the relation $nx_1 t=1$ along $\cala_n$. Thus the restriction of $f_n\circ \pr^{-1}$ onto $A_n$ coincides with the composition of imbeddings $h_n$ of $A_n$ into $\Delta^2(a,b)$ with the map $F: \Delta^2(a,b) \to \pp^2$, $F(x_1,t) 
= [t^2:t^3: \phi(x_1)]$. In the affine chart with $z\not=0$ we have $F(x_1,t) =
\left({t^2 \over \phi(x_1)}, {t^3 \over\phi(x_1)}\right)$, see [*Fig. 8*]{}.

Note that $$F\mid_{\cala_n} = f_n\circ \pr^{-1}\circ h_n^{-1}:\cala_n\to \cc\pp^2.
\eqno(4.5.2)$$

Remark. Denote by $P_n$ the compact curve $\{ x_1\cdot t= {1\over n}
\}$ in $\cc\pp^2$. For us it is important that $F$ is well- defined on $P_n\setminus \{ \vert x_1\vert >a\} $.

We interpret this picture in the following way. Annuli $\cala_n$ degenerate to a normal crossing of two disks, a “node” in the terminology of \[OM\], whereas maps $f_n\circ \pr^{-1}\circ h_n^{-1}:\cala_n \to \pp^2$ converge to a (holomorphic) map from $\cala_\infty$ to $\pp^2$ which is, in this case, a restriction of $F$ onto $\cala_\infty$, i.e., constantly $[0:0:1]$ on the one component $\{ t = 0\} $ of $A_\infty$, and, on the component $\{ x_1 = 0\} $, mapping $F$ is a parameterization $t\in \Delta(b) \mapsto (t^2, t^3)$ of curve $y^2=x^3$.

Note that on $A_n$ the contractiong circle is $\gamma_n = \{ \vert x_1\vert
 = \sqrt{{1\over n}}$, because $h_n(\gamma_n) = \{ \vert x_1\vert = \vert t
\vert = \sqrt{{1\over n}}$.

Let us finally construct stable curves $(C_n,u_n)$ and $(C_\infty,
u_\infty)$. Put $W_1 = T_1\setminus V$ and $W_2 = P_n\setminus \{ \vert
x_1 \vert > a \} $. Identify the boundaries of $W_1$ and $W_2$ by $h_n\circ \pr : \partial W_1 \to \partial W_2 $ to get the curve $C_n :=
W_1\sqcup W_2/\partial \sim \partial W_2$. Note that $C_\infty$ is a nodal curve with one nodal point and two components: torus and sphere.

The corresponding maps are defined as follows: $$u(q) = \cases f_n (q) & \text{ if }q\in W_1 \\
F(q) & \text{ if }q\in W_2 .
\endcases$$ Note that $u_n$ is well- defined due to relation (2).

The rest is obvious, i.e., one should take the parameterizations $\sigma_n:\Sigma
\to C_n$ satisfying the three following conditions:

\(a) Some fixed circle $\gamma $ should be mapped by $\sigma_n$ onto $h_n(
\gamma_n)\subset \cala_n$ (also $\sigma_\infty(\gamma)= 0\in \cala_\infty$).

\(b) Some fixed disk $\tilde V_2\subset \Sigma $, see [*Fig. 10*]{}, should be mapped by $\sigma_n$ onto the domain $P_n\cap \{ \vert t\vert >1\} \subset
W_2$ (also $\sigma_\infty(\tilde V_2 = P_\infty\cap
\{ \vert t\vert >1\} $.

\(c) Finally, some other domain $\tilde V_1\subset \Sigma$, which contains a handle, should be mapped onto $W_1$.

One should also take care that the structures $\sigma_n^*j_n$ should converge on compact subsets in $\Sigma \setminus \gamma$.

[Lecture 5]{}

[Gromov Compactness Theorem]{}

[5.1. Second A priori Estimate.]{}

Let $(X,J)$ be an almost complex manifold. In what follows the tensor $J$ is supposed to be only continuous, i.e., of class $C^0$. Fix some Riemannian metric $h$ on $X$. All norms and distances will be taken with respect to $h$. In particular, we have the following

Definition 5.1.1. [*A continuous almost complex structure $J$ is called [*uniformly continuous on $A \subset X$ with respect to $h$*]{}, if $\norm{J}_{L^\infty(A)} < \infty$ and for any $\epsi >0$ there exists $\delta=\delta(J, A, h) >0$ such that for any $x\in A$ one can find a $C^1$-diffeomorphism $\phi: B(x, \delta) \to B(0, \delta)$ from the ball $B(x, \delta)\deff \{y\in X \;:\; \dist_h(x,y) <\delta \}$ onto the standard ball in $\cc^n$ with the standard metric $h\st$ such that $$\norm{J -\phi^*J\st}_{L^\infty(B(x,\delta)\cap A)} +
\norm{h -\phi^*h\st}_{L^\infty(B
(x,\delta)\cap A)}
\le\epsi.
\eqno(5.1.1)$$* ]{}

Roughly speaking, this means that on the set $A$ we can $\norm\cdot _{L^\infty}
$-approximate $J$ by an integrable structure in $h$-metric balls of a radius independent of $x\in A$. The function $\mu(J,A,h)$ whose value at $\epsi>0$ is the biggest possible $\delta\le1$ with the above property is called the [*modulus of uniform continuity of $J$ on $A$*]{}. Note that every continuous almost complex structure $J$ is always uniformly continuous on [*relatively compact*]{} subsets $K \Subset X$.

Let $J^*$ be a continuous almost complex structure on $X$ and $A\subset X$ a subset. Assume that $J^*$ is uniformly continuous on $A$ and denote by $\mu_{J^*}= \mu( J^*,A, h)$ the modulus of uniform continuity of $J^*$ on $A$. Recall that in [*Lemma 2.4.1*]{} we proved the following

Proposition 5.1.1. [*(First A priori Estimate). *For every $p$ with $2< p<\infty$ there exists an $\eps_1 =\eps_1(\mu_{J^*}, A, h)$ (independent of $p$) and $C_p=C(p,\mu_{J^*}, A, h)$ such that for any continuous almost complex structure $J$ with $\norm{ J- J^*} _{L^\infty(A)}<\eps_1 $ and for every $J$-holomorphic map $u\in C^0\cap L^{1,2}(\Delta ,X)$, satisfying $u(\Delta )\subset A$ and $\norm{du} _{L^2( \Delta )} <\eps_1 $, one has the estimate $$\norm{du}_{L^p({1\over 2}\Delta )}\le C_p\cdot
\norm{du}_{L^2(\Delta )}.
\eqno(5.1.2)$$** ]{}

Definition 5.1.2. [*Define a [*cylinder*]{} $Z(a,b)$ by $Z (a,b) \deff S^1 \times [a,b]$, equipping it with coordinates $\theta \in [0,2\pi]$, $t\in[a,b]$, with the metric $ds^2= d\theta^2 + dt^2$ and the complex structure $J\st({\d \over \d\theta}) ={\d \over \d t}$. Denote $Z_i\deff Z(i-1,1)= S^1 \times [i-1,i]$.*]{}

Let $J^*$ be some continuous almost complex structure on $X$ and $A$ a subset of $X$ such that $J^*$ is uniformly continuous on $A$. Let $\mu_{J^*}$ denote the modulus of uniform continuity of $J^*$ on $A$.

Lemma 5.1.2. [*(Second A priori Estimate). *There exist constants $\gamma \in \,]\,0,1\,[$ and $\eps_2=\eps_2(\mu_{J^*}, A,h)>0$ such that for any $J$ with $\norm{ J- J^*} <\eps_2$ and every $J$-holomorphic map $u:Z(0,5)
\to X$ with $u(Z(0,5)) \subset A$ the condition $\norm{du}_{ L^2(Z_i)}<\eps_2$ for $i=1,\ldots,5$ implies $$\norm{du}^2_{L^2(Z_3)}\le {\gamma \over 2}\bigl(\norm{du}^2_{L^2(Z_2)} +
\norm{du}^2_{L^2(Z_4)}\bigr).
\eqno(5.1.3)$$** ]{}

Proof. Take $\eps_2>0$ small enough such that $\mu_{J^*}(\epsi_2) <
\epsi_1$, where $\eps_1$ is the constant from [*Lemma 5.1.1*]{}. Then for any $A' \subset A$ the condition $\diam(A') \le \epsi_2$ implies that $\osc(J^*,
A') \le \eps_1$. Due to [*Lemma 5.1.1*]{}, we may assume that $u(Z_i)\subset 
B$ for $i=2,3,4$, where $B$ is a small ball in $\rr^{2n} =\cc^n$ with the structure $J\st$. Moreover, we may assume that $\norm{ J^* - J\st}_{
L^\infty(B)} \le \eps_1$.

Find $v\in C^0\cap L^{1,2}(Z(1,4),\cc^n)$ such that $\dbar_{J\st}v=0$ and $\norm{du - dv}_{L^2(Z(1,4))}$ is minimal. We have $$\norm{  \dbar_{J\st}(u-v)}_{L^2(Z_i)} = \norm{(J\st-J(u))\d_y u
}_{L^2(Z_i)}
\le \norm{J\st - J}_{L^\infty(B)}\norm{du}_{L^2(Z_i)}.$$ So for $i=2,3,4$ we get $$\norm{du - dv}_{L^2(Z_i)}\le C\norm{J\st -J}_{L^\infty(B)}
\norm{ du
}_{L^2(Z(1,4))}.\eqno(5.1.4)$$ Now let us check the inequality (4.7) for $v$. Write $v(z)= \Sigma_{k=
-\infty} ^\infty v_k e^{k(t+i\theta)}$. Then $\norm{dv}^2_{L^2 (S\times \{
t\} )}=4\pi \Sigma_{k=-\infty}^\infty k^2| v_k|^2 e^{2kt}$. Since obviously $$\int_2^3e^{2kt}\le {\gamma_1\over 2}
\left( \int_1^2e^{2kt}dt+\int_3^4e^{2kt}dt \right)
%\eqno(3.8A)$$ for all $k \not=0$ with $\gamma_1={2\over e^2}$, one gets the required estimate for all holomorphic $v$.

Using (5.1.4) with $\norm{J\st - J}_{L^\infty}$ sufficiently small, we conclude that the estimate (5.1.3) holds for $u$ with appropriate $\gamma > \gamma_1$.

Corollary 5.1.3.

*Let $X$, $h$, $J^*$, $A$, and the constants $\eps_2$ and $\gamma$ be as in [*Lemma 5.1.2*]{}. Suppose that $J$ is a continuous almost complex structure on $X$ with $\norm{J-J^*}_{L^\infty(A)}
<\eps_2$ and $u \in C^0\cap L^{1,2}(Z(0,l),X)$ a $J$-holomorphic map such that $u(Z)\subset A$ and $\norm{du}_{ L^2(Z_i)}<\eps_2$ for any $i=1,\ldots
,l$. Let $\lambda>1$ be (the uniquely defined) real number with $\lambda = {\gamma \over 2} (\lambda^2+ 1)$.*

Then for $2\le k\le l-1$ one has $$\norm{du}^2_{L^2(Z_k)} \le \lambda^{-(k-2)} \cdot \norm{du}^2_{L^2(Z_2)}
+ \lambda^{-(l-1-k)} \cdot \norm{du}^2_{L^2(Z_{l-1})}.
\eqno(5.1.5)$$

Proof. The definition of $\lambda$ implies that for any $a_+$ and $a_-$ the sequence $y_k \deff a_+ \lambda^k + a_- \lambda^{-k}$ satisfies the recurrent relation $y_k = {\gamma\over2}(y_{k-1} + y_{k+1})$. In particular, so does the sequence $$A_k \deff
\msmall{ \lambda^{-(k-2)} - \lambda^{6-2l+ k-2}
\over 1- \lambda^{6-2l} } \norm{du}^2_{L^2(Z_2)}
+ \msmall{ \lambda^{-(l-1-k)} - \lambda^{6-2l+ l-1-k}
\over 1- \lambda^{6-2l} } \norm{du}^2_{L^2(Z_{l-1})},$$ which is determined by the values $A_2 = \norm{du}^2_{L^2(Z_2)}$ and $A_{l-1}
= \norm{du}^2_{L^2(Z_{l-1})}$.

We claim that for $2\le k\le l-1$ one has the estimate $\norm{du}^2 _{L^2(
Z_k)} \le A_k$, which is obviously stronger than (5.1.5). Suppose that there exists a $k_0$ such that $2\le k_0\le l-1$ and $\norm{du}^2 _{L^2(Z_{k_0})}>A_{k_0}$. Choose $k_0$ so that the difference $\norm{du }^2_{L^2(Z_{k_0})}-A_{k_0}$ is maximal. By [*Corollary 2.5.1*]{} and by our recurrent definition of $A_k$, we have $2< k_0 < l-1$ and $$\norm{du }^2_{L^2(Z_{k_0})}-A_{k_0}\le {\gamma \over 2}(
\norm{du }^2_{L^2(Z_{k_0+1})}-A_{k_0+1}+
\norm{du }^2_{L^2(Z_{k_0-1})}-A_{k_0-1})\le$$ $$\le {\gamma \over 2}2(\norm{du }^2_{L^2(Z_{k_0})}-A_{k_0}).$$ The second inequality follows from the fact that $\norm{du}^2_{L^2(
Z_{k_0})} - A_{k_0}$ is maximal. This gives a contradiction.

[5.2. Removal of Point Singularities.]{}

An immediate corollary of this estimate is the following improvement of the Sacks-Uhlenbeck theorem about removability of a point singularity, see \[S-U\] and \[G\].

Corollary 5.2.1.

*(Removal of Point Singularities). *Let $X$ be a manifold with a Riemannian metric $h$, $J$ a continuous almost complex structure, and $u:(\check\Delta, J\st)\to (X,J)$ a $J$ - holomorphic map from the punctured disk. Suppose that**

$J$ is uniformly continuous on $A \deff u(\check\Delta)$ $h$ and the closure of $A$ is $h$-complete;

there exists $i_0$ such that, for all annuli $R_i\deff\{ z\in \cc
:{1\over e^{i+1}}\le | z| \le {1\over e^i}\} $ with $i\ge i_0$, one has $\norm{du}^2_{L^2(R_i)}\le \eps_2$, where $\eps_2$ is defined in [*Lemma 5.1.2*]{}.

Then $u$ extends to the origin.

Condition is automatically satisfied if $A= u(\check\Delta)$ is relatively compact in $X$. Condition of “slow growth” is clearly weaker than just the boundedness of the area, see, e.g., \[S-U\], \[G\]. It is sufficient to have $\lim_{i\lrar\infty} \norm{du}^2_{L^2(R_i)} =0$, whereas boundedness of the area means $\sum_{i=1}^\infty \norm{du}^2_{L^2(R_i)} <\infty$.

Proof. The exponential map $\exp(t,\theta) \deff e^{-t+i\theta}$ defines a biholomorphism between the infinite cylinder $Z(0,\infty)$ and the punctured disk $\check\Delta$, identifying every annulus $R_i$ with the cylinder $Z(i,i+1)$. Applying [*Corollary 5.1.3*]{} to the map $u \scirc \exp$ on cylinders $Z(i_0,l)$ and setting $l\lrar \infty$, we obtain the estimate $$\norm{du}^2_{L^2(R_i)} \le \lambda^{-(i-i_0)} \cdot
\norm{du}^2_{L^2(R_{i_0})}, \qquad i>i_0.$$ Using this and [*Lemma 5.1.1*]{} we conclude that $\diam(u(R_i)) \le C\cdot
\lambda^{-i/2}$ for $i>i_0$. Since $\sum \lambda^{-i/2} <\infty$, $u$ extends continuously into $0\in \Delta$.

In the proof of the compactness theorem we shall use the following corollary of [*Corollary 2.5.1*]{}. Let $X$ be a manifold with a Riemannian metric $h$, $J$ a continuous almost complex structure on $X$, $A\subset X$ a closed $h$-complete subset such that $J$ is $h$-uniformly continuous on $A$. Furthermore, let $\{J_n\}$ be a sequence of almost complex structures uniformly converging to $J$, $\{l_n\}$ a sequence of integers with $l_n\to \infty$ and $u_n:Z(0,l_n)
\to X$ a sequence of $J_n$-holomorphic maps.

Lemma 5.2.2.

*Suppose that $u_n(Z(0,l_n))\subset A$ and $\norm{du_n}_{L^2 (Z_i)}\le 
\eps_2$ for all $n$ and $i\le l_n$. Take a sequence $k_n\to \infty$ such that $k_n<l_n-k_n\to \infty$. Then*

[*1)*]{} $\norm{du_n}_{L^2(Z(k_n,l_n-k_n))}\to 0$ and $\diam\bigl(u_n
(Z(k_n,l_n-k_n))\bigr)\to 0$;

[*2)*]{} if, in addition, all images $u_n(Z(0,l_n))$ are contained in some bounded subset of $X$, then there is a subsequence $\{ u_n \}$, still denoted $\{u_n \}$ such that both $u_n |_{Z(0,k_n)}$ and $u_n |_{Z(k_n,l_n)}$ converge in $L^{1,p}$-topology on compact subsets in $ \check \Delta \cong Z(0, +\infty )$ to $J$-holomorphic maps $u^+_\infty: \check \Delta \to X$ and $u^-_\infty:
\check \Delta \to X$. Moreover, both $u^+_\infty$ and $u^-_\infty$ extend to the origin and $u^+_\infty(0)= u^-_\infty(0)$.

Remarks. 1. The punctured disk $\check\Delta$ with the standard structure $J_{\Delta } {\d \over \d r}={1\over r}{\d \over \d \theta }$ is isomorphic to $Z(0,\infty )$ with the structure $J_Z{\d \over \d t}=-{\d \over
\d \theta }$ under a biholomorphism $(\theta ,t)\mapsto e^{-t+ \isl\theta}$. Thus, statement (2) of this corollary is meaningful.

2\. [*Lemma 5.2.2*]{} describes explicitly how the sequence of $J_n$-holomorphic maps of the cylinders of growing conformal radii converges to a $J$-holomorphic map of the standard node.

Lemma 5.2.3. [*There is an $\eps_3 =\eps_3(\mu_{J_\infty},A,h)$ such that for any continuous almost-complex structure $J$ on $X$ with $\norm{J 
- J_\infty}_{L^\infty}\le \eps_3 $ and any non-constant $J$-complex sphere $u: \cc\pp^1 \to X$, $u(\cc\pp^1)\subset A$ one has the inequalities $$\area(u(\cc\pp^1 ))\ge \eps_3
\qquad\text{and}\qquad
\diam(u(\cc\pp^1))\ge \eps_3.$$* ]{}

Proof. Let $\eps_1$ be the constant from [*Lemma 5.1.1*]{}. Suppose that $\area u(\cc\pp^1)=\norm{du}^2_{L^2(\cc\pp^2)}\le \eps_1^2$. Cover $\cc\pp^1$ by two disks $\Delta_1$ and $\Delta_2$. By (3.1) and the Sobolev imbedding $L^{1,p} \subset C^{0,1-{2\over p}}$, we obtain that $\diam(u(\Delta_1))$ and $\diam(u (\Delta_2) )$ are smaller than $const\cdot \eps_1$. Thus the diameter of the image of the sphere is smaller than $const\cdot \eps_1$.

Therefore, we can suppose that the image $u(S^2)$ is contained in the coordinate chart, i.e., in a subdomain in $\cc^n$, and the structures $J$ and $J_\infty$ are $L^\infty$-close to a standard one. Consider now $u:S^2\to U\subset \cc^n$ as a solution of the linear equation $$\d_xv(z) + J(u(z))\cdot\d_yv(z) = 0\eqno(5.2.1)$$ on the sphere. The operator $\dbar _J(v) = \d_xv(z) + J(u(z))\cdot
\d_yv(z)$ acts from $L^{1,p}(S^2, \cc^n)$ to $L^p(S^2, \cc^n)$ and is a small perturbation of the standard $\dbar$-operator. Note that the standard $\dbar$ is surjective and Fredholm. Thus small perturbations are also surjective and Fredholm, having the kernel of the same dimension. But the kernel of $\dbar$ consists of constant functions. Since all constants are in the kernel of (5.2.1), our $u$ should be a constant map.

We have proved that if the area or a diameter of a $J$-holomorphic map is sufficiently small then this map is constant.

Remark. The same statement is true for the curves of arbitrary genus $g$. In that case, in addition to the estimate (5.1.2), one should also use the estimate (5.1.3). This yields the existence of an $\eps $ which depends on $g$ (and, of course, on $X$, $J$, and $K$), but not on the complex structure on the parameterizing surface.

[5.3. Compactness for Curves with Free Boundary.]{}

In this section we give a proof of the Gromov compactness theorem for the curves with boundaries of fixed finite topological type and without boundary conditions on maps. The case of closed curves is obviously included in this.

Throughout this section we assume that the following setting holds.

*Let $X$ be a manifold with a Riemannian metric $h$, $J_\infty$ a continuous almost complex structure on $X$, $A\subset X$ an $h$-complete subset, $\{ C_n
\}$ a sequence of nodal curves parameterized by a real surface $\Sigma$ with parameterizations $\delta_n:\Sigma \to C_n$, and $u_n:(C_n,j_n) \to (X, J_n)$ a sequence of holomorphic maps. Further, $J_\infty$ is $h$-uniformly continuous on $A$, $J_n$ are also continuous and converge to $J_\infty$, $h$-uniformly on $A$, $u_n(C_n)\subset A$ for all $n$.*

Let us explain the main idea of the proof of [*Theorem 1.1*]{}. The Gromov topology on the space of stable curves over $X$ is introduced in order to recover natural convergence of sequences $(C_n,u_n)$ of complex curves of bounded area which do not converge in the “strong" (i.e., $L^{1,p}$-type) sense. The are two reasons for this. The first is that a sequence of (say, smooth) curves $C_n$ could diverge in an appropriate moduli space and the second is a phenomenon of “bubbling". In both cases one has to deal with the appearance of new nodes, i.e., with a certain degeneration of the complex structure on curves. The “model” situation of [*Lemma 5.2.2*]{} describes a convergence of “long cylinders” $u_n :Z(0, l_n) \to X$, $l_n \lrar \infty$, to a node $u_\infty: \cala_0 \to
X$. In our proof we cover curves $C_n$ by pieces which are either “long cylinders” converging to nodes or have the property that complex structures and maps “strongly” converge. Here the “strong” convergence means the usual one, i.e., the $C^\infty$-topology for complex structures, and the $L^{1,p}$-topology with some $p>2$ for maps. In fact, the strong convergence of maps is equivalent to the uniform one, i.e., the $C^0$-topology, and implies further regularity in the case when $J_n$ and $J_\infty$ have more smoothness. One consequence of this is that we remain in the category of nodal curves. Another is that we treat degeneration of a complex structure on $C_n$ and the “bubbling” phenomenon in a uniform framework of “long cylinders”.

For the proof we need some additional results.

Lemma 5.3.1. *For any $R>1$ there exists an $a^+= a^+(R)>0$ with the following property. For any cylinder $Z = Z(0,l)$ with $0 <l\le +\infty$ and any annulus $A \subset Z(0,l)$, which is adjacent to $\d_0 Z = S^1\times
\{0\}$ and has a conformal radius $R$, one has $Z(0,a^+) \subset A$.*

Proof. Without loss of generality we may assume that $l=+\infty$ and identify $Z$ with the punctured disk $\check\Delta$ via the exponential map $(t+ \isl \theta) \mapsto e^{ -t+ \isl \theta}$ such that $\d_0 Z$ is mapped onto $S^1= \d\Delta$.

Suppose that the statement is false. Then there would exist holomorphic imbeddings $f_n : A(1, R) \to \check\Delta$ and points $a_n \in \Delta \bs
f_n(A(1, R))$ such that $f_n(A(1, R))$ are adjacent to $\d\Delta$ and $a_n
\lrar a\in \d\Delta$. Passing to a subsequence, we may assume that $\{f_n\}$ converges uniformly on compact subsets in $A(1, R)$ to a holomorphic map $f: A(1, R) \to\Delta$.

If $f$ is not constant, then $f(A(1, R))$ must contain some annulus $\{ b
<|z|<1 \}$ with $b<1$. But then $\{ \sqrt{b} <|z|<1\} \subset f_n(A(1, R))$ for $n>\!>1$, which is a contradiction.

If $f$ is constant, then the diameter of images of the middle circle $\gamma\deff \{ |z|= \sqrt{R} \} \subset A(1, R)$ must converge to $0$. But $\diam(f_n(\gamma)) \ge \dist(0, a_n) \lrar1$. The obtained contradiction finishes the proof.

For the proof of [*Theorem 1.1*]{} we need a special covering of $\Sigma$ which will be constructed in the following theorem.

Theorem 5.3.2.

*Under the conditions of [*Theorem 1.1*]{}, after passing to a subsequence, there exist a finite covering $\calv$ of $\Sigma$ by open sets $V_\alpha$ and parameterizations $\sigma_n:\Sigma \to C_n$ such that*

[*(a)*]{} all $V_\alpha$ are either disks, or annuli or pants;

[*(b)*]{} for any boundary circle $\gamma_i$ of $\Sigma$ there is some annulus $V_\alpha$ adjacent to $\gamma_i$;

[*(c)*]{} $\sigma_n^*j_n\ogran_{V_\alpha}$ does not depend on $n$ if $V_\alpha$ is a disk, pants or an annulus adjacent to a boundary circle of $\Sigma$;

[*(d)*]{} all non-empty intersections $V_\alpha \cap V_\beta$ are annuli, where the structures $\sigma^*j_n$ are independent of $n$;

[*(e)*]{} if $a$ is a node of $C_n$ and $\gamma^n_a=\sigma_n\inv (a)$ the corresponding circle, then $\gamma_a^n=\gamma_a$ does not depend on $n$ and is contained in some annulus $V_\alpha$, containing only one such “contracting" circle for any $n$; moreover, the structures $\sigma_n^*j_n\ogran_{V_\alpha\bs \gamma_a}$ are independent of $n$;

[*(f)*]{} if $V_\alpha$ is an annulus and $\sigma_n( V_\alpha)$ doesn’t contains nodes, then the conformal radii of $\sigma_n( V_\alpha)$ converge to some positive $R_\alpha^\infty>1$ or to $+\infty$;

[*(g)*]{} if for initial parameterizations $\delta_n$ and fixed annuli $A_i$, each adjacent to the boundary circle $\gamma_i$ of $\Sigma$, the structures $\delta_n^* j_n\ogran_{A_i}$ do not depend on $n$, then the new parameterizations $\sigma_{n}$ can be taken equal to $\delta_{n}$ on some subannuli $A'_i \subset A_i$ also adjacent to $\gamma_i$.

Proof. We shall prove the properties [*(a)–(f)*]{}. The property [*(g)*]{} will follow from [*Lemma 5.3.3*]{} below.

There are four cases where the existence of such a covering is obvious. If all $C_n$ are disks or annuli without nodal points, there is nothing to prove. In the third case each $C_n$ is a sphere, and we cover it by two disks.

In the forth case each $C_n$ is a torus without marked points. Then any complex torus can be represented by the form $\cc{\bigm/}(\zz+
\tau \zz)$ with $|\re\tau| \le {1\over 2}$ and $\im\tau > {1\over 2}$. Considering the map $z\in \cc \mapsto e^{2\pi\isl z} \in \check\cc \deff
\cc\bs\{0\}$, we represent $(T^2,j)$ as the quotient $\cc\bigm/ \{ z \sim
\lambda^2 z\}$ with $\lambda = e^{\pi\isl\tau}$, so that $|\lambda|<
e^{-\pi/2}< {1\over 3}$. The annuli $\{ {|\lambda| \over2} <|z| <1\}$ and $\{{|\lambda|^2 \over2} <|z| <|\lambda| \}$ form the needed covering.

In all remaining cases we start with the construction of appropriate graphs $\Gamma_n$ associated with some decomposition of $C_n$ into pants. [*Lemma 5.1.2*]{} and a non-degeneration of the complex structure $j_n$ on $C_n$ shows that lengths of all boundary circles of all non-exceptional components $C_{n,i}$ of $C_n$ are uniformly bounded from above. At this point we make the following

Remark. The Collar Lemma from \[Ab\], Ch.II, §3.3 yields the existence of the universal constant $l^*$ such that for any simple geodesic circles $\gamma'$ and $\gamma''$ on $C_{n,i}$ the conditions $\ell(\gamma') <l^*$ and $\ell( \gamma'') <l^*$ imply $\gamma' \cap \gamma'' = \emptyset$. We shall call geodesic circles $\gamma$ with $\ell(\gamma) <l^*$ [*short geodesics*]{}.

The fact that $(C_n,u_n)$ are $J_n$-complex and of bounded area shows that $C_n$ have a uniformly bounded number of components. Indeed, the number of exceptional components, which are spheres and disks, is bounded by the energy (see [*Lemma 5.2.3*]{}), and the number of boundary circles of $C_n$ is equal to that of $\Sigma$. Furthermore, the operation of contracting a circle on $\Sigma$ to a nodal point either diminishes the genus of some component of $C_n$ or increases the number of components. Thus, the number of nodal points on $C_n$ and the total number of marked points on its components are also uniformly bounded. This implies that the number of possible topological types of components is finite.

In this situation the Teichmüller theory (see \[Ab\], Ch.II, §3.3) states the existence of decomposition of every non-exceptional component $C_{n,i}\bs
\mapo$ into pants with the following properties:

Every short geodesic is a boundary circle of some pants of the decomposition.

The intrinsic length of every boundary circle is bounded from above by a (uniform) constant depending only on an upper bound of lengths of boundary circles and possible topological types of $C_{n,i} \bs \mapo$.

Having decomposed all $C_{n,i} \bs \mapo$ into pants, we associate with every curve $C_n$ its graph $\Gamma_n$. As was noted above, the number of vertices and edges of $\Gamma_n$ is uniformly bounded. Thus, after passing to a subsequence, we can assume that all $\Gamma_n$ are isomorphic to each other (as marked graphs). Denote this graph by $\Gamma$. Now, the parameterizations $\sigma_n: \Sigma \to C_n$ can be found in such a way that the decompositions of $C_{n,i} \bs\mapo$ into pants define the same set $\bfgamma =\{ \gamma
_\alpha \}$ of circles on $\Sigma$ and induce the same decomposition $S\bs
\cup_\alpha \gamma_\alpha = \cup_j S_j$ with the graph $\Gamma$.

By our construction of the graph $\Gamma$, each edge of $\Gamma$ corresponds either to a circle in $\Sigma$ contracted by every parameterization $\sigma_n$ to a nodal point, or to a circle mapped by every $\sigma_n$ onto a geodesic circle separating two pants. Furthermore, each tail of $\Gamma$ corresponds to a boundary circle of $\Sigma$. Thus, we shall use the same notation $\gamma
_\alpha$ for an edge or a tail of $\Gamma$ and for the corresponding circle on $\Sigma$. If $\sigma_n(\gamma_\alpha)$ is a boundary circle of some pants $C_j$, then the intrinsic length $\ell_{n,\alpha}= \ell_n (\gamma_\alpha)$ of $\sigma_n(\gamma_\alpha)$ is well- defined. This happens in the following two cases:

[*a)*]{} $\sigma_n(\gamma_\alpha)$ separates two pants, or else

[*b)*]{} $\gamma _\alpha$ is a boundary circle $\Sigma$ and the irreducible component of $C_n$ attached to $\sigma_n(\gamma_\alpha)$ is not a disk with a single nodal point. Note that the appearance of these two cases is independent of $n$.

By our choice of $\gamma_\alpha$, the lengths $\ell_n (\gamma_\alpha)$ are uniformly bounded from above. Passing to a subsequence, we may assume that for any fixed $\alpha$ the sequence $\{ \ell_{n,\alpha} \}$ converges to $\ell_{\infty,\alpha}$.

As one can expect, the condition $\ell_{n, \alpha} \lrar 0$ means that the circle $\gamma_\alpha$ is shrunk to a nodal point on the limit curve. We shall prove the statement of the theorem by induction in the number $N$ of those circles $\gamma_\alpha$ for which $\ell_{\infty,\alpha} =0$.

The case $N=0$, where there are no such circles, is easy. Passing to a subsequence, we may assume that the Fenchel-Nielsen coordinates $(\bfell_n,
\bfvartheta_n)$ of any non-exceptional component $C_{n,i}$ of $C$ converge to the Fenchel-Nielsen coordinates $(\bfell_\infty, \bfvartheta_\infty)$ of some smooth curve $C_{\infty, i}$ with marked points. Gluing together appropriate pairs of marked points, we obtain a nodal curve $C_\infty$, which admits a suitable parameterization $\sigma_\infty: \Sigma \to C_\infty$ and has the same graph $\Gamma$. [*Lemma 5.1.3*]{} shows that for $n>\! >1$ the curves $C_n$ can be obtained from $C_\infty$ by deformation of the transition functions for the intrinsic local coordinates on non-exceptional components of $C_\infty$. Note that such a deformation can be realized as a deformation of the operator $j_\infty$ of a complex structure on $C_\infty$, localized in small neighborhoods of circles $\sigma_\infty(\gamma_\alpha)$, see Fig. 10. In the case where $\gamma_\alpha$ is a boundary circle we may additionally assume that the annulus, where $j_n$ changes, lies away from $\gamma_\alpha$. Now the existence of the covering with desired properties is obvious.

Let us now consider the general case where the number $N$ of “shrinking circles” is not zero. Take a circle $\gamma_\alpha$ with $\ell_\infty
(\gamma _\alpha) =0$. Let $S_j$ be pants adjacent to $\gamma_\alpha$. Consider the intrinsic coordinates $\rho_\alpha$ and $\theta_\alpha$ at $\sigma_n(\gamma_\alpha)$ and the annuli $$\eqalign{
A_{n,\alpha,j} &\deff \{\, (\rho
_\alpha, \theta_\alpha) \in \sigma_n(S_j) \;:\; 0\le \rho_\alpha \le {\pi^2
\over \ell_{n,\alpha} } - {2\pi \over a^*} \;\}
\cr
A^-_{n,\alpha,j} &\deff
\{\, (\rho_\alpha, \theta_\alpha) \in \sigma_n(S_j) \;:\;
0\le \rho_\alpha \le {\pi^2 \over \ell_{n,\alpha} } - {2\pi \over a^*}-1
\;\},
}$$ adjacent to $\sigma_n( \gamma_\alpha)$. Note that ${\pi^2 \over \ell_{n,
\alpha} }- {2\pi \over a^*}$ (resp. ${\pi^2 \over \ell_{n,\alpha} }- {2\pi
\over a^*}-1$) is the logarithm of the conformal radius of $A_{n, \alpha, j}$ (resp. of $A^-_{n, \alpha,j}$). Consequently, we can use [*Lemma 4.3.2*]{} to show that these annuli are well-defined.

If $\gamma_\alpha$ is a boundary circle, we set $C^-_n \deff C_n \bs A^-_{n,
\alpha,j}$. Otherwise $\gamma_\alpha$ separates two pants, say $S_j$ and $S_k$. Then we define in a similar way the annuli $A_{n,\alpha, k}$ and $A^-
_{n,\alpha,k}$, set $A_{n,\alpha} \deff A_{n, \alpha,j} \cup A_{n, \alpha,
k}$ and $A^-_{n,\alpha} \deff A^-_{n,\alpha,j} \cup A^-_{n, \alpha, k}$ and put $C^-_n \deff C_n \bs A^-_{n,\alpha}$.

The parameterizations $\sigma_n: \Sigma \to C_n$ can be chosen in such a way that the annuli $\sigma_n\inv(A^-_{n,\alpha,j})$ (resp. $\sigma_n\inv(A^-
_{n, \alpha,k})$) define the same annulus $A^-_{ \alpha,j}$ (resp. $A^-
_{\alpha, k}$) on $\Sigma$. Let $\gamma^-_{ \alpha,j}$ (resp. $\gamma^-
_{\alpha,k}$) denote its boundary circles different from $\gamma_\alpha$. Thus, the curves $C^-_n$ are parameterized by a real surface $\Sigma^- \deff
\Sigma \bs A^-_{ \alpha,j}$ (resp. $\Sigma^- \deff \Sigma \bs (A^-_{\alpha,
j} \cup A^- _{\alpha, k}$), and the restrictions of $\sigma_n$ can be chosen as parameterization maps. Thus, the decompositions of components of $C_n$ into pants define the combinatorial type of decompositions of components of $C^-_n$ into pants. Moreover, the corresponding graph $\Gamma^-$ will be the same for all $C^-_n$. It coincides with $\Gamma$ if $\gamma_\alpha$ is a boundary circle. Otherwise $\Gamma^-$ can be obtained from $\Gamma$ by replacing the edge corresponding to $\gamma_\alpha$ by two new tails for two new boundary components.

Take some non-exceptional component $C^-_{n,i}$ of $C^-_n$ and decompose it into pants according to graph $\Gamma^-$ in the canonical way, so that the boundary circles of the obtained pants are geodesic. Note that even if the constructed pants are in combinatorial one-to-one correspondence with the pants of $C_n$, the intrinsic metric on $C^-_{n,i}$ and the obtained geodesics circle $\gamma^-_\beta$ differ from the corresponding objects on $C_{n,i}$

Nevertheless, we claim that for the obtained decomposition of $C^-_n$ the intrinsic lengths are uniformly bounded from above (possibly by a new constant) and that the sequence $\{ C^-_n \}$ has fewer “shrinking circles” than $\{ C_n \}$. The meaning of the above construction is the following. The curves $C^-_n$ are obtained from $C^-_n$ by cutting off the annuli $A^-_{n,\alpha,j}$ (and resp. the annuli $A^-_{n,\alpha,j}$). These annuli are sufficiently long so that one “shrinking circle” disappears, but not too long so that the complex structures of the curves $C^-_n$ remain non-degenerating near the boundary.

Indeed, the complex structures on $C^-_n$ do not degenerate at the boundary circle $\gamma^-_{ \alpha,j}$ (resp. at $\gamma^-_{\alpha,k}$), because $C^-_n$ contain annuli $A_{n,\alpha,j} \bs A^-_{n,\alpha,j}$ (resp.$A_{n,\alpha,k} \bs A^- _{n,\alpha,k}$) of the constant conformal radius $R=e>1$. This means that the lengths $\ell^-_n(\gamma^-_{\alpha,j})$ of $\sigma_n( \gamma^- _{\alpha,j})$ (resp. $\ell^-_n(\gamma^-_{\alpha,k})$ of $\sigma_n(\gamma^- _{\alpha,k})$) with respect to the intrinsic metrics on $C^-_n$ are uniformly bounded.

On the other hand, the lengths $\ell^-_n(\gamma^-_{\alpha,j})$ (resp.$\ell^-_n(\gamma^-_{\alpha,k})$) are also uniformly bounded from below by a positive constant. Otherwise, by [*Lemma 3.4*]{}, after passing to a subsequence, there would exist annuli $A_n\subset C^-_n$ of infinitely increasing radii $R_n$, adjacent to $\sigma_n(\gamma^-_{\alpha,j})$ (resp.to $\sigma_n(\gamma^-_{ \alpha,k})$). The superadditivity of the logarithm of the conformal radius of annuli, see \[Ab\], Ch.II, §1.3, shows that the conformal radius $R^+_n$ of the annulus $A^-_{n,\alpha,j} \cup A_n$ satisfies the inequality $\log R^+_n \ge {\pi^2 \over \ell_{n,\alpha} }- {2\pi \over
a^*}-1 + \log R_n$, which contradicts [*Lemma 4.3.2*]{}, part [*i)*]{}.

Now we estimate the intrinsic lengths of boundary circles and the number of “shrinking circles” on $C^-_n$. Denote $\ell_n \deff \ell_n(\gamma_\alpha)$, where $\gamma_\alpha$ is the circle on $\Sigma$ used in the above constructions. Compute the width $L_n$ of $A_{n,\alpha,j} \bs A^- _{n, \alpha,
j}$ the intrinsic metric on $C_n$. Using $\ell_n \lrar0$, we obtain $$\eqalign{
L_n &= \int_{\rho={\pi^2 \over \ell_n }- {2\pi \over a^*}-1}
^{ {\pi^2 \over \ell_n }- {2\pi \over a^*}}
\left(\msmall{ {\ell_n \over 2\pi} \over \cos {\ell_n \rho \over 2\pi}}
\right) d\rho =
\left[ \log\;\cotan \left( {\pi \over 4} -{\ell_n \rho \over 4\pi} \right)
\right]_{\rho={\pi^2 \over \ell_n }- {2\pi \over a^*}-1}
^{{\pi^2 \over \ell_n }- {2\pi \over a^*}}
\cr
&= \log \msmall{ \cotan{\ell_n  \over2a^*} \over
\cotan({\ell_n \over 2a^*} + {\ell_n \over4\pi}) }
\approx \log \msmall{ {\ell_n  \over2a^*} + {\ell_n \over4\pi} \over
{\ell_n \over2a^* } } =
\log \left( 1 + \msmall{ a^* \over 2\pi } \right) >0.
}$$

Thus we can find annular neighborhoods $A_n\subset C^-_n$ of $\sigma_n(\gamma_\alpha)$ with constant conformal radius $R>1$. Let $\gamma^-_{n, \alpha} \subset C^-_n$ be the geodesic circle corresponding to $\gamma_\alpha$. It is the unique circle, which is homotopic to $\gamma_{n,
\alpha} = \sigma_n( \gamma_\alpha)$ and is geodesic the intrinsic metric on $C^-_n$. Let $\ell^-_{n,\alpha}$ denote its $C^-_n$-intrinsic length. Using the monodromy argument as in the proof of [*Lemma 5.1.2*]{}, we see that every $A_n$ can be isometrically imbedded into the annulus $A^+_n \deff (-{\pi^2\over \ell^-
_{n,\alpha}} , {\pi^2\over \ell^-_{n,\alpha}}) \times S^1$ with coordinates $
-{\pi^2\over \ell^-_{n,\alpha}} <\rho < {\pi^2\over \ell^-_{n,\alpha}}$, $0\le
\theta \le 2\pi$ and with metric $\Bigl({\ell^-_{n,\alpha} \over 2\pi} / 
\cos{\ell^-_{n,\alpha} \rho\over 2\pi}\Bigr)^2 (d\rho^2 + d\theta^2)$ of conformal radius $e^{2\pi^2/\ell^-_{n,\alpha}}$. The monotonicity of conformal radius of annuli yields the upper bound $\ell^-_{n,\alpha} \le {2\pi^2 \over \log\,R}$.

The same argumentation shows that if $\ell_{\infty,\alpha} \not=0$, then $\ell^- _{n,\alpha}$ also do not vanish. Indeed, if $\ell^-_{n,\alpha}\lrar 0$, then we could find annular neighborhoods $A'_n\subset C^-_n \subset C_n$ of $\gamma_{n,\alpha}$ with infinitely increasing radii $R'_n$. Using the homotopic equivalence of $\gamma^-_{n,\alpha}$ with $\sigma_n(\gamma_\alpha)$ as above, we would obtain the estimate $\ell_{n,\alpha} \le {2\pi^2 \over \log\,R'_n}
\lrar 0$, which is a contradiction.

Thus, we have shown that $C^-_n$, with the intrinsic metric and defined by the $\Gamma^-$ decomposition, have uniformly bounded lengths of marked circles and less “shrinking circles” than $C_n$. Using induction, we may assume that the postulated covering of $\Sigma^-$ and parameterizations of $C^-_n$ by $\Sigma^-$ exist. Since $C_n \bs C^-_n$ is an annulus of increasing conformal radius, the statement of the theorem is valid for $C_n$.

Lemma 5.3.3. *Let $C_n$ be a sequence of complex annuli with structures $j_n$, $\Sigma$ some fixed annulus and $\delta_n: \Sigma \to C_n$ parameterizations. Suppose that for some fixed annuli $A_1, A_2 \subset
\Sigma$ adjacent to the boundary circles of $\Sigma$ restrictions $\delta_n^*j_n|_{A_i}$ do not depend on $n$.*

Then one can find parameterizations $\sigma_n: \Sigma \to C_n$ such that $\sigma_n$ coincide with $\delta_n$ on some (possibly smaller) annuli $A'_i$, are also adjacent to the boundary circles of $\Sigma$ restrictions, and that

if conformal radii $R_n$ of $C_n$ converge to $R_\infty <\infty$, then $\sigma_n^*j_n$ converge to some complex structure;

if conformal radii $R_n$ of $C_n$ converge to $\infty$, then for some circle $\gamma \subset \Sigma$ structures $\sigma_n^*j_n$ converge on compact subsets $K \Subset \Sigma \bs \gamma$ to a complex structure of disjoint union of two punctured disks. Moreover, such a $\gamma$, an arbitrary imbedded circle generating $\pi_1(\Sigma)$, can be chosen.

Proof. Without loss of generality we may assume that $\Sigma =
A(1,10)$, $A_1=A(7,10)$, $A_2=A(1,4)$ and that a given circle $\gamma$ lies in $A(3,7)$. Let $\delta_n : \Sigma \to C_n$ be the given parameterizations. There exist biholomorphisms $\phi_n: C_n \to A(r_n,1)$ with $r_n\inv=R_n$ being the conformal radii of $C_n$ such that $\phi_n (\delta_n(A_1))$ is adjacent to $\{ |z|=1\} = \d\Delta$ and $\phi_n (\delta_n(A_2))$ is adjacent to $\{ |z|=r_n\}$. Define $\phi'_n(z) \deff {r_n \over \phi_n(z)}$.

Recall that structures $\delta_n^*j_n|_{A_i}$ are independent of $n$. We call this structure $j$. Consider maps $\phi_n \scirc \delta_n : (A_1, j) \to
A(r_n,1) \subset \Delta$ and $\phi'_n \scirc \delta_n : (A_2, j) \to A(r_n,1)
\subset \Delta$. Passing to a subsequence we can suppose that $r_n \lrar
r_\infty <1$ and that maps $\phi_n \scirc \delta_n$, $\phi'_n \scirc \delta_n$ converge on $A_1\cup A_2$ to holomorphic maps $\psi: (A_1\cup A_2, j) \to
\Delta$ and $\psi': (A_1\cup A_2, j) \to \Delta$, respectively. This means that maps $(\phi_n \scirc \delta_n, \phi'_n \scirc \delta_n): (A_1 \cup A_2)
\to \Delta^2$ take values in $\{(z,z') \in \Delta^2 \;:\; z{\cdot} z'= r_n
\}$ and converge to the map $(\psi, \psi'): (A_1 \cup A_2) \to \Delta^2$ with values in $\{(z,z') \in \Delta^2 \;:\; z{\cdot}z' = r_\infty \}$.

Arguments from the proof of [*Lemma 5.3.1*]{} show that $\psi(A_1)$ and $\psi'(A_2)$ are annuli adjacent to $\d\Delta$. This implies that for $n>\!>
1$ there exist diffeomorphisms $(\psi_n, \psi'_n): \Sigma \to \{(z,z') \in
\Delta^2 \;:\; z{\cdot}z' = r_n \}$ such that $\psi_n \equiv \phi_n \scirc
\delta_n$ on $A(9,10)$, $\psi'_n(z) \equiv \phi'_n \scirc \delta_n(z)$ for $z\in A(1,2)$, and $(\psi_n, \psi'_n)$ converge to $(\psi, \psi')$ on $\Sigma$. Moreover, we may assume that $|\psi_n(t)|=|\psi'_n(t)|= \sqrt{r_n}$ for any $t\in \gamma$. This means that $(\psi_n, \psi'_n)(\gamma)$ lies on the middle circle $\{ (z,z'): |z|= \sqrt{r_n},\; z'= {r_n\over z} \}$ of $\{(z,z') \in \Delta^2 \;:\; z{\cdot}z' = r_n \}$.

Set $\sigma_n \deff \phi_n\inv \scirc \psi_n: \Sigma \to C_n$. Then, obviously, $\sigma_n \equiv \delta_n$ on $A(1,2)$ and on $A(9,10)$, and $\sigma_n^*j_n
= (\psi_n,\psi')_n^* J\st \lrar (\psi,\psi')^* J\st$, where $J\st$ denotes the standard complex structure on $\Delta^2$.

Proof of Theorem 4.1.1. Let $\{(C_n, u_n)\}$ be the sequence from the hypothesis of the theorem. Then the condition [*c)*]{} of the theorem and [*Lemma 5.3.1*]{} provide the existence of parameterizations $\delta_n: \Sigma \to
C_n$ and annuli $A_i$, adjacent to each boundary circle $\gamma_i$, such that $\delta_n^* j_{C_n}$ are constant in every $A_i$. Thus we may assume that such $\delta_n$ are given.

Take a covering $\calv= \{V_\alpha\}$ and parameterizations $\sigma_n$ as in [*Theorem 4.3*]{}. With every such covering we can associate the curves $C_{\alpha,n} \deff \sigma_n(V_\alpha)$, the parameterizations $\sigma_{\alpha,
n} \deff \sigma_n\ogran_{V_\alpha}: V_\alpha \to C_{\alpha,n}$ and the maps $u_{\alpha,n} \deff u_n \ogran_{C_{\alpha, n}} : C_{\alpha, n} \to X$. Consider the following type of convergence of sequences $\{ (C_{\alpha, n}, u_{\alpha,
n}, \sigma_{\alpha, n} )\}$ with the fixed $\alpha$:

**

[A)]{} $C_{\alpha, n}$ are annuli of infinitely growing conformal radii $l_n$ and the conclusions of [*Lemma 5.2.2*]{} hold;

[B)]{} every $C_{\alpha, n}$ is isomorphic to the standard node $\cala_0=
\Delta \cup_{ \{0\} } \Delta$ such that the compositions $V_\alpha \buildrel
\sigma _{\alpha, n} \over \lrar C_{\alpha, n} \buildrel \cong \over \lrar
\cala_0$ define the same parameterizations of $\cala_0$ for all $n$; furthermore, the induced maps $\ti u_{\alpha, n}: \cala_0 \to X$ strongly converge;

[C)]{} the structures $\sigma_n^*j\vph_n \ogran_{V_\alpha}$ and the maps $u_{\alpha, n}\scirc \sigma _{\alpha, n}: V_\alpha \to X$ strongly converge.

Here the strong convergence of maps is the one in the $L^{1,p}$-topology on compact subsets for some $p>2$ (and hence for all $p<\infty$), and the convergence of structures means the usual $C^\infty $-convergence.

Suppose that there is a subsequence, still indexed by $n\to \infty$ such that for any $V_\alpha$ we have one of the convergence types [*A)–C)*]{}. Then the sequence of global maps $\{ (C_n , u_n, \sigma _n )\}$ converges in the Gromov topology which gives us the proof, as well as a precise description of the convergence picture.

Otherwise, we want to find a refinement of our covering $\calv$ and parameterizations $\sigma_n$ which have the needed properties. We shall proceed by induction, estimating an area of pieces of coverings of $\Sigma$. To do this, we fix $\eps>0$ satisfying $\eps \le {\eps_1^2 \over2 }$ with $\eps_1$ from [*Lemma 5.1.1*]{}, $\eps \le {\eps_2^2 \over2 }$ with $\eps_2$ from [*Lemma 5.1.2*]{} and $\eps \le {\eps_3\over 3}$ with $\eps_3$ from [*Lemma 5.2.3*]{}. First consider the

[*Special case: $\area(u_n(\sigma_n(V_\alpha))) \le \eps$ for any $n$ and any $V_\alpha \in \calv$*]{}. We can consider every $V_\alpha$ separately. If the structures $\sigma_n^* j_n |_{V_\alpha}$ are constant, then some subsequence of $u_n \scirc \sigma_n$ strongly converges due to [*Corollary 2.5.1*]{}.

If structures $\sigma_n^* j_n |_{V_\alpha}$ are not constant, then $V_\alpha$ must be an annulus. Fix biholomorphisms $\phi_n: Z(0,l_n) \buildrel \cong
\over \lrar \sigma_n(V_\alpha)$. If $l_n \lrar \infty$, [*Lemma 5.2.2*]{} shows that (and describes how!) an appropriate subsequence of $u_n \scirc \phi_n$ converges to a $J_\infty$-holomorphic map of a standard node. Otherwise, we can find a subsequence, still denoted $(C_n, u_n)$, for which $l_n \lrar l_\infty
<\infty$ and $u_n \scirc \phi_n$ converge to a $J_\infty$-holomorphic map of $Z(0,l_\infty)$ in $L^{1,p}$-topology on compact subsets $K \Subset Z(0,l_\infty)$ for any $p<\infty$. To construct refined parameterizations $\ti\sigma _{\alpha,
n} : V_\alpha \to C_{\alpha, n}= \sigma_n(V_\alpha)$, we use property [*(d)*]{} from [*Theorem 5.3.2*]{} and apply [*Lemma 5.3.1*]{}.

Thus, we reach one of the convergence types [*A)–C)*]{} which gives the proof in this [*Special case*]{}.

[*General case.*]{} Suppose that the theorem is proved for all sequences of $J_n$-holomorphic curves $\{(C_n, u_n)\}$ with parameterizations $\delta_n: \Sigma \to C_n$ which satisfy the additional condition $\area(u_n(C_n)) \le (N-1) \eps$ for all $n$. We see this as the hypothesis of the induction in $N$, so that our [*Special case*]{} is the base of the induction.

Assume that there exists a subsequence, still indexed by $n\to \infty$ such that for every $V_\alpha$ and for the curves $C_{\alpha, n} = \sigma_n(
V_\alpha )$ the statement of the theorem holds. This means that there exist refined coverings $V_\alpha= \cup_i V _{\alpha, i}$ and new parameterizations $\ti\sigma _{\alpha, n}: V_\alpha \to C_{\alpha, n}$ such that the $\ti\sigma_n$ coincide with the $\sigma_n$ near the boundary of every $V_\alpha$ and such that for the curves $C_{\alpha, i,n} \deff \ti\sigma_n (V _{\alpha ,i})$ we have convergence of one of the types [*A)–C)*]{}. Then we can glue $\ti\sigma
_{\alpha, n}$ together to global parameterizations $\ti\sigma_n: \Sigma \to
C_n$ and set $\wt\calv \deff \{ V _{\alpha, i} \}$, obtaining the proof.

In particular, due to the inductive hypothesis, this is also true for any $V_\alpha$ such that $\area(u_n(\sigma_n(V_\alpha))) \le (N-1) \eps$ for all $n$.

This implies that it is sufficient to consider only those $V_\alpha$ for which $(N-1) \eps\le \area(u_n(\sigma_n( V_\alpha ))) \le N \eps$ for all $n$. Obviously, it is sufficient to show the desired property only for such a piece of covering, say for $V_1$. To construct the refined parameterizations $\ti\sigma_{1,n}$ and the covering $V_1 = \cup_i V _{1, i}$, we consider four cases.

. Then we can realize $(V_1, \sigma_n^*j_n)$ as a constant bounded domain $D$ in $\cc$. Hence we can consider $u_n \scirc \sigma_n: V_\alpha \to
X$ as holomorphic maps $u_n :D \to (X,J_n)$. Now we use the “patching construction” of Sacks-Uhlenbeck \[S-U\].

Fix some $a>0$. Denote $D_{-a} \deff \{ z\in D\;:\; \Delta(z,a) \subset D\}$. Find a covering of $D_{-a}$ by open sets $U_i \subset D$ with $\diam(U_i) <a$ such that any $z\in D$ lies in at most 3 pieces $U_i$. Then for any $n$ there exists at most $3N$ pieces $U_i$ with $\area( u_n(U_i)) > \eps$. Taking a subsequence, we may assume that the set of such “bad” pieces $U_i$ is the same for all $n$. Repeat successively the same procedure for ${a\over2}$, ${a\over4}$, and so on, and then take a diagonal subsequence. We obtain at most $3N$ “bad” points $y^*_1, \ldots, y^*_l$ such that a subsequence of $u_n$ converges in $D\bs \{y^*_1, \ldots,y^*_l\}$ strongly, i.e., in the $L^{1,p}$-topology on compact subsets $K \Subset D\bs \{y^*_1, \ldots,y^*_l\}$. These “bad” points $y^*_1, \ldots, y^*_l$ are characterized by the property $$\text{\ \ for any $r>0$\ \ }
\area(u_n(\Delta(y^*_i, r)) >\eps \text{\ \ for $n$ all sufficiently big}.
\eqno(5.3.1)$$

Remark. As we shall now see, every such point is the place where the “bubbling phenomenon” occurs. Therefore we shall call $y^*_i$ [*bubbling points*]{}. The characterization property of a bubbling point is (4.1).

If there are no bubbling points, i.e., $l=0$, then the chosen subsequence $u_n$ converges strongly and we can finish the proof by induction.

Otherwise, we consider the first point $y^*_1 \in D$. Take a disk $\Delta
(y^*_1,\varrho)$ which doesn’t contain any other bubbling points $y^*_i$, $i>1$.

Then for any $n$ we can find the unique $r_n$ such that

\(1) $r_n \le {\varrho \over 2}$ and $\area(u_n(\Delta(x, r_n)))\le\eps$ for any $x\in \barr\Delta (y^*_1,{\varrho \over 2})$;

\(2) $r_n$ is maximal (1).

Then $r_n\lrar 0$, because otherwise for $r^+ \deff {\sf lim\,sup}\; r_n >0$ and for some subsequence $n_k \lrar \infty$ with $r_{n_k} \lrar r^+$ we would have $$\area\bigl(u_{n_k}(\Delta(y^*_1, r^+))\bigr) \le \eps,$$ which would contradict (5.3.1).

Lemma 5.3.4. [*For every $n>\!>1$ there exists $x_n\in \barr 
\Delta(y^*_1, {\varrho \over2})$, such that $x_n\to y^*_1$ and $\area( u_n( 
\Delta(x_n, r_n))) =\eps$.*]{}

Proof. If not, then for some subsequence $n_k\lrar \infty$ and every $x\in \barr\Delta (y^*_1, {\varrho \over2})$ we would have $\area(u_{n_k}(
\Delta (x, r_{n_k})))< \eps $. Since $r_n \lrar0$, this would contradict the maximality of $r_n$. In particular, there exists the postulated sequence $\{ x_n \}$.

If $x_n$ do not converge to $y^*_1$, then after going to a subsequence we would find $y'=\lim_{n\to \infty } x_n\not= y^*_1$. By our construction, $y'$ does not coincide with any other bubbling point $y^*_i$. Take $a>0$ such that $\Delta(y', a)$ contains no bubbling point. Then by [*Corollary 2.5.1*]{} some subsequence $u_{n_k}$ would converge to some $u' \in L^{1,p}_\loc (\Delta(y',
a), X)$ in strong $L^{1,p}(K)$-topology for any compact subset $K \Subset
\Delta(y', a)$ and any $p<\infty$. In particular, for sufficiently small $b<a$ we would have $\area(u_{n_k}(\Delta(y', b))) \lrar \area(u'( \Delta(y',
b))) < \eps$, which would contradict the choice of $r_n$ and $x_n$.

Using constructed $r_n$ and $x_n$, define maps $v_n:\Delta (0,{\varrho \over
2r_n}) \to (X,J_n)$ by $v_n(z)\deff u_n(x_n +r_n z)$. By the definition of $r_n$ we have $$\area(v_n(\Delta (x ,1))\le \eps \ \ \text{for all $x\in \Delta
(0,{\textstyle{\varrho \over 2r_n}}-1)$}.
\eqno(5.3.2)$$ On the other hand, $\area(v_n(\Delta(0,1)) = \area(u_n( \Delta (x_n, r_n)) =
\eps$ by [*Lemma 5.3.4*]{}. Thus $v_n$ converge (after going to a subsequence) on compact subsets in $\cc$ to a nonconstant $J_\infty $-holomorphic map $v_\infty$ with finite energy. Consequently, $v_\infty$ extends onto $S^2$ by the removable singularity theorem of [*Corollary 5.2.1*]{}.

Since $v_\infty$ is nonconstant, $\norm{ dv_\infty }^2_ {L^2( S^2)}= \area(
v_\infty(S^2))\ge 3\eps$ by [*Lemma 5.2.3*]{} and the choice of $\eps$. Choose $b>0$ in such a way that $$\area(v_\infty (\Delta (0,b)) =
\norm{ dv_\infty}_{L^2(\Delta (0,b))}^2 \ge 2\eps.
\eqno(5.3.3)$$ By [*Corollary 2.5.1*]{} this provides that $$\norm{du_n}_{L^2(\Delta (x_n,br_n))}^2=
\norm{ dv_n}_{L^2(\Delta (0,b))}^2 \ge\eps.
\eqno(5.3.4)$$

For $n>\!>$ we consider the coverings of $V_1$ by 3 sets $$\textstyle\mathsurround=0pt
V^{(n)}_{1,1} \deff V_1 \bs \barr\Delta (y^*_1, {\varrho\over2}),
\qquad
V^{(n)}_{1,2}\deff \Delta (y^*_1, \varrho) \bs \barr \Delta (x_n, br_n),
\qquad
V^{(n)}_{1,3} \deff \Delta (x_n, 2br_n).$$ Fix $n_0$ sufficiently big. Denote $V_{1,1} \deff V^{(n_0)}_{1,1}$, $V_{1,2}
\deff V^{(n_0)}_{1,2}$, and $V_{1,3} \deff V^{(n_0)}_{1,3}$. There exist diffeomorphisms $\psi_n: V_1 \to V_1$ such that $\psi_n: V_{1,1} \to V^{(n)}
_{1,1}$ is identity, $\psi_n: V_{1,2} \to V^{(n)}_{1,2}$ is a diffeomorphism and $\psi_n: V_{1,3} \to V^{(n)}_{1,3}$ is biholomorphic the complex structures, induced from $C_{1,n}$.

Thus, we have constructed the covering $\{ V_{1,1}, V_{1,2}, V_{1,3} \}$ of $V_1$ and parameterizations $\sigma'_n \deff \sigma_{1,n} \scirc \psi_n: V_1
\to C_{1, n}$ such that the conditions of [*Theorem 5.3.2*]{} are satisfied. Moreover, $\area(u_n(\sigma'_n(V_{1,i}))) \le (N-1)\eps$ due to inequality (5.3.4). Consequently, we can apply the inductive assumptions for the sequence of curves $\sigma'_n(V_{1,i})$ and finish the proof by induction.

[*Case 2):*]{} $V_1$ is a cylinder, structures $\sigma_n^*j_n|_{V_1}$ vary with $n$, but conformal radii of $(V_1, \sigma_n^*j_n)$ are bounded uniformly in $n$. Applying [*Lemma 5.3.3*]{}, we can assume that structures $\sigma_n^*
j_n$ converge to a structure of an annulus with finite conformal radius. The constructions of [*Case 1)*]{} are used here with the following minor modifications. First, we find the set of the bubble points $ y^*_i\in V_1$, using the same patching construction and the characterization (5.3.1). Then we find diffeomorphisms $\phi_n : V_1 \to V_1$ such that [*a)*]{} $\phi_n$ converge to the identity map $\id: V_1 \to V_1$; [*b)*]{} $\phi_n$ are identical in fixed (independent of $n$) annuli adjacent to the boundary circles of $V_1$; [*ċ)*]{} $\phi_n$ preserve every bubble point, $\phi_n(
y^*_i) = y^*_i$; and finally [*d)*]{} for the “corrected” parameterizations $\ti\sigma_n \deff \sigma_n \scirc \phi_n$ the structures $\ti\sigma_n^*j_n|
_{V_1}$ are constant in a neighborhood of every bubble point $y^*_i$. Then we repeat the rest of the constructions of [*Case 1)*]{} using the new parameterizations $\ti\sigma_n$.

[*Case 3): Every $C_{1,n} = \sigma_n(V_1)$ is isomorphic to the standard node $\cala_0$*]{}. Fix identifications $C_{1,n} \cong \cala_0$ such that the induced parameterization maps $\sigma_{1, n} : V_1 \to \cala_0$ are the same for all $n$. Represent $\cala_0$, and hence every $C_{1,n}$, as the union of two discs $\Delta'$ and $\Delta''$ with identification of the centers $0\in
\Delta'$ and $0\in \Delta''$ into the nodal point of $\cala_0$, still denoted by $0$. Let $u'_n :\Delta' \to X$ and $u''_n :\Delta'' \to X$ be the corresponding “components” of the maps $u_{1,n} : C_{1,n} \to X$. Find the common collection of bubbling points $y^*_i$ for both maps $u'_n :\Delta' \to X$ and $u''_n :\Delta'' \to X$. If there are no bubble points, then we obtain the convergence type [*B)*]{} and the proof can be finished by induction. Otherwise consider the first such point $y^*_1$ which lies, say, on $\Delta'$. If $y^*_1$ is distinct from the nodal point $0 \in
\Delta'$, then we simply repeat all the constructions from [*Case 1)*]{}.

It remains to consider the case $y^*_1=0 \in \Delta'$. Now one should modify the argumentations of [*Case 1*]{} in the following way. Repeat the construction of the radii $r_n\lrar 0$, the points $x_n \lrar y^*_1=0$, and the maps $v_n: \Delta(0, {\varrho \over 2r_n}) \to X$, $v_\infty: S^2 \to
X$ from [*Case 1*]{}. Set $R_n \deff |x_n|$, so that $R_n$ is the distance from $x_n$ to point $0=y^*_1 \in \Delta'$. After rescaling $u_n$ to the maps $v_n$, the point $0\in \Delta'$ will correspond to the point $z^*_n \deff
-{x_n \over r_n}$ in the definition domain $\Delta(0, {\varrho \over 2r_n})$ of the map $v_n$. We will now consider two subcases.

[*Subcase 3$'$): The sequence ${R_n\over r_n}$ is bounded*]{}. This is equivalent to boundedness of the sequence $ z^*_n$. Going to a subsequence we may assume that the sequence $z^*_n$ converges to a point $z^*\in \cc$. This point will be a nodal one for $(S^2, v_\infty)$. As above, $v_\infty$ is nonconstant and $\norm{ dv _\infty} ^2 _{L^2( S^2)}=
\area( v_\infty( S^2) )\ge 3\eps$. Choose $b>0$ in such a way that $$\norm{ dv_\infty}_{L^2(\Delta (0,b))}^2 \ge 2\eps
\eqno(5.3.5)$$ and $b \ge 2|z^*|+2$. Due to [*Corollary 2.5.1*]{} for $n>\!>1$ we obtain the estimate $$\norm{du'_n}_{L^2(\Delta' (x_n, br_n))}^2 =
\norm{ dv_n}_{L^2(\Delta (0,b))}^2 \ge \eps.
\eqno(5.3.6)$$ Here $\Delta' (x, r)$ denotes the subdisc of $\Delta'$ with the center $x$ and the radius $r$. Furthemore, for $n>\!>1$ we have the relation $z^*_n \in \Delta( 0, b-1)$, or equivalently, $0 \in  \Delta' (x_n, (b-1)
r_n))$.

Define the coverings of $\cala_0$ by four sets $$\mathsurround=0pt
\matrix\format\l\ \ &\l\\
W^{(n)}_1 \deff  \Delta'
\bs \barr\Delta' (0, {\textstyle{\varrho\over2}}),
&
W^{(n)}_2 \deff \Delta' (0, \varrho)
\bs \barr \Delta' (x_n, br_n),
\cr
\noalign{\vskip5pt}
W^{(n)}_3 \deff \Delta' (x_n, 2br_n)
\bs \barr\Delta'(0,{\textstyle {r_n \over 2}}),
&
W^{(n)}_4 \deff \Delta' (0, r_n) \cup
\Delta'',
\endmatrix$$ and lift them to $V_1$ by putting $V^{(n)}_{1,i} \deff \sigma\inv_{1,n}(
W^{(n)}_i)$. Choose $n_0 >\!> 0$ such that $z^*_{n_0} \in \Delta( 0, b-1)$ and the relation (5.3.6) holds. Set $V_{1,i} \deff V^{(n_0)} _{1, i}$. Choose diffeomorphisms $\psi_n: V_1 \to V_1$ such that $\psi_n: V_{1,1} \to V^{(n)}
_{1,1}$ is the identity map, $\psi_n: V_{1,2} \to V^{(n)}_{1,2}$ and $\psi_n:
V_{1,3} \to V^{(n)}_{1,3}$ are diffeomorphisms, and $\psi_n: V_{1,4} \to
V^{(n)}_{1,4}$ corresponds to isomorphisms of nodes $W^{(n)}_4 \cong
\cala_0$. Set $\sigma'_n \deff \sigma_n \scirc \psi_n$. The choice above can be done in such a way that the refined covering $\{ V_{1,i} \}$ of $V_1$ and parameterization maps $\sigma'_n: V_1 \to C_{1,n}$ have the properties of [*Theorem 5.3.2*]{}. Moreover, relations (5.3.2) and (5.3.6) imply the estimate $\area(
u_n (\sigma'_n(V_{1,i})) \le (N-1)\,\eps$. This provides the inductive conclusion for [*Subcase 3$'$)*]{}.

[*Subcase 3$''$): The sequence $R_n \over r_n$ increases infinitely*]{}. This means that the sequence $ z^*_n$ is not bounded. Nevertheless $R_n \lrar
0$ since $x_n \lrar 0$. We proceed as follows. Repeat the construction of the raduis $b$ from [*Case 1)*]{}. For $n>\!>0$ define the coverings of $\cala_0$ by six sets $$\mathsurround=0pt
\matrix\format\l\ \ &\l\\
W^{(n)}_1 \deff  \Delta'
\bs \barr\Delta' (0, {\varrho\over2}),
&
W^{(n)}_2 \deff \Delta' (0, \varrho)
\bs \barr \Delta' (x_n, 2R_n),
\cr\noalign{\vskip5pt}
W^{(n)}_3 \deff  \Delta' (x_n, 4R_n)
\bs \bigl(\barr \Delta' (x_n, {R_n \over 6})
\cup \barr \Delta' (0,{R_n\over 6}) \bigr)
&
W^{(n)}_4 \deff \Delta' (0, {R_n\over 3}) \cup \Delta'',
\cr\noalign{\vskip5pt}
W^{(n)}_5 \deff \Delta' (x_n, {R_n \over 3})
\bs \barr\Delta'(x_n, br_n),
&
W^{(n)}_6 \deff \Delta' (0, 2br_n ),
\endmatrix$$ and lift them to $V_1$ by putting $V^{(n)}_{1,i} \deff \sigma\inv_{1,n}(
W^{(n)}_i)$. Choose $n_0 >\!> 0$ such that $R_{n_0} >\!> br_{n_0}$ , and set $V_{1,i} \deff V^{(n_0)} _{1, i}$. Choose diffeomorphisms $\psi_n: V_1 \to
V_1$ such that $\psi_n: V_{1,1} \to V^{(n)} _{1,1}$ is the identity map, $\psi_n: V_{1,2} \to V^{(n)}_{1,2}$, $\psi_n: V_{1,4} \to V^{(n)}_{1,4}$ and $\psi_n: V_{1,5} \to V^{(n)}_{1,5}$ are diffeomorphisms, and finally, $\psi_n: V_{1,6} \to V^{(n)}_{1,6}$ corresponds to isomorphisms of nodes $W^{(n)}_6 \cong \cala_0$. Set $\sigma'_n \deff \sigma_n \scirc \psi_n$. Again, this choice can be done in such a way that $\{ V_{1,i} \}$ and parameterization maps $\sigma'_n: V_1 \to C_{1,n}$ have the properties of [*Theorem 5.3.2*]{}. As above, we get the estimate $\area(u_n (\sigma'_n(
V_{1,i} )) \le (N-1)\,\eps$ due to (5.3.2). Thus we get the inductive conclusion for [*Subcase 3$''$)*]{} and can proceed further.

[*Case 4):*]{} $V_1$ is a cylinder, structures $\sigma_n^*j_n|_{V_1}$ vary with $n$, and conformal radii of $(V_1, \sigma_n^*j_n)$ converge to $+\infty$. Using [*Lemma 5.3.3*]{}, we can assume that structures $\sigma_n^*
j_n$ satisfy property of this lemma.

Fix biholomorphisms $\sigma_n(V_1) \cong Z(0, l_n)$. If $\area(u_n(Z(a-1,
a))) \le \eps$ for any $n$ and any $a \in [1, l_n]$, then [*Lemma 5.2.2*]{} shows that $u_n: \sigma_n(V_1) \to X$ converge to a $J_\infty$-holomorphic map from a node.

If not, then, after passing to a subsequence, we can find a sequence $\{a_n
\}$ with $a_n \in [1, l_n]$ such that $\area(u_n(Z(a_n \allowbreak -1, a_n)))
\ge \eps$. If $a_n$ is bounded, say $a_n \le a^+$, then we cover $Z(0,
l_n)$ by the sets $V_{1,1} \deff Z(0, a^+ +2)$ and $V_{1,2} \deff Z(a^+ +1,
l_n)$. If $l_n - a_n$ is bounded, say $l_n - a_n \le a^+$, then we cover $Z(0,
l_n)$ by the sets $V_{1,1} \deff Z(0, l_n - a^+ +2)$ and $V_{1,2} \deff
Z(l_n -a^+ +1, l_n)$. In the remaning case, when both $a_n$ and $l_n - a_n$ increase infinitely, we cover $Z(0, l_n)$ by 3 sets $V_{1,1}^{(n)} \deff
Z(0,a_n-1)$, $V_{1,2}^{(n)} \deff Z(a_n-2, a_n+1)$, and $V_{1,3}^{(n)} \deff
Z(a_n, l_n)$. Cover $V_1$ by 2 or, respectively, 3 successive cylinders $V_{1,
i}$ in an obvious way. Find diffeomorphisms $\psi_n : V_1 \to V_1$ identical in the neighborhood of the boundary of $V_1$ and such that $\psi_n(V_{1,i}) =
\sigma_n ^{-1} V^{(n)} _{1,i}$. Define the new parameterizations $\sigma'_n
\deff \sigma_n \scirc \psi_n$. Note that we may additionally assume that if the conformal radius of $\sigma'_n(V_{1,i})$ is independent of $n$ then the structure $\sigma'_n{}^*j_n \ogran_{V_{1,i}}$ is also independent of $n$.

By this construction we obtain the following property of the covering $\{ V_{1,i}\}$ and new parameterizations $\sigma'_n$. For any $V_i$ we have either the estimate $$\area(u_n(\sigma'_n(V_{1,i}))) \le (N-1)\,\eps.$$ or the structures $\sigma'_n{}^*j_n|_{V_{1,i}}$ do not depend on $n$. Thus we reduce our case to the situation which is covered either by the inductive assumption or by [*Case 1)*]{}.

The proof of the theorem can be finished by induction. The fact that the limit curve $(C_\infty, u_\infty)$ remains stable over $X$ is proved in [*Lemma 5.3.5*]{} below.

Remark. Here we explain the meaning of the constructions used in the proof. We start with [*Case 1)*]{}, where $J_n$-holomorphic maps from a fixed domain $D\subset \cc$ are treated. Bubbling points $y^*_i$ appear in this case as those where the strong convergence of maps $u_n: D \to (X, J_n)$ fails. The patching construction of Sacks and Uhlenbeck insures that the “convergence failure” set is finite and insures an effective estimate on the number of bubbling points by the upper bound of the area, $l \le 3N$ in our situation. The characterization property (5.3.2) of bubbling points is essentially due to Sacks and Uhlenbeck; the only difference is that we use the area of the map $u$ (which is equivalent to the energy of $u$, i.e., $L^2$-norm of $du$) , whereas in \[S-U\] the $\norm{du}_{L^\infty}$ is used. The next step, the construction of maps $v_n$ as the rescaling of the $u_n$ and the existence of the limit $v_\infty$ is also due to Sacks and Uhlenbeck.

Due to the explicit construction of the map $v_\infty$, it is useful to imagine the curve $(S^2, v_\infty)$ as a “bubbled sphere” and $y^*_1$ as the point where the “bubbling” occurs. Moreover, one obtains natural partitions (one for each $n>\!>0$) of $D$ into three pieces: $D$ minus a fixed small neighborhood of $y^*_1$; disks $(\Delta(x_n, br_n), u_n)$ representing pieces $(\Delta(0, b), v_\infty)$ and approximating a sufficiently big part $(\Delta(0, b), v_\infty)$ of the bubbled sphere and the “part between”.

These latter “parts between” appear to be the annuli of infinitely growing conformal radii, the situation considered in [*Case 4)*]{}. Since neither outer nor inner boundary circle should be preferred in some way, we consider them as long cylinders $C_n = Z(0,l_n)$ with $l_n \lrar \infty$, in the spirit of [*Definition 3.2*]{}. [*Lemma 5.2.2*]{} provides a “good” convergence model for long cylinders, stated above as convergence type [*A)*]{}. If such a convergence for a sequence $(C_n, u_n)$ fails, then there must exist subannuli $A_n \subset C_n$ of a constant conformal radius for which $\area(u_n(A_n)) \ge \eps$.

In both cases — a constant domain $D$ or long cylinders — we proceed by cutting the curves into smaller pieces. We arrive at a situation that is simpler in the following way. The obtained curves either converge or have the upper bound for the area smaller in the fixed constant $\eps$. Thus, the use of induction leads finally to a decomposition of the curves into pieces for which one of the convergence types [*A)–C)*]{} holds. The possibility of gluing these final pieces together is insured by the fact that the partitions above are represented by appropriate coverings satisfying the conditions of [*Theorem 5.3.2*]{}.

Considering curves with nodes, additional attention should be paid to the case where bubbling appears at a nodal point. This situation is considered in [*Case 3)*]{}. The constructed points $x_n$ and radii $r_n$ describe the “center” and the “size” of energy localization of the bubbling, represented by the sequence of the maps $v_n \lrar v_\infty$. So the convergence picture depends on whether the energy localization occurs near the nodal point ([*Subcase 3$'$)*]{}) or away from it ([*Subcase 3$''$)*]{}). As a result, the nodal point can either remain on the “bubbled” sphere $(S^2, v_\infty)$ or move into the “part between”, which is represented by long cylinders.

In [*Subcase 3$'$)*]{} we remove neighborhoods of the nodal point from the disks $(\Delta(0, b), v_n)$ and thus get four pieces of converging instead of three as in [*Case 1)*]{}. In [*Subcase 3$''$)*]{} the situation is more complicated, because we must take into account the position of the nodal point in the long cylinders — the “parts between”. Thus, we must consider now the sequence of cylinders with one marked point, i.e., the sequence of pants. The vanishing $R_n\to 0$ and $r_n / R_n \to$ mean that conformal structure of those pants is not constant and converges to one of the spheres with three punctures.

In order to have the covering pieces with the convergence types [*A)–C)*]{}, we choose an appropriate refinement of the covering. After that, we obtain two sequences of long cylinders, describing the appearance of two new nodal points. The first one corresponds to the part between the “original” nodal point and bubbled sphere and is represented by $V_5$, whereas the other one, represented by $V_2$, lies on the other side of the “original” nodal point. In addition, we fix a neighborhood of the “original” nodal point which has a constant complex structure and is topologically an annulus with the disc $\Delta''$ attached to the nodal point. To satisfy the requirements of [*Theorem 5.3.2*]{}, we cover the neighborhood by two pieces, the pants $V_3$ and the piece $V_4$ parameterizing the nodes $W^{(n)}_4$. This explains the appearance of six pieces of covering in [*Subcase 3$''$)*]{}.

Lemma 5.3.5. *The limit curve $(C_\infty, u_\infty)$ constructed in the proof of [*Theorem 1.1*]{} is stable over $X$.*

Proof. If $(C_\infty, u_\infty)$ is unstable over $X$, then either $C_\infty$ is a torus with $u_\infty$ constant, or $C_\infty$ should have a component $C' \subset C_\infty$ such that $\ti C'$ is a sphere with at most two marked points, and $u_\infty(C')$ is a point.

The case of a constant map from a torus is easy to exclude. In fact, in this case all $C_n$ must also be tori with $\area(u_n(C_n))$ sufficiently small for $n>\!>1$. Cover every $C_n$ by an infinite cylinder $Z(-\infty, +\infty)$ and consider compositions $\ti u_n: Z(-\infty, +\infty) \to X$ of $u_n$ with the covering maps. Since $\area(u_n(C_n))\approx 0$, [*Corollary 5.2.1*]{} can be applied to show that every $\ti u_n$ extends to a $J_n$-holomorphic map from $S^2$ to $X$. Consequently, $\area(\ti u_n(Z(-\infty, +\infty)))$ must be finite. On the other hand, $\area(u_n(C_n))>0$ due to the stability condition; hence $\area(\ti u_n(Z(-\infty, +\infty)))$ must be infinite. This contradiction excludes the case of a torus.

The same argumentations are valid in the case, where $C_\infty$ is the sphere with no marked points. Then the curves $C_n$ are also parameterized by the sphere $S^2$. The condition of instability means that $\area(u_\infty(C_\infty))=0$. Due to [*Corollary 2.5.1*]{}, $\area(u_n(C_n))$ must be sufficiently small for $n>\!>1$. Now [*Lemma 5.2.3*]{} and the stability of $(C_n,u_n)$ exclude this possibility.

Now consider the cases where the limit curve $C_\infty$ has a “bubbled” component $C'$, which is the sphere with one or two marked points. If $C'$ has one marked point, then $C'$ must appear as a “bubbled” sphere $(S^2, v_\infty)$ in the constructions of [*Cases 1)–3)*]{} in the proof of [*Theorem 1.1*]{}. But these constructions yield only non-trivial “bubbled” spheres, for which $v_\infty \not= const$. Thus, such a component $C'$ must be stable.

In the remaining case, a component $C'$ with two marked points, we consider a domain $U\subset C_\infty$, which is the union of the component $C'$ and neighborhoods of the marked point on $C'$. If $C'$ is an unstable component, then $\area(u_\infty(C'))=0$, and we can achieve the estimate $\area( u_\infty(
U))< \eps$ taking $U$ sufficiently small. Set $\Omega \deff \sigma_\infty
\inv(U)$, where $\sigma_\infty: \Sigma \to C_\infty$ is the parameterization of $C_\infty$. Let $\gamma_1$ and $\gamma_2$ be the pre-images of marked points on $C'$. Then $\Omega$ must be a topological annulus, and $\gamma_i$, $i=1,2$, disjoint circles generating the group $\pi_1 (\Omega) = \zz$. Further, $C'$ must be a “bubbling” component of $C_\infty$, i.e., at least for one of the circles $\gamma_i$, $i=1,2$, the images $\sigma_n(\gamma_i)$ are not nodal points of $C_n$ but smooth circles.

If these are both circles $\gamma_1$ and $\gamma_2$, then $U_n \deff 
\sigma_n (\Omega)$ would satisfy the conditions of [*Lemma 5.2.2*]{}. In this case we should have the convergence type [*A)*]{}, and hence the limit piece $\sigma_\infty(\Omega)$ should be isomorphic to the node $\cala_0$.

In the case where only one circle, say $\gamma_1$, corresponds to nodal points on $C_n$, and for the other one the images $\sigma_n(\gamma_2)$ are smooth circles, then the domains $\sigma_n(\Omega)$ must be isomorphic to the node $\cala_0$. Furthermore, due to the condition $\area(u_\infty(\sigma_\infty(
\Omega))) < \eps$, we would have $\area(u_n(\sigma_n (\Omega))) < \eps$. Consequently, we would have the convergence type [*B)*]{}, and the unstable component $C'$ could not appear.

[Appendix 3]{}

[Compactness with Totally Real Boundary Conditions.]{}

[A3.1. Curves with Boundary on Totally Real Submanifolds.]{}

In this section we consider the behavior of complex curves over an almost complex manifold $(X,J)$ with a boundary on totally real submanifold(s). As in the “interior" case, we need to allow some type of boundary singularity.

Definition A3.1.1.

*The set $\cala^+ \deff \{ (z_1,z_2) \in 
\Delta^2\;:\; z_1 \cdot z_2=0, \im z_1\ge 0, \im z_2\ge 0\}$ is called the [*standard boundary node*]{}. A curve $\barr C$ with boundary $\d C$ is called a [*nodal curve with boundary*]{} if*

$C$ is a nodal curve, possibly not connected;

$\barr C=C\cup \d C$ is connected and compact;

every boundary point $a\in \d C$ has a neighborhood homeomorphic either to the half-disk $\Delta^+ \deff \{ z \in \Delta
\;:\; \im z\ge0 \}$, or to the standard boundary node $\cala^+$.

In the last case $a\in \d C$ is called a [*boundary nodal point*]{}, whereas nodal points of $C$ are called [*interior nodal points*]{}. Both boundary and interior nodal points are simply called nodal points.

Definition A3.1.2. [*Let $(X,J)$ be an almost complex manifold. A pair $(\barr C, u)$ is called a [*curve with boundary over $(X,J)$*]{} if $\barr
C=C\cup \d C$ is a nodal curve with boundary, and $u: \barr C \to (X,J)$ is a continuous $L^{1,2}$-smooth map, which is holomorphic on $C$.* ]{}

A curve $(C,u)$ with boundary is stable if the same condition as in [*Definition 1.5*]{} on the automorphism groups of compact irreducible components is satisfied.

Remark. One can see that $\barr C$ has a uniquely defined real analytic structure such that the normalization $\barr C{}^{\sf nr}$ is a real analytic manifold with a boundary. More precisely, the pre-image of every boundary nodal point $a_i$ consists of two points $a'_i$ and $a''_i$. The normalization map $s: \barr C{}^{\sf nr} \to \barr C$ glues pairs $(a'_i, a''_i)$ of points on $\barr C{}^{\sf nr}$ into nodal points $a_i= s(a'_i)= s(a''_i)$ on $\barr
C$. This implies that the notion of a $L^{1,p}$-smooth map, $p>2$, as well as a continuous $L^{1,p}$-smooth map $u: \barr C \to X$ is well- defined.

Definition A3.1.3.

*We say that a real oriented surface with boundary $(\Sigma, \d\Sigma)$ [*parameterizes*]{} a nodal curve with boundary $C$ if there is a continuous map $\sigma :\barr\Sigma \to \barr C$ such that*

if $a\in C$ is an interior nodal point, then $\gamma_a \deff \sigma\inv
(a)$ is a smooth imbedded circle in $\Sigma$;

if $a\in\d C$ is a boundary nodal point, then $\gamma_a\deff \sigma
\inv (a)$ is a smooth imbedded arc in $\Sigma$ with end points on $\d\Sigma$, transversal to $\d\Sigma$ at this point;

if $a,b \in \barr C$ are distinct (interior or boundary) nodal points, then $\gamma_a\cap \gamma_b= \emptyset$;

$\sigma :\barr\Sigma \bs \bigcup_{i=1}^N\gamma_{a_i}\to \barr C \bs \{
a_1,\ldots ,a_N\} $ is a diffeomorphism, where $a_1,\ldots ,a_N$ are all (interior and boundary) nodal points of $\barr C$.

Recall that a real subspace $W$ of a complex vector space is called [*totally real*]{} if $W \cap \isl W =0$. Similarly, a $C^1$-immersion $f:W \to X$ is called [*totally real*]{} if for any $w\in W$ the image $d f(T_wW)$ is a totally real subspace of $T_{f(w)}X$.

Let $(\barr C, u)$ be a stable curve with boundary over an almost complex manifold $(X,J)$ of a complex dimension $n$.

Definition A3.1.4.

*We say that $(\barr C, u)$ satisfies the [*totally real boundary condition $\mib W$ of type $\bfbeta$*]{} if*

$\bfbeta= \{\beta_i\}$ is a collection of arcs with disjoint interiors, which defines a decomposition of the boundary $\d C= \cup_i \beta_i$; moreover, we assume that every boundary nodal point is an endpoint for four arcs $\beta_i$;

${\mib W} = \{(W_i, f_i)\}$ is a collection of totally real immersions $f_i: W_i \to X$, one for every $\beta_i$;

there are given continuous maps $u^{(b)}_i :\beta_i \to W_i$ [*realizing conditions $\mib W$*]{}, i.e., $f_i \scirc u^{(b)}_i = u|_{\beta_i}$.

Remarks. 1. We shall consider (immersed) totally real submanifolds only of [*maximal real dimension*]{} $n= \dimc X$.

2\. If $\bfbeta$ is a collection of arcs as above, a parameterization $\sigma: \barr \Sigma \to \barr C$ induces a collection of arcs $\sigma\inv(
\bfbeta) \deff \{ \sigma\inv(\beta_i)\;:\; \beta_i \in \bfbeta \} $ with the properties similar to of [*Definition A3.1.4*]{}. Thus, $\sigma\inv(
\beta_i)$ have disjoint interiors, $\cup_i\sigma\inv(\beta_i) = \d\Sigma$, and for any boundary node $a\in \barr C$ every endpoint of the arc $\beta_a=
\sigma\inv(a)$ is an endpoint of two arcs $\sigma\inv(\beta_i)$. Since $\bfbeta$ is completely determined by $\sigma\inv(\bfbeta)$, we shall denote both collections simply by $\bfbeta$ and shall not distinguish them when considering boundary conditions.

[A3.2. A priori Estimates near a Totally Real Boundary.]{}

A totally real boundary condition is a suitable elliptic boundary condition for an elliptic differential operator $\dbar$ of the Cauchy-Riemann type. In particular, all statements about “inner" regularity and convergence for complex curves remain valid near “totally real" boundary. As in the “inner” case, to get some “uniform” estimate at boundary one needs $W$ to be “uniformly totally real”.

Definition A3.2.1.

*Let $X$ be a manifold with a Riemannian metric $h$, $J$ a continuous almost complex structure, $W$ a manifold, and $A_W
\subset W$ a subset. We say that an immersion $f: W \to X$ is [*$h$-uniformly totally real along $A_W$ with a lower angle $\alpha = \alpha(W,
A_W, f) > 0$*]{},*

$df: TW \to TX$ is $h$-uniformly continuous along $A_W$;

for any $w \in A_W$ and any $\xi \not =0 \in T_wW$ the angle $\angle_h\bigl(Jdf(\xi), df(T_wW) \bigr) \ge \alpha$.

We start with an analog for the First A priori Estimate. Define the half-disks $\Delta^+(r) \deff \{ z\in \Delta(r) \;:\; \im z \ge 0 \}$ with $\Delta^+ = \Delta^+(1)$ and $\Delta^- \deff \{ z \in \Delta \; :\; \im z \le
0 \}$. Set $\beta_0 \deff (-1, 1) \subset \d\Delta^+$. Let $X$ be a manifold with a Riemannian metric $h$, $A\subset X$ a subset, $J^*$ a continuous almost complex structure, $f: W \to X$ a totally real immersion and $A_W
\subset W$ a subset.

Lemma A3.2.1.

*Suppose that $J^*$ is $h$-uniformly continuous on $A$ with the uniform continuity modulus $\mu_{J^*}$ and that $f : W \to X$ is $h$-uniformly totally real along $A_W$ with a lower angle $\alpha_f>0$ and the uniform continuity modulus $\mu_f$. Then for every $2< p<\infty $ there exists an $\eps^b_1 =\eps^b_1(\mu_{J^*}, \alpha_f, \mu_f )$ (independent of $p$) and $C_p= C(p, \mu_{J^*}, \alpha_f, \mu_f )$ such that the following holds.*

If $J$ is a continuous almost complex structure on $X$ with $\norm{ J- J^*}
_{L^\infty(A)} <\eps^b_1$, and if $u\in C^0 \cap L^{1,2}(\barr\Delta^+ ,X)$ is a $J$-holomorphic map with $u(\Delta) \subset A$ and with the boundary condition $u|_{\beta_0} =f \scirc u^b$ for some continuous $u^b: \beta_0
\to A_W \subset W$, then the condition $\norm{du}_{ L^2( \Delta^+ )}
<\eps^b_1$ implies the estimate $$\norm{du}_{L^p(\Delta^+({1\over 2}) )}\le C_p\cdot
\norm{du}_{L^2(\Delta^+ )}.\eqno(A3.2.1)$$

Proof. Suppose additionally that $\diam(u(\Delta^+))$ is sufficiently small. Then we may assume that $u(\Delta^+)$ is contained in some chart $U
\subset \cc^n$ such that $\norm{ J- J\st} _{L^\infty(U)}$ is also small enough. Let $z=(z_1, \ldots, z_n)$ be $J\st$-holomorphic coordinates in $U$ such that $u(0)= \{z_i= 0\}$. Making an appropriate diffeomorphism of $U$, we may additionally assume that $W_0 \deff f(W) \cap U$ lies in $\rr^n$ and that $J = J\st$ along $W_0$.

Consider a trivial bundle $E \deff \Delta \times \cc^n$ over $\Delta$ with complex structures $J\st$ and $J_u \deff J \scirc u$. We can consider $u$ as a section of $E$ over $\Delta^+$ satisfying the equation $\dbar_{J_u} u\deff \d_x u+ J_u \d_y u=0$. Over $\beta_0$ we obtain a $J_u$-totally real subbundle $F \deff \beta_0 \times \rr^n$ such that $u(\beta_0) \subset F$. Let $\tau$ denote a complex conjugation in $\Delta$ as well as a complex conjugation in $E$ with respect to $J\st$. Extend $J_u$ on $E|_{\Delta^-}$ as the composition $- \tau\scirc J_u \scirc \tau$. This means that for $z\in Delta^-$ we obtain $$J_u(z): v \mapsto  \tau v \in E_{\tau z}
\mapsto J_u(\tau z)(\tau v) \in E_{\tau z}
\mapsto  -\tau J_u(\tau z)(\tau v) \in E_z .
\eqno(A3.2.2)$$

Since $J_u=J \scirc u$ coincides with $J\st$ along $\beta_0$, this extension is also continuous. Further, for $v\in L^1( \Delta^+, E)$ define the extension $\ex(v)$ by setting $v(z)\deff \tau v(\tau z)$. This gives a continuous linear operator $\ex: L^p( \Delta^+, E) \to L^p( \Delta^+, E)$ for any $1\le p\le \infty$. An important property of operator $\ex$ is that if $v \in L^{1,p} (\Delta^+, E)$ with $1\le p\le \infty$ (resp. $v\in C^0(
\barr \Delta^+ )$) satisfies the boundary condition $v|_{\beta_0} \subset
F$, then $\ex v \in L^{1,p}(\Delta, \cc^n)$ (resp. $\ex v \in C^0 (\barr
\Delta)$). Let us denote by $L^{1,p} (\Delta^+, E,F)$ (resp. by $C^0
(\Delta^+, E,F) $) the corresponding spaces of $v$ with the boundary condition $v|_{\beta_0} \subset F$.

Since for $v\in L^{1,1}(\Delta^+, E)$ holds $\d_x(\tau v) =\tau (\d_x v)$ and $\d_y(\tau v) = - \tau (\d_y v)$, we obtain $\dbar_{J_u} (\ex v) =\ex (\dbar_{J_u} v)$ for any $v \in L^{1,p} (\Delta^+, E,F)$. In particular, for $\ti u \deff \ex u \in C^0 \cap L^{1,2} (\Delta, E)$ we have $\dbar_{J_u} \ti u=0$.

From this point we can repeat the steps of the proof of [*Lemma 2.4.1*]{}.

Remark. We shall refer to the construction of a complex structure $J_u$ in $E$ over $\Delta^-$ and (resp. of a section $\ti u$ of $E$ over $\Delta^-$) as an [*extension of $J \scirc u$ (resp. of $u$) by the reflection principle*]{}.

Let $X$ be a manifold with a Riemannian metric $h$, $J^*$ a continuous almost complex structure on $X$, $A\subset X$ a closed $h$-complete subset such that $J^*$ is $h$-uniformly continuous on $A$, and $f_0 :W \to X$ an immersion, which is $h$-uniformly totally real on some closed $f^*h$-complete subset $A_W\subset W$.

Corollary A3.2.2.

*Let $\{ J_n \}$ be a sequence of continuous almost complex structures on $X$ such that $J_n$ converge $h$-uniformly on $A$ to $J$, $f_n: W \to X$ a sequence of totally real immersion such that $df_n$ converge $h$-uniformly on $A_W$ to $df_0$, and $u_n\in C^0\cap L^{1,2}(
\Delta^+ ,X)$ a sequence of $J_n$-holomorphic maps such that $u_n(\Delta^+)
\subset A$, $\norm{du_n}_{L^2 (\Delta^+ )} \le \eps^b_1$, $u_n(0)$ is bounded in $X$, and $u_n| _{\beta_0} = f_n \scirc u^b_n$ for some continuous $u^b_n
:\beta_0 \to A_W$.*

Then there exists a subsequence $u_{n_k}$ where $L^{1,p}_\loc (\Delta^+
)$-converge to a $J$-holomorphic map $u_\infty$ for all $p < \infty$.

Here $\beta_0 = (-1,1) \subset \d \Delta^+$ and $L^{1,p}_\loc (\Delta^+
)$-convergence means $L^{1,p} (\Delta^+(r) )$-convergence for all $r<1$, i.e., convergence up to boundary component $\beta_0$.

Proof. As a statement itself, the proof of [*Corollary A3.2.2*]{} copies the one of [*Corollary 2.5.1*]{} with appropriate modifications and using the reflection principle.

[A3.3. Long Strips and the Second A priori Estimate.]{}

Consider now an analog of the Second A priori Estimate. An analog of “long cylinders” is now “long strips" satisfying appropriate boundary conditions.

Definition A3.3.1. [*Define a [*strip $\Theta(a,b) \deff (a,b)
\times [0,1]$*]{} with the complex coordinate $\zeta \deff t - i\theta$, $t \in (a,b)$, $ \theta \in [0,1]$. Define also $\Theta_n \deff \Theta(n-1,n)$, $\d_0 \Theta(a,b) \deff (a,b) \times \{0\}$, and $\d_1 \Theta(a,b) \deff (a,b)
\times \{1\}$.* ]{}

We are interested in maps $u: \Theta(a,b) \to X$, which are holomorphic with respect to the complex coordinate $\zeta$ on $\Theta(a,b)$ and a continuous almost complex structure $J$ on $X$, which satisfy the boundary conditions $$u\ogran_{\d_0\Theta(a,b)} = f_0 \scirc u^b_0,
\hskip6em
u\ogran_{\d_1\Theta(a,b)} = f_1 \scirc u^b_1,$$ with some $J$-totally real immersions $f_{0,1} :W_{0,1} \to X$ and continuous maps $u_{0,1} :\d_{0,1}\Theta(a,b) \to W_{0,1}$. First we consider the special linear case.

Lemma A3.3.1. [*Let $W_0$ and $W_1$ be $n$-dimensional totally real subspaces in $\cc^n=(\rr^n,J\st )$. Then there exist a constant $\gamma_W=
\gamma(n, W_0, W_1)$ with $0< \gamma_W <1$ such that for any holomorphic map $u: \Theta(0,3) \to \cc^n$ with the boundary conditions $$u(\d_0 \Theta(0,3)) \subset W_0 \qquad
u(\d_1 \Theta(0,3)) \subset W_1
\eqno(A3.3.1)$$ we have the following estimate: $$\int_{\Theta_2} |du|^2 dt\,d\theta \le \msmall{\gamma_W\over2}
\left( \int_{\Theta_1} |du|^2 dt\,d\theta +
\int_{\Theta_3} |du|^2 dt\,d\theta \right).
\eqno(A3.3.2)$$* ]{}

Proof. Let $L^{1,2}_W([0,1], \cc^n)$ be a Banach manifold of those $v(\theta) \in L^{1,2}([0,1], \cc^n)$, $v(0) \in W_0$ and $v(1) \in W_1$. Consider a nonnegative quadratic form $Q(v) \deff \int_0^1 |\d_\theta v(
\theta)|^2 d\theta$. Since $Q(v) + \norm{v} ^2_{L^2} =\norm{v}^2 _{L^{1,2}}$ and the imbedding $L^{1,2}_W([0,1], \cc^n) \hook L^2([0,1],\cc^n)$ is compact, we can decompose $L^{1,2}_W([0,1], \cc^n)$ into a direct Hilbert sum of eigenspaces $\ee_\lambda$ of $Q$ $\norm{v}^2_{L^2}$. This means that $v_\lambda$ belongs to $\ee_\lambda$ iff for any $w\in L^{1,2}_W([0,1],
\cc^n)$ $$\int_0^1 \<
\d_\theta v_\lambda(\theta), \d_\theta w(\theta) \> d\theta = \int_0^1
\lambda\< v_\lambda(\theta), w(\theta) \> d\theta ,\eqno(A3.3.3)$$ where $\<\cdot,\cdot \>$ denotes a standard [*$\rr$-valued*]{} scalar product in $\cc^n$. Integrating by parts yields $$\int_0^1 \< \d^2_{\theta\theta} v_\lambda(\theta)
+ \lambda v_\lambda(\theta), w(\theta) \> d\theta + \<\d_\theta v_\lambda(
\theta), w(\theta) \>|_{\theta=1} - \<\d_\theta v_\lambda(
\theta), w(\theta) \>|_{\theta=0} =0.$$ This implies that $v_\lambda$ belongs to $\ee_\lambda$ $\d^2_{\theta\theta} v_\lambda(\theta) + \lambda v_\lambda( \theta)=0$, $\d_\theta v_\lambda(1) \perp W_1$, and $\d_\theta v_\lambda(0) \perp W_0$. Since $J\st$ is $\<\cdot, \cdot\>$-orthogonal, we can conclude that $J\d_\theta v_\lambda(\theta) \in \ee_\lambda$.

Positivity and compactness of $Q$ $\norm\cdot_{L^2}$ imply that all $\ee_\lambda$ are finite dimensional and empty for $\lambda<0$. Further, since $\d_\theta v=0$ for any $v\in \ee_0$, $\ee_0$ consists of constant functions with values in $W_0 \cap W_1$.

Now let $u: \Theta(0,3) \to \cc^n$ be a holomorphic map with the boundary condition (5.2). We can represent $u$ in the form $u(t, \theta)= \sum_\lambda
u_\lambda(t,\theta)$ with $u_\lambda(t, \cdot) \deff \pr_\lambda (u(t,\cdot))
\in L^{1,2}([0,3],\ee_\lambda)$. Since $J\d_\theta$ is an endomorphism of every $\ee_\lambda$, every $u_\lambda(t,\theta)$ is also holomorphic. In particular, $u_0$ is also holomorphic and constant in $\theta$. Thus $u_0$ is constant.

Since $u$ is harmonic, i.e., $(\d^2_{tt} + \d^2_{\theta \theta})u =0$, it follows that $\d^2_{tt} u_\lambda(t,\theta)\allowbreak = \lambda u_\lambda(t,\theta)$. For $\lambda > 0$ this yields $u_\lambda(t,\theta) = e^{+\sqrt \lambda t}
v^+_\lambda(\theta) + e^{-\sqrt\lambda t} v^-_\lambda(\theta)$ with $v^\pm
_\lambda(\theta) \in \ee_\lambda$. Fixing an orthogonal $\rr$-basis of $\ee_\lambda $ $v^i_\lambda$, we write every $u_\lambda$ in the form $$u_\lambda(t,\theta)= \sum_i (a^i_\lambda e^{+\sqrt \lambda t} +
b^i_\lambda e^{-\sqrt \lambda t}) v^i_\lambda(\theta)$$ with real constants $a^i_\lambda$, $b^i_\lambda$. Since $u_0(t,\theta)$ is constant, $\norm{du}^2_{L^2(\Theta_k)} = 2\norm{\d_\theta u}^2_{L^2(\Theta_k} = 
\sum_{\lambda,i} 2\lambda \int_{k-1}^k (a^i_\lambda e^{+\sqrt\lambda t}
+b^i_\lambda e^{-\sqrt\lambda t})^2 d\theta $. Here we used $(5.4)$ and $L^2$-orthonormality of $v^i_\lambda $. This leads us to the problem of determining the smallest possible constant $\gamma$ for the inequality $$\int_1^2 (ae^{\alpha t} + be^{-\alpha t})^2 dt
\le \msmall{\gamma\over2} \left(
\int_0^1 (ae^{\alpha t} + be^{-\alpha t})^2 dt +
\int_2^3 (ae^{\alpha t} + be^{-\alpha t})^2 dt \right)
\eqno(A3.3.4)$$ with $a,\,b\in\rr$ for given $\alpha>0$. Integration gives $$a^2 e^{3\alpha}\msmall{ e^\alpha - e^{-\alpha} \over 2\alpha}
+ b^2 e^{-3\alpha}\msmall{ e^\alpha - e^{-\alpha} \over 2\alpha}
+2ab \le$$ $$\le \msmall{\gamma\over2} \left(
a^2 e^{3\alpha}\msmall{ (e^\alpha - e^{-\alpha}) (e^{2\alpha} + e^{-2\alpha})
\over 2\alpha}+
b^2 e^{-3\alpha}\msmall{ (e^\alpha
- e^{-\alpha}) (e^{2\alpha} + e^{-2\alpha})
\over 2\alpha}
+ 4ab \right)$$ or equivalently $$a^2 e^{3\alpha}\msmall{ (e^\alpha - e^{-\alpha})
(e^{2\alpha} + e^{-2\alpha} -2/\gamma ) \over 2\alpha}
+
b^2 e^{-3\alpha}\msmall{ (e^\alpha - e^{-\alpha})
(e^{2\alpha} + e^{-2\alpha} -2/\gamma) \over 2\alpha}$$ $$+ 4ab(1-1/\gamma)
\ge 0$$ The determinant of the last quadratic form in $a,b$ is $$\left(\!\!\msmall{ (e^\alpha - e^{-\alpha})
(e^{2\alpha} + e^{-2\alpha} -2/\gamma )
\over 2\alpha}\!\!\right)^{\!\!2} - 4(1-1/\gamma)^2
=
4\left(\!\!\msmall{ \sh \alpha (\ch\,2\alpha -1/\gamma )
\over \alpha} \!\!\right)^{\!\!2} - 4(1-1/\gamma)^2$$ $$\ge 4(\ch\,2\alpha -1/\gamma )^2 -4(1-1/\gamma)^2 =
4(\ch\,2\alpha -1)(\ch\,2\alpha +1 -2/\gamma).$$ Thus, the inequality (5.5) holds for every $a,b \in\rr$ provided $\gamma \ge
{2 \over 1 + \ch\,2\alpha}<1$.

Note that there exists a minimal positive eigenvalue $\lambda_1>0$ of $Q$. Thus, the estimate (5.3) holds for $\gamma_W \deff {2 \over 1 + \ch(2
\sqrt{\lambda_1} )}<1$.

Remark. A dependence of $\gamma_W$ as a function of $\lambda_1=
\lambda_1(W_0, W_1)$ shows that $\gamma_W<1$ can be chosen the same for all pairs $(\wt W_0,\wt W_1)$ sufficiently close to $(W_0, W_1)$, provided $\dim
(\wt W_0 \cap \wt W_1) = \dim (W_0 \cap W_1)$. Vice versa, if we perform a sufficiently small deformation of $(W_0, W_1)$ into $(\wt W_0, \wt W_1)$ with $\dim (\wt W_0 \cap \wt W_1) < \dim (W_0 \cap W_1)$, then some $v \in
\ee_0(W_0, W_1)$ will wander into an eigenvector $\ti v \not\in \ee_0 (\wt
W_0, \wt W_1) = \wt W_0 \cap \wt W_1$, but with a sufficiently small eigenvalue $\lambda_1(\wt W_0,\wt W_1) >0$, so that the best possible $\gamma_{\wt W}$ will be arbitrarily close to $1$. Thus the uniform separation of $\gamma_W$ from 1 under a small perturbation of $(W_0, W_1)$ is equivalent to a uniform separation of $\lambda_1(W_0, W_1)$ from 0, and is equivalent to stability of $\dim(W_0 \cap W_1)$.

Another phenomenon, also connected with spectral behavior, is that for [*harmonic*]{} $u(t, \theta)$ from $\ee_0= \{\,const\, \}$ does not follow $u_0 = const$, but merely $u_0(t, \theta) = v_0 + v_1 t$ with $v_0,
v_1 \in \ee_0$, so that the inequality (5.3) is not true. This leads to a much more complicated bubbling phenomenon with energy loss for harmonic and harmonic-type maps, compare \[S-U\], \[P-W\], \[Pa\].

Note also that if $W_0$ and $W_1$ are [*affine*]{} totally real subspaces of $\cc^n$, then the inequality (5.3) for holomorphic $u: \Theta(0,3)\to \cc^n$ with boundary condition (5.2) is in general not true. An easy example is a natural imbedding $u:\Theta(0,3) \hook \cc$, with the nonconstant component $u_0 \equiv
u$ and with $\int_{\Theta(0,1)} |du|^2 = \int_{\Theta (1,2)} |du|^2 = \int_{
\Theta(2,3)} |du|^2 \not=0$. In general, those are $n$-dimensional totally real affine planes $W_0$ and $W_1$ in $\cc^n$ with an empty intersection. This can happen if $W_0$ and $W_1$ are parallel or [*skew*]{}. The latter means that the corresponding vector spaces $V_0$ and $V_1$ ($W_i = V_i + w_i$ for some $w_i \in \cc^n$) are different. In both cases the intersection $V_0 \cap
V_1$ is not zero, because otherwise $W_0 \cap W_1 \not= \emptyset$ by dimensional argumentation.

The considerations above show which properties should be controlled in order to obtain a reasonable statement in the nonlinear case.

Definition A3.3.2.

*Let $X$ be a manifold with a Riemannian metric $h$, $f_0 : W_0 \to X$ and $f_1 : W_1 \to X$ immersions and $A_0 \subset
W_0$, $A_1 \subset W_1$ subsets. We say that [*$f_0 : W_0 \to X$ and $f_1
:W_1 \to X$ are $h$-uniformly transversal along $A_0$ and $A_1$ with parameters $\delta >0$ and $M$*]{} if for any $x_0 \in A_0$ and $x_1 \in A_1$ the following holds:*

either $\dist_h(f_0(x_0), f_1(x_1)) > \delta$;

or there exists $x'_i \in A_i$ with $f_0( x'_0) =f_1(x'_1)$ such that $$\dist_h(x_0,x'_0) + \dist_h(x_1,x'_1) \le M\,\dist_h(f_0(x_0), f_1(x_1)).$$

Remark. Roughly speaking, the condition excludes the appearance of points, where $W_0$ and $W_1$ are “asymptotically parallel or skew" and ensures a uniform lower bound for the angle between $W_0$ and $W_1$.

Now let $X$ be a manifold with a Riemannian metric $h$, $J^*$ a continuous almost complex structure on $X$, $f_0 : W_0 \to X$ and $f_1 : W_1 \to X$ immersions, and $A\subset X$, $A_0 \subset W_0$, $A_1 \subset W_1$ subsets. Suppose that $J^*$ is $h$-uniformly continuous on $A$, $df_i: TW_i \to TX$ are $h$-uniformly continuous on $A_i$ and that $f_i$ are $h$-uniformly transversal along $A_i$ with parameters $\delta= \delta(f_0,f_1)>0$ and $M=
M(f_0,f_1)$.

Lemma A3.3.2.

*There exist constants $\eps^b_2 =\eps^b_2 (\mu_{J^*},
\, f_0,\, f_1,\, \delta,\, M) >0$ and $\gamma^b = \gamma^b(\mu_{J^*},\, f_0,\,
f_1,\, \delta,\, M) <1$ such that for any continuous almost complex $\ti J$ with $\norm{\ti J-J^*}_{L^\infty(A)}<\eps_2^b$, any immersions $\ti f_i :W_i
\to X$ with $\dist(\ti f_i, f_i)_{C^1(A_i)} <\eps_2^b$, and any $\ti J
$-holomorphic map $u \in C^0 \cap L^{1,2}( \Theta(0,5), X)$ with $u(
\Theta(0,5)) \subset A$, $u|_{\d_i\Theta(0,5)} = f_i \scirc u^b_i$ for some continuous $u^b_i : \d_i\Theta(0,5) \to A_i \subset W_i$ the conditions*

$\norm{ du}_{ L^2(\Theta_i)} <\eps_2^b$;

$\ti f_i: W_i \to X$ are $h$-uniformly transversal along $A_i$ with the same parameters $\delta$ and $M$

imply the estimate $$\norm{du}^2_{L^2(\Theta_3)}\le \msmall{\gamma^b \over 2} \cdot
\left( \norm{du}^2_{L^2(\Theta_2)} +  \norm{du}^2_{L^2(\Theta_4)} \right)
\eqno(A3.3.5).$$

Proof. Suppose the statement of the lemma is false. Then there should exist a sequence of continuous almost complex structures $J_k$ with $\norm{
J_k- J^*} _{L^\infty (A)} \lrar 0$, a sequence of immersions $f_{k,i}: W_i
\to X$ with $f_{k,i} \lrar f_i$ in $C^1(A_i)$ such that $f_{k,i} :W_i \to X$ are $h$-uniformly transversal with the same parameters $\delta$ and $M$, and a sequence of $J_k$-holomorphic maps $u_k \in C^0\cap L^{1,2}(\Theta(0,5),
X)$ with $u_k(\Theta(0,5)) \subset A$ and $u_k |_{\d_i\Theta(0,5)} = f_{k,i}
\scirc u^b_{n,i}$ for some continuous $u^b_{n,i} : \d_i\Theta(0,5) \to A_i
\subset W_i$ such that $\norm{du_k}^2_{L^2(\Theta(0,5))} \lrar 0$ and $$\norm{du_k}^2_{L^2(\Theta_3)}\ge \msmall{\gamma_k \over 2} \cdot
\left( \norm{du_k}^2_{L^2(\Theta_2)} +  \norm{du_k}^2_{L^2(\Theta_4)} \right)
\eqno(A3.3.6)$$ with $\gamma_k = 1- 1/k$. [*Lemmas 5.3.1*]{} and [*A3.2.1*]{} provide that in this case

$\diam_h(u_k( \Theta(1,4))) \allowbreak \lrar 0$.

Since $f_{k,i} :W_i \to X$ are $h$-uniformly transversal with the same parameters $\delta$ and $M$, there should exist sequences $x_k \in A$, $x_{k,
0} \in A_0$, and $x_{k,1} \in A_1$ such that $x_k = u_0(x_{k,0}) =u_1(x_{k,
1})$ and $u_k(\Theta(1,4)) \subset B(x_k, r_k)$ with $r_k \lrar 0$. The $h
$-uniform continuity of $J^*$ implies that there exist $C^1$-diffeomorphisms $\phi_k: B(x_k, r_k) \to B(0, r_k) \subset \cc^n$ with $\norm{J_k - \phi^*_k
J\st}_{L^\infty(B(x_k, r_k))} + \norm{h -\phi^*_k h\st} _{L^\infty(B(x_k,
r_k))} \lrar 0$.

Using $\phi_k$, we transfer our situation into $B(0, r_k) \subset \cc^n$ and rescale it. Namely, we set $\alpha_k \deff \norm{du_k}_{L^2(\Theta_3)}$ and define diffeomorphisms $\psi_k \deff {1\over \alpha_k} \scirc \phi_k: B(x_k,
r_k) \to B(0, R_k) \subset \cc^n$ with $R_k \deff \alpha_k\inv \cdot r_k$. Note that by [*Lemmas 5.3.1*]{} and [*A3.2.1*]{} we have $\alpha_k = 
\norm{du_k} _{L^2(\Theta(2,3))} \le C \diam_h(u_k( \Theta(1,4))) \le C' r_k$, so that $R_k$ are uniformly bounded from below.

In $B(0, R_k)$ we consider Riemannian metrics $h_k \deff \alpha^{-2}_k \cdot
\psi_{k\,*}h_k$ (pushed forward and $\alpha^{-2}_k$-rescaled $h_k$), almost complex structures $J^*_k \deff \psi_{k\,*} J_k$ and $J^*_k
$-holomorphic maps $u^*_k \deff \psi_k \scirc u_k : \Theta(1,4) \to
B(0,R_k)$. Note that here we consider $h$ as a metric tensor; thus, multiplying $h$ by $\alpha^{-2}$, we increase $h$-norms and $h$-distances in $\alpha^{-1}$ and not in $\alpha^{-2}$ times.

Then $\norm{du^*_k}_{L^2(\Theta_3, h_k)}=1$, $\norm{du^*_k}^2_{L^2(\Theta_2,
h_k)} + \norm{du^*_k}^2_{L^2(\Theta_4, h_k)} \le {2k\over k-1}$, and $$\norm{J^*_k - J\st}_{L^\infty(B(0, R_k), h_k)} =
\norm{J_k - \phi^*_kJ\st}_{L^\infty(B(x_k, r_k), h)} \lrar 0.$$ The last equality uses the obvious relation $$\msmall{|F(\xi)|_{\alpha^{-2} \cdot h} \over |\xi|_{\alpha^{-2} \cdot h} }
= \msmall{\alpha\inv \cdot |F(\xi)|_h \over \alpha\inv \cdot |\xi|_\cdot h }
= \msmall{|F(\xi)|_h \over |\xi|_h }$$ for any linear $F: T_xX \to T_xX$ and $\xi\not = 0\in T_xX$. In a similar way we also obtain $\norm{h_k - h\st}_{L^\infty(B(0, R_k), h_k)}\lrar 0$.

Going to a subsequence, we may additionally assume that the tangent spaces $d\psi_k \scirc df_i (T_{x_{k,i}} W_i)$, $i=1,2$, converge at some spaces $W_i^* \subset \cc^n$. Since $W_i$ are uniformly totally real, $W_i^*$ are also totally real linear subspaces in $\cc^n$. Since the maps $df_i : TW_0
\to TX$ are uniformly continuous on $A_i \subset W_i$, $f_{k,i} \lrar f_i$ in $C^1(A_i)$, and since $r_k \lrar 0$, the images $W^*_{k,i} \deff \psi_k
\scirc f_{k,i}(B_{W_i}(x_{k,i}, r_k))$ of the balls $B_{W_i}(x_{k,i}, r_k)
\subset W_i$ are imbedded submanifolds of $\cc^n$ with $0 \in W^*_{k,i}$ which converge to $W^*_i$ in Hausdorff topology. Moreover, we can consider $W^*_{k,i}$ as graphs of maps $g_{k,i}$ from subdomains $U_{k,i} \subset
W^*_i \cap B(0,R_k)$ to $W^*_i {}^\perp$ and for any fixed $R \le \inf
\{R_k\}$ the restrictions $g_{k,i} |_{W^*_i \cap B(0,R)}$ converge to zero map from $W^*_i \cap B(0,R)$ to $W^*_i {}^\perp$.

The a priori estimates for the maps $u^*_k : \Theta(1,4) \to \cc^n$ provide that for any $p<\infty$ the maps $u^*_k$ converge in weak- $L^{1,p}$-topology to some $J\st$-holomorphic map $u^*: \Theta(1,4) \to \cc^n$. Furthermore, since $u^*_k$ satisfy totally real boundary conditions $u^*_k|_{\d_i\Theta(1,4)}
\subset W^*_{k,i}$, the same is true for $u^*$, i.e., $u^*|_{\d_i\Theta(1,4)}
\subset W^*_i$. Nice behavior of $W^*_{k,i}$ provides that on $\Theta_3$ we also have a strong convergence; hence $\norm{du^*}_{L^2(\Theta_3)} = \lim
\norm{du^*_k}_{L^2(\Theta_3)} =1$. In particular, $u^*$ is not constant. On the other hand, $\norm{du^*}^2_{L^2(\Theta_2)} + \norm{du^*}^2_{L^2(\Theta_4)}
\le \lim \norm{du^*_k}^2_{L^2(\Theta_2)} + \norm{du^*_k}^2_{L^2(\Theta_4)} \le
2$. The obtained contradiction to [*Lemma A3.3.1*]{} shows that [*Lemma A3.3.2*]{} is true.

Let $X$, $h$, $J$, $A$, $f_i: W_i \to X$, $A_i$, and the constant $\eps^b_2$ and $\gamma^b$ be as in [*Lemma A3.3.2*]{}. Suppose that $\ti J$ is a continuous almost complex structure on $X$ with $\norm{\ti J-J}_{L^\infty(A)} <\eps^b_2$, and $\ti f_i: W_i \to X$ are totally real immersions with $\dist(\ti f_i, f_i)
_{C^1(A_i)} \le \eps^b_2$ such that $f_i$ are $h$-uniformly transversal along $A_i$ with the same parameters $\delta$ and $M$ as $f_i: W_i \to X$.

Corollary A3.3.3.

*Let $u \in C^0\cap L^{1,2}(\Theta(0,l),X)$ be a $\ti J$-holomorphic map such that $u(\Theta(0,l))\subset A$, $u|_{\d_i
\Theta(0,l)} = \ti f_i \scirc u^b_i$ for some continuous $u^b_i: \d_i\Theta(
0,l) \to A_i \subset W_i$ and $\norm{du}_{ L^2(Z_k)}<\eps_2$ for any $k=1,\ldots,l$.*

Let $\lambda_b>1$ be the (uniquely defined) real number with $\lambda_b 
= {\gamma^b \over 2}(\lambda_b^2+ 1)$. Then for $2\le k\le l-1$ the following holds: $$\norm{du}^2_{L^2(\Theta_k)}
\le \lambda_b^{-(k-2)} \cdot \norm{du}^2_{L^2(\Theta_2)} +
\lambda_b^{-(l-1-k)} \cdot \norm{du}^2_{L^2(\Theta_{n-1})}.
\eqno(A3.3.7)$$

Proof. It is the same as in [*Lemma 5.3.5*]{}.

The immediate corollary of this estimate is a lower bound of energy on a nonconstant “infinite strip”.

Lemma A3.3.4. [*Let $X$, $h$, $J$, $A$, $f_i: W_i \to X$, $A_i$, the constant $\eps^b_2$ and $\gamma^b$, also a structure $\ti J$, and immersions $\ti f_i : W_i \to X$ be the same as in [*Corollary A3.3.3*]{}. Let $u \in C^0\cap L^{1,2}(\Theta(-\infty, +\infty),X)$ be a nonconstant $\ti
J$-holomorphic map such that $u(\Theta(-\infty, +\infty))\subset A$ and $u|_{\d_i\Theta(0,l)} = \ti f_i \scirc u^b_i$ for some continuous $u^b_i :
\d_i\Theta(-\infty, +\infty) \to A_i \subset W_i$. Then $\norm{du}_{ L^2(
\Theta_k)} > \eps^b_2$ for some $k$. In particular, $\norm{du}_{ L^2(
\Theta(-\infty, +\infty))} > \eps^b_2$.* ]{}

Proof. [*Corollary A3.3.3*]{} provides that if $\norm{du}_{L^2(
\Theta_k)} \le \eps^b_2$ for all $k$, then $\norm{du}_{ L^2(\Theta_k)} 
\allowbreak =0$, i.e., $u$ is constant.

Another consequence of [*Corollary A3.3.3*]{} is a generalization of Gromov’s result about removability of boundary point singularity, see \[G\]. An important improvement is the fact that the statement remains valid also when one has [*different*]{} boundary conditions on the left and on the right of a singular point. One can see such a point $x$ as a [*corner point*]{} for the corresponding complex curve. A typical example appears in symplectic geometry where one takes Lagrangian submanifolds as boundary conditions.

Define the punctured half-disk by setting $\check\Delta^+ \deff \Delta^+\bs
\{0\}$. Define $I_- \deff (-1,0) \subset \d\check\Delta^+$ and $I_+ \deff (0,+1) \subset \d\check\Delta^+$.

Corollary A3.3.5.

*(Removal of boundary point singularities). *Let $X$ be a manifold with a Riemannian metric $h$, $J$ a continuous almost complex structure, $f_i : W_i \to X$, $i=1,2$, totally real immersions and $A_i \subset W_i$ subsets. Let $u:(\check\Delta^+, J\st)
\to (X,J)$ be a holomorphic map. Suppose that**

$J$ is uniformly continuous on $A \deff u(\check\Delta)$ $h$, and closure of $A$ is $h$-complete;

$u$ satisfies boundary conditions of the form $u|_{I_+} = f_0
\scirc u^b_+$ and $u|_{I_-} = f_1 \scirc u^b_-$ with some continuous $u^b_+ :
I_+ \to A_0 \subset W_0$ and $u^b_- : I_- \to A_1 \subset W_1$;

$f_i$ are $h$-uniformly totally real on $A_i$ and $h$-uniformly transversal along $A_i$;

there exists $k_0$ such that for all half-annuli $R^+_k\deff \{
z\in \Delta^+ :{1\over e^{\pi(k+1)}}\le | z| \le {1\over e^{\pi k}}\}$ with $k\ge k_0$ one has $\norm{du}^2_{L^2(R^+_k)}\le \eps^b_2$, $\eps^b_2$ as in [*Lemma A3.3.2*]{}.

Then $u$ extends to origin $0\in \Delta^+$ as an $L^{1,p}$-map for some $p>2$.

Proof. Using the holomorphic map $\exp : \Theta(0,\infty) \to
\check\Delta^+$, $\exp(\theta + \isl t) \deff e^{\pi(-t + \isl\theta)}$, we can reduce our situation to the case of the holomorphic map $u^* \deff u
\scirc \exp$ form “infinite strip” $\Theta(0,\infty)$. By [*Corollary A3.3.3*]{}, for $k\ge k_0$ we obtain the estimate $\norm{ du^*} _{L^2(\Theta_k)} \le
\lambda_b^{-(k-k_0)/2} \norm{ du^*}_{L^2(\Theta_{k_0})}$ with some $\lambda_b
>1$. This is equivalent to the estimate $\norm{ du} _{L^2(R^+_k)} \le
\lambda_b^{-(k-k_0)/2} \norm{ du}_{L^2(R^+_{k_0})}$. [*Lemmas 5.3.1*]{} and [*A3.2.1*]{} and the scaling property of $L^p$-norms provide the estimate $$\norm{ du}_{L^p(R^+_k)} \le C e^{-k(\log\lambda_b/2 + \pi(2/p-1))}.\eqno(A3.3.8)$$ Thus, $du \in L^p(\Delta^+)$ for any $p$ with $\log\lambda_b/2 > \pi(1-2/p)$, which means $p < {4\pi \over 2\pi - \log \lambda_b}\cdot$

Remark. Unlike the “inner" and smooth boundary cases, it is possible that the map $u$, as in [*Corollary A3.3.5*]{}, is not $L^{1,p}$-regular in the neighborhood of a “corner point” $0\in \Delta^+$ for some $p>2$. For example, the map $u(z) = z^\alpha$ with $0<\alpha <1$ satisfies totally real boundary conditions $u(I_+) \subset \rr$, $u(I_-) \subset e^{\alpha\pi \isl}\rr$ and is $L^{1,p}$-regular only for $p< p^* \deff {2\over1-\alpha}\cdot$

As in the “inner” case, for the proof of the boundary compactness theorem we need a description of a convergence of a “infinitely long strip”. Let $X$ be a manifold with a Riemannian metric $h$, $J$ a continuous almost complex structure, $A\subset X$ a closed $h$-complete subset such that $J$ is $h$-uniformly continuous on $A$, and let $\{J_n\}$ be a sequence of almost complex structures converging $h$-uniformly on $A$ to $J$. Also let $f_0: W_0
\to X$ and $f_1: W_1\to X$ be immersions, $A_i \subset W_i$ subsets such that $df_i$ are uniformly $h$-uniformly totally real on $A_i$ and $f_i$ are $h$-uniformly transversal along $A_i$. Let $f_{n,i} : W_i \to X$ be totally real immersions, which $C^1$-converge to $f_i$ on $A_i$ such that $f_{n,0}$ and $f_{n,1}$ are $h$-uniformly transversal along $A_i$ with uniform in $n$ parameters $\delta$ and $C^*$. Finally, let $\{l_n\}$ be a sequence of integers with $l_n\to \infty$, and $u_n: \Theta(0,l_n) \to X$ a sequence of $J_n$-holomorphic maps, satisfying boundary conditions $u_n|_{\d_i\Theta(0,
l_n)} = f_{n,i}\scirc u^b_{n,i}$ with some continuous $u^b_{n,i}:
\d_i\Theta(0,l_n) \to A_i \subset W_i$.

Lemma A3.3.6.

*In the described situation, suppose additionally that $u_n(\Theta(0,l_n)) \allowbreak \subset A$ and $\norm{du_n} _{ L^2( 
\Theta_k) }\le \eps^b_2$ for all $n$ and $k\le l_n$. Take a sequence $k_n\to
\infty$ such that $k_n<l_n-k_n\to \infty$. Then*

[*1)*]{} $\norm{du_n}_{L^2(\Theta(k_n,l_n-k_n))}\to 0$ and $\diam\bigl(u_n (\Theta(k_n,l_n-k_n))\bigr)\to 0$.

[*2)*]{} There is a subsequence $\{ u_n \}$, still denoted $\{ u_n \}$ such that both $u_n |_{\Theta(0,k_n)}$ and $u_n |_{\Theta(k_n,l_n)}$ converge in $L^{1,p}
$-topology on compact subsets in $\check\Delta^+ \cong \Theta(0, +\infty)$ to $J^*$-holomorphic maps $u^-_\infty$ and $u^+_\infty$. Moreover, both $u^+_\infty$ and $u^-_\infty$ extend to the origin and $u^+_\infty(0)=
u^-_\infty(0)$.

Proof. It follows from the above considerations.

[A3.4. Gromov Compactness for Curves with Totally Real Boundary Conditions.]{}

Let us turn to the Gromov Compactness Theorem for curves with boundary on totally real submanifolds. To give a precise statement we need to modify the definition of the Gromov convergence ([*Definition 4.1.6*]{}). The reason to do this is the following. Considering [*open*]{} curves $C_n$ with changing complex structures, we want to fix some kind of a common “neighborhood of infinity" $i_n: C^*\hookrightarrow C_n$ of every $C_n$. Thus, we can imagine that all changes of complex structure take place “outside of infinity”, i.e., in a relatively compact part $C_n \bs i_n(C^*) \Subset C_n$. This is done to insure that $C_n$ do not approach infinity in an appropriate moduli space.

On the other hand, it is more natural to consider curves $(\barr C_n, u_n)$ with totally real boundary conditions as compact objects without “infinity”. In fact, in this case the behavior of $u_n$ near the boundary $\d C_n$ can be controlled. The obtained a priori estimates near a “totally real boundary” can be viewed as a part of such a “control”. So for curves with totally real boundary conditions we can hope to extend the Gromov convergence up to the boundary.

Further, as in the “inner case”, an appropriate modification of the Gromov convergence in this case should allow boundary bubbling and the appearance of boundary nodes. This means, however, that the structure of the boundary can change during the approach to the limit curve and cannot be considered as fixed. Instead, one should fix a type of boundary conditions. We shall consider the following general situation.

Let $u_n: \barr C_n \to X$ be a sequence of stable $J_n$-complex curves over $X$ with parameterizations $\delta_n: \barr\Sigma \to \barr C_n$. Also let $\bfbeta =\{ \beta_i \}_{i=1}^m$ be a collection of arcs $\beta_i$ in $\d
\Sigma$ such that $\cup_{i=1}^m \beta_i= \d\Sigma$ and that the interiors of $\beta_i$ are mutually disjoint and do not intersect pre-images of boundary nodal points of $C_n$. Let further $\{ W_i\} _{i=1}^m$ be a collection of real $n$-dimensional manifolds, $f_{n,i} : W_i \to X$ a sequence of totally real immersions and $u^b_{n,i}:
\beta_{n,i} \to W_i$ a sequence of continuous maps from $\beta_{n,i} \deff
\delta_n( \beta_i)$. Then ${\mib W}_n \deff \{ (W_i,
f_{n,i} )\} _{i=1}^m$ are totally real boundary conditions on $(\barr C_n,
u_n)$ of the same type $\bfbeta$.

Definition A3.4.1.

*In the situation above we say that the sequence of boundary conditions ${\mib W}_n$ of the same type $\bfbeta$ [*converges $h$-uniformly transversally*]{} to $J^*$-totally real boundary conditions ${\mib W}$ on subsets $A_i \subset W_i$ if*

${\mib W}= \{ (W_i, f_i) \}_{i=1}^m$, where $f_i: W_i \to X$ are $J^*$-totally real immersions;

$f_{n,i}$ converge to $f_i$ in $C^1$-topology and this convergence is $h$-uniform on $A_i$;

for any $n$ immersions $\{f_{n,i}\}_{i=1}^m$ are mutually $h$-uniformly transversal along $A_i$ with parameters $\delta>0$ and $M$, and these parameters are independent of $n$.

Note that condition implies that the limit immersions $f_i$ are also mutually $h$-uniformly transversal along $A_i$ with the same parameters $\delta>0$ and $M$.

Definition A3.4.2.

*We say that the sequence $(\barr C_n, u_n)$ [*converges up to the boundary*]{} to a stable $J^*$-holomorphic curve $(\barr
C_\infty, u_\infty)$ over $X$ if the parameterizations $\sigma_n: \barr\Sigma
\to \barr C_n$ and $\sigma_\infty: \barr\Sigma \to \barr C_\infty$ can be chosen in such a way that the following holds:*

$u_n\scirc \sigma_n$ converges to $u_\infty\scirc \sigma_\infty$ in $C^0( \barr\Sigma, X)$-topology;

if $\{ a_k \}$ is the set of the nodes of $C_\infty$ and $\{ \gamma_k
\}$, $\gamma_k \deff \sigma_\infty\inv(a_k)$ are the corresponding circles and arcs in $\barr\Sigma$, then on any compact subset $K\comp \barr\Sigma \bs
\cup_k\gamma_k$ the convergence $u_n\scirc \sigma_n\to u_\infty\scirc
\sigma_\infty$ is $L^{1,p}(K, X)$ for all $p< \infty$;

for any compact subset $K\comp \barr\Sigma \bs \cup_k\gamma_k$ there exists $n_0=n_0(K)$ such that $ \sigma_n^{-1}(\{ a_k \}) \cap K= \emptyset$ for all $n\ge n_0$ and complex structures $\sigma_n^*j_{C_n}$ smoothly converge to $\sigma_\infty^*j_{C_\infty}$ on $K$.

Theorem A3.4.1.

*Fix a metric $h$ on $X$, and an $h$-complete subset $A\subset X$, and subsets $A_i \subset W_i$. Suppose that*

[*a)*]{} $J_n$ are continuous almost complex structures on $X$, converging $h$-uniformly on $A$ to a continuous almost complex structure $J^*$;

[*b)*]{} $u_n(C_n) \subset A$ and $\area [u_n (C_n)]\le M$ with a constant $M$ independent of $n$;

[*c)*]{} ${\mib W}_n \deff \{(W_i, f_{n,i}) \}_{i=1}^m$ are totally real boundary conditions of the same type $\bfbeta = \{ \beta_i \}_{i=1}^m$ such that ${\mib W}_n$ converge $h$-uniformly transversally to a boundary condition ${\mib W}= \{(W_i, f_i) \}_{i=1}^m$ on subsets $A_i \subset W_i$;

[*d)*]{} immersions $f_i: W_i \to (X,J^*)$ are $h$-uniformly totally real along $A_i$;

[*e)*]{} there exist maps $u^b_{i,n}: \beta_i \to  A_i\subset W_i$, realizing boundary conditions ${\mib W}_n$.

Then there exits a subsequence of $\{( \barr C_n, u_n )\}$, still denoted $\{( \barr C_n, u_n )\}$, and parameterizations $\sigma_n: \barr \Sigma
\to \barr C_n$ such that $(C_n, u_n, \sigma_n)$ converges up to the boundary to a stable $J^*$-holomorphic curve $(\barr C_\infty, u_\infty, \sigma_\infty)$ over $X$.

If, in addition, $A_i \subset W_i$ are $f_i^*h$-complete, then the limit curve $(\barr C_\infty, u_\infty)$ satisfies real boundary conditions $\mib
W$ with maps $u^b_i: \beta_i \to A_i \subset W_i$.

Our main idea of the proof is to apply arguments used in the demonstration of [*Theorem 1.1*]{}. To realize this, we use the following trick. We replace every pair $(C_n, u_n)$ by a triple $(C^d_n, \tau_n, u^d_n)$, where $C^d_n$ is the [*Schottky double*]{} of $C_n$ with an antiholomorphic involution $\tau_n$ and $u^d_n: C_n^d \to X$ a $\tau_n$-invariant map. Then we shall show the changes of all the constructions in the proof to make them $\tau_n$-invariant in an appropriate sense. In particular, the convergence $(C^d_n, \tau_n, u^d_n) \lrar (C^d_\infty, \tau_\infty, u^d_\infty)$ will be equivalent to the convergence $(C_n, u_n) \lrar (C_\infty, u_\infty)$.

We start with a construction of the Schottky double of a nodal curve $\barr C$ with boundary. Take two copies $\barr C^+ \equiv \barr C$ and $\barr C^-$ of $\barr C$. Equip $\barr C^-$ with the opposite complex structure, so that the identity map $\tau: \barr C^+ \to \barr C^-$ now becomes antiholomorphic. Glue $\barr C^+$ and $\barr C^-$ together along their boundaries, identifying $\d C^+$ and $\d C^-$ by means of the identity map $\tau: \d C^+ \buildrel
\cong \over \lrar \d C^-$. The union $C^d \deff \barr C^+ \cup_{\d C} \barr
C^-$ obeys the unique structure of a closed nodal curve compatible with imbeddings $\barr C^\pm \hook C^d$. The boundary $\d C$ becomes the fixed point set of $\tau$.

The map $\tau$ induces an antiholomorphic involution of $C^d$ which we also denote by $\tau$. We call the obtained curve $C^d$ the [*Schottky double*]{} of $\barr C$. Note that every boundary nodal point $a_i\in \d C$ defines a $\tau
$-invariant nodal point $a_i$ on $C^d$, whereas an inner nodal point $b_i \in
C$ defines a pair of nodal points $b_i^\pm$ on $C^d$ interchanged by $\tau$. If $\sigma: \barr\Sigma \to \barr C$ is a parameterization of $\barr C$, then we obtain in an obvious way the double $\Sigma^d$ with the involution $\tau:
\Sigma^d \to \Sigma^d$ and the parameterization $\sigma^d: \Sigma^d \to C^d$ compatible with the involutions.

Remark. The introduced notation $C^d$ for the [*Schottky double*]{} of a nodal curve $\barr C$ with boundary coincides with that for the [*holomorphic double*]{}, used in [*Section 2*]{}. Since in the present section only the Schottky double is considered, this should not lead to confusion.

Suppose, additionally, that an almost complex structure $J$ on $X$ and a $J$-holomorphic map $u: \barr C \to X$ are given. Suppose, also, that the curve $(\barr C,u)$ satisfies the totally real boundary conditions $\mib W$ of type $\bfbeta$. In particular, $\bfbeta$ defines a certain system of arcs $\{\beta_i\}$ on $\d C$. In order to take into account the type of boundary conditions, we fix the ends of $\beta_i$ which are not boundary nodal points of $\barr C$ and declare these points as marked points of $C^d$. Note that these and the nodal points are the only “corner” points of $(\barr C,
u)$. The latter means that in a neighborhood of these points the map $u$ cannot be $L^{1,p}$-smooth for all $p< \infty$. The example in the [*Remark*]{} following [*Corollary A3.3.5*]{} explains the notion of a “corner point”. Considering the Schottky double, we shall always equip $C^d$ with this set of marking points. Note also that every boundary circle of $\barr C$ contains at least one nodal or marked point as above.

For $(\barr C,u)$ as above, we extend the $J$-holomorphic map $u: \barr C \to
X$ to a map $u^d: C^d \to X$ by setting $u^d(x) \deff u(\tau(x))$ for $x\in
C^-$. By the construction, $u^d$ is $\tau$-invariant, $u^d \scirc \tau= u^d$, but $u^d$ [*is not $J$-holomorphic*]{} (with the only trivial exception $u
\equiv const$). However, the analysis already described in this section provides necessary $L^{1,p} $-estimates for $u^d$, at least for some $p^*>2$.

In the situation of [*Theorem A3.4.1*]{}, such an exponent $p^*>2$ can be chosen to be the same for all curves $(\barr C_n, u_n)$. This depends only on the topology of $\barr C_n$ and the geometry of immersions $f_n: W_n \to X$. In particular, every $u_n^d$ is continuous.

The next step of the proof is to find a $\tau_n$-invariant decomposition of $C^d_n$ into pants. This implies that the corresponding graph $\Gamma_n$ becomes $\tau_n$-invariant. In the construction which follows we shall use the fact that $\tau_n$ is an isometry on the union of the non-exceptional components of $C^d_n$. This is provided by the uniqueness of the intrinsic metric.

Lemma A3.4.2. *Let $C$ be a nodal curve with boundary, $\sigma: \barr
\Sigma \to \barr C$ a parameterization, and $\{ x_i \}_{i=1}^m$ a set of marked points on the boundary $\d C$. Let $C^d$ be the Schottky double of $C$ with the antiholomorphic involution $\tau$.*

Then there exists a $\tau$-invariant decomposition of $C^d\bs \mapo$ into pants such that the intrinsic length of corresponding boundary circles is bounded by a constant $l^+$ depending only on genus $g$ of $\Sigma^d$ and the number of marked points $m$.

Moreover, every [*short*]{} geodesic appears as a boundary circle of some pants of the decomposition.

Remark. Recall (see [*Remark*]{} on page 25) that a closed geodesic $\gamma$ is called [*short*]{} if $\ell(\gamma) < l^*$, where $l^*$ is the universal constant $l^*$ with the following property. For any simple closed geodesics $\gamma'$ and $\gamma''$ on the conditions $\ell(\gamma') <l^*$ and $\ell( \gamma'') <l^*$ imply $\gamma' \cap \gamma'' = \emptyset$.

Proof. Since the genus of the parameterizing real surface $\Sigma^d$ and the number of marked points is fixed, we obtain a uniform upper bound on the possible genera and the number of marked points of non-exceptional components of $C^d$, as well as on the number of exceptional components. This implies that there exists a decomposition of every non-exceptional component $C_i$ of $C^d$ into pants $S_\alpha$ such that the intrinsic length of boundary circles of $S_\alpha$ is bounded by the constant $l^+$ depending only on $g$ and $m$. The idea of the proof of our lemma is to show that the construction of such a decomposition, given in \[Ab\], Ch.II, §3.3, can be modified to produce a $\tau$-invariant decomposition.

Let us first describe the construction itself, say, for a given smooth curve $C^*$ with marked points $\{x_i\}$ of non-exceptional type. The procedure is done inductively by choosing at every step a non-trivial simple closed geodesic $\gamma_{J^*} \subset C^* \bs\mapo$, disjoint from an already chosen geodesic $\gamma_j$, $j<J^*$. Moreover, at every step there exists a geodesic $\gamma_{J^*}$ as above whose intrinsic length is bounded by a constant $l^+_{J^*}$ depending only on the genus of $C^*$, the number of marked points, and the maximum of the lengths of the already chosen geodesics $\gamma_j$, $j<J^*$.

Take any non-exceptional component $C^d_i$ of $C^d$. Two cases can occur: either $C^d_i$ is $\tau$-invariant, or $\tau \bigl( C^d_i \bigr)$ is another component $C^d_{i'}$. Two separate cases are distinguished: $C^d_i$ intersects the boundary $\d C$ (first case) or not (second one).

The existence of $\tau$-invariant decomposition into pants for every pair of non-exceptional components $C^d_i$ and $\tau(C^d_i) \not= C^d_i$ is obvious. We choose an appropriate decomposition of $C^d_i$ and transfer it on $\tau(
C^d_i)$ by means of $\tau$.

It remains to consider the case of a $\tau$-invariant non-exceptional component $C^d_i$.

Suppose that at some step we have already chosen a $\tau$-invariant set $\{\gamma_1, \ldots, \allowbreak \gamma_{J^*-1}\}$ of simple disjoint geodesics on $C^d_i \bs \mapo$. Take a simple geodesic $\gamma$ of the length $\ell(\gamma) \le l^+_{J^*}$, where $l^+_{J^*}$ is the upper bound introduced above. By the construction of the double $C^d$, the fixed point set of $\tau$ on $C^d_i$ is $C^d_i \cap \d C$ and is non-empty. Denote $C_i \deff C \cap
C^d_i$, so that $C^d_i \cap \d C= \d C_i$. Note that any boundary circle of $C_i$ contains at least one marked point of $C^d_i$. Consequently, it has an infinite length the intrinsic metric on $C^d_i \bs\mapo$. Thus the chosen geodesic $\gamma$ cannot lie on $\d C_i$. Only three cases can happen.

[*Case 1.*]{} $\gamma$ is disjoint from $\d C_i$. Then $\gamma$ lies either in $C_i$ or in $\tau (C_i)$. In any case, $\gamma\cap \tau(\gamma) = \emptyset$. Thus, we can set $\gamma_{J^*} = \gamma$ and $\gamma_{J^*+1}= \tau(\gamma)$, obtaining the $\tau$-invariant set $\{\gamma_1, \ldots, \gamma _{J^*+1}\}$ of simple disjoint geodesics. This will be discussed in the next two steps of our construction.

[*Case 2.*]{} $\gamma \cap \d C_i \not=0$ and $\gamma$ is $\tau$-invariant. We set $\gamma_{J^*}=\gamma$ and proceed inductively. Note that in this case $\gamma \cap \d C_i$ consists of 2 points, in which $\gamma$ is orthogonal to $\d C_i$.

The last expression means that we move along corresponding arcs in the prescribed order, as it is shown on Fig. 11. Note that only one part of $C^d_i$ is drawn, namely $C_i$. The rest of the picture is symmetric the involution $\tau$. Thus we can see only half of the geodesics in classes $[\ti\gamma_i]$, $i=2,3,4$.

Each of the classes $[\ti\gamma_k]$ is either represented by a closed geodesic or corresponds to a curve which winds around some marked point of $C^d_i$. To shorten notation, we say in the last case that the class $[\ti\gamma_i]$ corresponds to a marked point of $C^d_i$.

If one of the classes $[\ti\gamma_k]$, $k=1,2,3$, is represented by the geodesic $\ti\gamma_k$, which is different and disjoint from the already chosen geodesics $\gamma_j$, $j<J^*$, then we can set $\gamma_{J^*} =
\ti\gamma_k$. If $k=1$ we also set $\gamma_{J^*} = \ti\gamma_1$ and $\gamma_{J^*+1} = \tau(\ti\gamma_1)$. Then we proceed inductively.

To finish the proof it remains to consider the following situation. Under the conditions of [*Case 3*]{}, each of the classes $[\ti\gamma_k]$, $k=1,2,3$, either corresponds to a marked point or is represented by a closed geodesic $\ti\gamma_k$, which intersects or coincides with one from the already chosen geodesics $\gamma_j$, $j<J^*$.

We state that, in fact, a proper intersection cannot happen, i.e., each class $[\ti\gamma_k]$, $k=1,2,3$, either corresponds to a marked point or is represented by an already chosen geodesic $\gamma_j$, $j<J^*$. To show this we note that $\gamma_j \cap \tau(\gamma) =\emptyset$ for all $j<J^*$. Otherwise, we could have a contradiction with the conditions $\gamma_j \cap
\gamma =\emptyset$ and $\tau$-invariance of the set of the geodesics $\gamma_j$, $j<J^*$. Consequently, each class $[\ti\gamma_k]$ is represented by a circle $\alpha_k \subset C^d_i \bs\mapo$, $k=1,2,3,4,$ with $\alpha_k
\cap \gamma_j = \emptyset$.

Now assume that the proper intersection of $\ti\gamma_k$ and some $\gamma_j$, $j<J^*$ occurs. Let $\ell_k\deff \ell(\ti\gamma_k)$ be the intrinsic metric of $\ti\gamma_k$. As in the proof of [*Lemma 4.3.2*]{} construct the annulus $A= \{ (\rho, \theta) : |\rho| < {\pi^2 \over \ell} \} \times \{0\le
\theta \le 2\pi\}$ with the metric $({\ell_k \over 2\pi} / \cos {\ell_k \rho
\over 2\pi})^2 (d\rho^2 + d\theta^2)$ and an isometric covering of $C^d_i
\bs\mapo$ by $A$, which sends the geodesic $\beta_k \deff \{ \rho=0\} \subset
A$ onto $\ti\gamma_k \subset C^d_i$. Find a lift of $\gamma_j$ to a geodesic line $L_j \subset A$ with $L_j\cap \beta_k \not= \emptyset$ and a lift of a circle $\alpha_k$ to a circle $\ti\alpha_k \subset A$ homotopic to $\beta_k$. Then the intersection $L_j \cap \beta_k$ must consist of exactly one point, and, consequently, the homology intersection index $[L_j] \cdot [\beta_k]$ is equal to $\pm1$. This would imply that $[L_j]\cdot [\ti\alpha_k] =[L_j] 
\cdot [\beta_k] \not =0$ and consequently $L_j \cap \ti\alpha_k \not= 
\emptyset$. But this would contradict $\gamma_j \cap \alpha_k =\emptyset$.

Summing up, we see that in the situation we are considering we must have the picture of [*Fig. 11*]{}. Namely, both geodesics $\gamma$ and $\tau(\gamma)$ lie in a $\tau $-invariant domain $\Omega$ on $C^d_i$ with four components of the boundary; these components of $\d\Omega$ are either marked points or geodesics corresponding to the classes $[\ti\gamma_1]$, $[\tau(\ti\gamma_1)]$, $[\ti
\gamma_2]$, $[\ti\gamma_3]$; finally, every boundary circle of $\Omega$ is one of the geodesic $\gamma_j$. We conclude that the class $[\ti\gamma_4]$ is represented by a $\tau$-invariant geodesic $\ti\gamma_4$, which can be chosen at this step of the construction of $\tau$-invariant decomposition of $C^d_i$ into pants.

Note that by construction for the intrinsic length of $\gamma_{J^*}$ we obtain $\ell(\gamma_{J^*}) \le 2\ell(\gamma) \le 2l^+_{J^*}$. This means that in our construction we do not lose control of the intrinsic length of the chosen geodesics. This shows the existence of a constant $l^+$ stated in the lemma.

Finally, the definition of a [*short*]{} geodesic provides that the geodesic $\gamma$ in [*Case 3*]{} above cannot be short. This implies that the set of short geodesics on $C^d$ is disjoint. Since the involution $\tau$ is an isometry, the set of short geodesics on $C^d$ is also $\tau$-invariant. Thus, in our construction of decomposition into pants we can start with this set of geodesics. This shows the last statement of the lemma.

Remark. To explain the meaning of [*Lemma A3.4.2*]{}, let us consider pants $S$ with a complex structure $J_S$ and an antiholomorphic involution $\tau$ acting on $S$. It is easy to see that only two types of such an action, illustrated by Figs. 8a) and 8b), are possible.

In the first case, Fig. 12a), the involution $\tau$ interchanges two boundary components $\gamma_1$ and $\gamma_2$ of $S$ and leaves the third one $\gamma_3$ invariant. The fixed point set $\beta$ of $\tau$ is a geodesic arc with both ends on the $\tau$-invariant boundary component $\gamma_3$. This case includes subcases where some boundary components of $S$ are not geodesics but marked points (punctures). In particular, if $\gamma_3$ is a marked point, then the set $\beta$ is an (infinite) geodesic line with both ends approaching $\gamma_3$. The set $\beta$ divides $S$ into two parts, $S^+$ and $S^-$ (see Fig. 12a)), which are interchanged by $\tau$. Topologically, each part $S^\pm$ is an annulus.

In the second case, Fig. 12b), all three boundary components $\gamma_1$, $\gamma_2$, and $\gamma_3$ are invariant. The fixed point set of $\tau$ consists of geodesic arcs $\beta_1$, $\beta_2$, and $\beta_3$. These are the shortest simple geodesics between $\gamma_2$ and $\gamma_3$, resp., $\gamma_3$ and $\gamma_1$ and resp., $\gamma_2$ and $\gamma_3$. If some boundary component of $S$ is not a geodesic but a marked point, then corresponding arcs have ends of infinite length approaching this boundary component. The arcs $\beta_k$, $k=1,2,3$, divide $S$ into two parts, $S^+$ and $S^-$ (see Fig. 12b)), which are interchanged by $\tau$. In this case, each part $S^\pm$ is topologically a disc.

We call pieces $S^\pm$ [*half-pants of the first or second type*]{}, respectively. Note that in both cases $\tau$-invariant arcs $\beta$ or $\beta_i$ are orthogonal to corresponding boundary circles $\gamma_j$.

Return to the situation of a nodal curve $\barr C$ with boundary and marked points. Let $C^d$ be the Schottky double and $\tau$ the antiholomorphic involution. Suppose that $C^d \bs\mapo$ is non-exceptional. Use [*Lemma A3.4.2*]{} and find a $\tau$-invariant decomposition into pants $C^d = \cup_j
S_j$. Set $S_j^+ \deff S_j \cap \barr C$. Then we obtain a decomposition $\barr C= \cup_j S^+_j$ such that the pieces $S^+_j$ are either pants (which means $S^+_j = S_j$), or half-pants of the first or second type. This decomposition is a suitable one for the situation of Gromov convergence up to the boundary of curves with totally real boundary conditions. In particular, we obtain arcs $\beta_{j,k}$ as $\tau$-fixed point sets of $S^+_j$, which define a decomposition $\d C= \cup_{j,k} \beta_{j,k}$ of the boundary of $\barr C$. The collection $\bfbeta' \deff \{ \beta_{j,k} \}$ of these arcs satisfies condition of [*Definition 5.4*]{}, but it can be different from the collection $\bfbeta =\{ \beta_i \}$ which was given. The reason is that in the construction of the pants-decomposition $C^d = \cup_j S_j$ we can subdivide original arcs $\beta_i \in \bfbeta$ into smaller pieces, so that every arc $\beta_i \in \bfbeta$ is a union of arcs $\beta_{j,k}$ from $\bfbeta'$. This means compatibility of $\bfbeta$ and $\bfbeta'$.

The next step is to establish an analog of [*Theorem 5.3.2*]{}. Assume that the hypothesis of [*Theorem A3.4.1*]{} is fulfilled. For each curve $C_n$ denote by $C^d_n$ its Schottky double and by $\tau_n$ the corresponding involution.

Lemma A3.4.3. *In the situation above, after passing to a subsequence, there exist parameterizations $\sigma^d_n: \Sigma^d \to C^d_n$, a finite covering $\calv$ of $\Sigma^d$ by open sets $\{ V_\alpha \}$, and a set $\{x^*_1, \ldots, x^*_m\}$ of marked points on $\Sigma$ such that the conditions [*(a), (c)–(f)*]{} of [*Theorem 4.2*]{} and the following additional conditions [*(b’)*]{} and [*(h)*]{} are satisfied:*

[*(b’)*]{} $\sigma_n\{x^*_1, \ldots, x^*_m\}$ is the set of marked points on $C^d_n$ corresponding to the decomposition of the boundary $\d C_n$ into arcs $\beta_{n,i}$; moreover, each such point $x^*_j$ lies in a single piece of covering $V_\alpha$ which is a disc;

[*(h)*]{} there exists an involution $\tau : \Sigma^d \to \Sigma^d$ which is compatible with the covering $\calv$ and with parameterizations $\sigma^d_n$, i.e.,$\calv$ is $\tau$-invariant and $\tau_n \scirc \sigma^d_n = \sigma^d_n
\scirc \tau$. In particular, each marked point $x^*_i$ of $\Sigma^d$ is fixed by $\tau$.

Remark. The condition [*(g)*]{} of [*Theorem 5.3.2*]{} is trivial in this case, because $C^d_n$ and $\Sigma^d$ are closed.

Proof. One can use the proof of [*Theorem 5.3.2*]{} with minor modifications. Note that the starting points of that proof were the intrinsic metric on non-exceptional components of nodal curves $C_n$ and the decomposition of $C_n$ into pants. Now the existence of a $\tau$-invariant decomposition of the curves $C^d_n$ into pants is provided by [*Lemma A3.4.2*]{}, whereas the $\tau$-invariance of the intrinsic metrics follows from the fact that any (anti)holomorphic isomorphism of curves with marked points is an isometry the intrinsic metric. Thus the constructions of the proof of [*Theorem 5.3.2*]{} yield $\tau$-invariant objects. Condition [*(b’)*]{} does not cause much difficulty.

Now we are ready to finish the

Proof of Theorem A3.4.1. As was mentioned, our main idea is to modify the construction used in the proof of [*Theorem 4.1.1*]{} to make it $\tau$-invariant. The main work has already been done. We have the necessary a priori estimates, the construction of a $\tau$-invariant pants-decomposition of the double $C^d_n$ of the curve $\barr C_n$ and the appropriate covering $\calv$ of the real surface $\Sigma^d$ parameterizing the doubles $C^d_n$.

As in the proof of [*Theorem 4.1.1*]{}, we consider the curves $C_{ \alpha ,n}
\deff \sigma^d_n(V_\alpha)$. Due to the presence of the involutions $\tau_n$, the geometric situation in now different. This involves new phenomena and needs additional considerations and constructions. In particular, the pieces $C_{
\alpha ,n}$ are divided into two groups depending on whether they are disjoint from the boundary $\d C_n$ or intersect it. In the last case $C_{
\alpha ,n}$ is $\tau_n$-invariant. In this case we shall use the notation $C^+_{ \alpha ,n} \deff C_{ \alpha,n} \cap C_n$ for the part of $C_{ \alpha
,n}$ lying in $C_n$. In addition, we denote $V^+_\alpha \deff V_\alpha \cap
\Sigma$. Then $V_\alpha$ appear as the union of domains $V^+_\alpha$ and $\tau(V^+_\alpha)$, interchanged by $\tau$. Similarly, it is true for $C_{ \alpha,n}$.

To prove the theorem, we want to construct a refined covering $\wt\calv$ of $\Sigma$ and refined parameterizations $\ti\sigma_n: \Sigma \to C_n$ such that for every $V_\alpha \in \wt\calv$ the sequence $(C_{ \alpha ,n},
u_{ \alpha ,n})$ with $C_{ \alpha ,n} \deff \ti\sigma_n(V_\alpha)$ one the following convergence types holds:

In the case [*B$''$)*]{} the strong convergence of maps $\ti u^+_n: \cala^+
_0 \to X$ is the one in the $L^{1,p^*}$-topology for some $p^*>2$ [*up to the boundary intervals containing the nodal point*]{}. An equivalent requirement is the usual $L^{1,p^*}$-convergence of the doubles $\ti u^d_n: C_{\alpha, n}
\to X$ on compact subsets of $C_{\alpha, n} \cong \cala_0$.

To obtain a desired refinement, we use the same inductive procedure as in the proof of [*Theorem 1.1*]{}. To insure convergence near the boundary $\d C_n$, we take a new value for the constant determining the inductive step. We choose a positive $\eps^b$ such that $\eps^b \le \eps$ and all a priori estimates of this section are valid for maps with area.$\le 3\eps^b$. This will yield the convergence of type [*A)–C)*]{} for sequences of curves with totally real boundary conditions and with the upper bound $\eps^b$ on the area.

In fact, essential modifications of the constructions of [*Theorem 1.1*]{} are needed only if the covering piece $V_\alpha$ is $\tau$-invariant. Indeed, if $V_\alpha$ is not $\tau$-invariant, then we can apply all the argumentations and constructions used in [*Cases 1)–4)*]{} in the proof of [*Theorem 1.1*]{}, and then “transfer” them onto $\tau(V_\alpha)$ by means of $\tau$. This gives the inductive step preserving $\tau$-invariance.

Hence, it remains to consider the situation when the covering piece $V_\alpha$ is $\tau$-invariant. As in [*Theorem 1.1*]{}, we must consider four cases.

*Case 1$_b$): $C_{\alpha, n}$ have constant complex structure different from the one of the standard node.*

Case 2$_b$): $C_{\alpha, n}$ are annuli of changing conformal radii $R_n$ such that $R_n \to R <\infty$.

Case 3$_b$): $C_{\alpha, n}$ are isomorphic to the standard node, so that $C^+_{\alpha, n}$ are isomorphic to the standard boundary node $\cala_0^+$.

Case 4$_b$): $C_{\alpha, n}$ are annuli of infinitely growing conformal radii $R_n$.

The subindex $(\cdot)_b$ indicates that we will consider the cases where $V_\alpha$ intersects the [*b*]{}oundary of $\Sigma$. As was mentioned, the last property is equivalent to the fact that $V_\alpha$ is $\tau$-invariant. References to [*Cases 1)–4) *without**]{} the subindex will mean the corresponding parts of the proof of [*Theorem 1.1*]{}.

[*Case 1$_b$)*]{}. Without loss of generality we may assume that $V_\alpha$ is a domain with a fixed complex structure and a fixed antiholomorphic involution $\tau$, and that $u_{\alpha, n}: V_\alpha \to X$ is a sequence of $\tau$-invariant maps which are (anti)holomorphic outside the set of $\tau
$-invariant points of $V_\alpha$. If we have the convergence of type [*C)*]{}, there is nothing to do. Otherwise, we fix a $\tau$-invariant metric on $V_\alpha$ compatible with the complex structure. Repeating the constructions from [*Case 1)*]{}, we distinguish the “bubbling” points $y^*_1, \ldots, y^*_l$ where the strong convergence fails.

Take the first point $y^*_1$. Suppose $y^*_1$ is disjoint from $\d\Sigma$. Then we may assume that $y^*_1\in V^+_\alpha$. Thus, we can repeat the rest of the constructions from [*Case 1)*]{}. The only correction needed at this point is that the neighborhood $\Delta( y^*_1, \varrho)$ of $y^*_1$ must be small enough and lie in $V^+_\alpha$. Transferring all these constructions into $\tau(V_\alpha)$, we realize the inductive step preserving the $\tau$-invariance.

It remains to consider the case, where $y^*_1 \in \d\Sigma$. This means that $y^*_1$ is $\tau$-invariant. Let $z$ be a holomorphic coordinate in a neigborhood of $y^*_1$ on $V_\alpha$ such that $z=0$ in $y^*_1$, the involution $\tau$ corresponds to the conjugation $z \mapsto \bar z$ and $\im
z >0$ in $\Sigma$. Find the sequences $r_n\lrar0$ of the radii and $x_n \to
y^*_1$, using the constructions from [*Case 1)*]{}. Note that the sequence $\tau(x_n)$ has the same property. Thus, replacing some points $x_n$ by $\tau(x_n)$, we may additionally assume that all $x_n$ lie in $\barr
V^+_\alpha$. Let $v_n:\Delta (0,{\varrho \over 2r_n}) \to (X,J_n)$ be the rescalings of maps $u_n$ defined by $v_n(z) \deff u_n(x_n + {z\over r_n})$. Argumentations of [*Case 1*]{} show that there exists the limit $v_\infty:
\cc \to X$ of (a subsequence of) $\{ v_n \}$ which extends to a map $v_\infty
: S^2 \to X$.

Denote by $\rho_n$ the distance from $x_n$ to $\d\Sigma$ and by $\ti x_n$ the point on $\d\Sigma$ closest to $x_n$. Then $x_n = \ti x_n + \isl \rho_n$ in the coordinate $z$ introduced above. In addition, $\lim \ti x_n = y^*_1$. We consider two subcases according to the possible behavior of $\rho_n$ and $r_n$.

[*Subcase 1$'_b$): $\bigl\{{ \rho_n \over r_n }\bigr\}$ is bounded*]{}. Passing to a subsequence, we may assume that ${\rho_n \over r_n}$ converges. Fix an upper bound $b$ for the sequence ${\rho_n \over r_n}$. In particular, $b \ge \lim {\rho_n \over r_n}$.

For $n>\!>1$ define maps $v_n:\Delta (0,{\varrho \over 2r_n}-b) \to (X,J_n)$ and $\ti v_n:\Delta (0,{\varrho \over 2r_n}-b) \to (X,J_n)$ setting $v_n(z)
\deff u_n (x_n +r_n z)$ and $\ti v_n(z)\deff u_n(\ti x_n +r_n z)$, respectively. Then every $\ti v_n$ is the shift of the map $v_n$ by $\isl{\rho_n\over r_n}$, i.e., $\ti v_n(z)= v_n\bigl(z + \isl {\rho_n\over
r_n}\bigr)$. The arguments of [*Case 1)*]{} show that $v_n$ converge on compact subsets of $\cc$ to a non-constant map. Consequently, $\ti v_n$ also converge on compact subsets of $\cc$ to a non-constant map $\ti v_\infty: \cc
\to X$. Moreover, since $\area( \ti v_\infty( \cc))$ is finite, $\ti v_\infty$ extends to a map $\ti v_\infty: S^2 \to X$. By the choice of $\eps^b$, $\area(\ti v_\infty (S^2)) \ge 3\eps^b$. Changing the choice of the constant $b$, we can additionally assume that $\area(\ti v_\infty( \Delta(0,b)) \ge
2\eps^b$. Then for all sufficiently big $n$ we obtain $$\area(\ti v_n(\Delta(0,b))  \ge \eps^b.
\eqno(A3.4.1)$$

For $n>\!>1$ we define the coverings of $V_\alpha$ by three sets $$V^{(n)}_{\alpha,1} \deff  V_\alpha
\bs \barr\Delta (0, {\textstyle{\varrho\over2}}),
\qquad
V^{(n)}_{\alpha,2} \deff \Delta (0, \varrho)
\bs \barr \Delta (\ti x_n, br_n),
\qquad
V^{(n)}_{\alpha,3} \deff \Delta (\ti x_n, 2br_n).$$ Fix $n_0$ sufficiently big. Denote $V_{\alpha,1} \deff V^{(n_0)}_{\alpha,1}$, $V_{\alpha,2} \deff V^{(n_0)}_{\alpha,2}$ and $V_{\alpha,3} \deff V^{(n_0)}
_{\alpha,3}$. There exist diffeomorphisms $\psi_n: V_1 \to V_1$ such that $\psi_n: V_{\alpha,1} \to V^{(n)}_{\alpha,1}$ is an identity, $\psi_n:
V_{\alpha, 2} \to V^{(n)}_{\alpha,2}$ is a diffeomorphism, and $\psi_n:
V_{\alpha,3} \to V^{(n)}_{\alpha,3}$ is biholomorphic the complex structures, induced from $C_n$ by means of $\sigma^d_n$. Note that the sets $V^{(n)}_{\alpha,i}$ are $\tau$-invariant. Moreover, we can choose the maps $\psi_n$ in such a way that $\psi_n$ are also $\tau$-invariant.

The covering $\{ V_{\alpha,1}, V_{\alpha,2}, V_{\alpha,3} \}$ of $V_1$ and parameterizations $\ti\sigma_n \deff \sigma_{\alpha,n} \scirc \psi_n: V_1 \to
C_{\alpha, n}$ satisfy the conditions of [*Lemma A3.4.3*]{}. Moreover, inequality (A3.2.1) implies $\area(u_n(\ti\sigma_n(V_{\alpha,i}))) \le
%%%%%%%%%%%%%%%%%%%%%%%question%%%%%%%%%%%%%%%%%%%%%
(N-1)\eps^b$. Consequently, we can apply the inductive assumptions for the sequence of curves $\ti\sigma_n(V_{\alpha,i})$ and finish the proof by induction.

[*Subcase 1$''_b$): $\bigl\{{ \rho_n \over r_n}\bigr\}$ is unbounded*]{}. Passing to a subsequence, we may assume that ${\rho_n \over r_n}$ increases infinitely. However, $\rho_n \lrar 0$ since $x_n \lrar y^*_1 \in \d\Sigma$.

Define maps $v_n:\Delta (0,{\varrho \over 2r_n}) \to (X,J_n)$, setting $v_n(z)
\deff u_n(x_n +r_n z)$. As in [*Case 1)*]{}, $v_n$ converge on compact subsets of $\cc$ to a non-constant map $v_\infty: \cc \to X$, which extends to a map from the whole sphere $S^2$. Choose $b>0$ satisfying (A3.2.1). For $n>\!>1$ we define the coverings of $V_\alpha$ by five sets $$\mathsurround=0pt
\matrix\format\l\ \ &\l\\
V^{(n)}_{\alpha,1} \deff  V_\alpha
\bs \barr\Delta (0, {\textstyle{\varrho\over2}}),
&
V^{(n)}_{\alpha,2} \deff \Delta(0, \varrho) \bs
\barr \Delta(0, 2\rho_n)
\cr
\noalign{\vskip5pt}
\rlap{$
V^{(n)}_{\alpha,3} \deff \Delta(0, 4\rho_n) \bs
\bigl(\barr \Delta (x_n, br_n) \cap  \barr \Delta (\tau(x_n), br_n) \bigr)
$}&
\cr
\noalign{\vskip5pt}
V^{(n)}_{\alpha,4} \deff \Delta (x_n, 2br_n),
&
V^{(n)}_{\alpha,5} \deff \Delta(\tau(x_n), 2br_n).
\endmatrix$$ Fix $n_0$ sufficiently big. Denote $V_{\alpha,i} \deff V^{(n_0)}_{\alpha,i}$, $i=1,\ldots,5$. Then for every $n>\nobreak\!>1$ there exists a diffeomorphism $\psi_n: V_1 \to V_1$ with the following properties:

$\psi_n$ maps $V_{\alpha,i}$ onto $V^{(n)}_{\alpha,i}$ diffeomorphically;

$\psi_n: V^{(n)}_{\alpha,1} \to V^{(n)}_{\alpha,1}$ is the identity;

$\psi_n: V_{\alpha, 2} \to V^{(n)}_{\alpha,2}$ and $\psi_n:
V_{\alpha,3} \to V^{(n)}_{\alpha,3}$ are diffeomorphisms;

$\psi_n: V_{\alpha,3} \to V^{(n)}_{\alpha,3}$ is biholomorphic the complex structures, induced from $C_n$ by means of $\sigma^d_n$; and, finally

$\psi_n$ are $\tau$-invariant: $\tau \scirc\psi_n =\psi_n \scirc\tau$. Note that the last property is obtained due to the fact that the sets $V^{(n)} _{\alpha, i}$ are $\tau$-invariant. The remaining constructions are the same as in [*Subcase 1$'_b$)*]{}.

[*Case 2$_b$)*]{}. Consider the parameterizations $\sigma_n: V_\alpha \to
C_{\alpha,n}$. Without loss of generality we may assume that the complex structures $\sigma_n^* j_n\ogran_{V_\alpha}$ are constant near the boundary $dV_\alpha$ and converge to some complex structure. If we have the convergence of type [*C)*]{}, i.e., the strong convergence, there is nothing to do. Otherwise, there exists only a finite set of points $\{y^*_1,\ldots, y^*_l\}$ where the strong convergence fails. Changing the parameterizations $\sigma_n$, we may additionally assume that the structures $\sigma_n^* j_n \ogran
_{V_\alpha}$ are constant in the neighborhood of these points. Then we repeat the argumentations of [*Case 1$_b$)*]{}.

[*Case 3$_b$)*]{}. Fix identifications $C_{\alpha,n} \cong \cala_0$ such that every $C_{\alpha,n}^+$ is mapped onto $\cala_0^+$ and the induced parameterization maps $\sigma_{\alpha, n} : V_\alpha \to \cala_0$ are the same for all $n$ and $\tau$-invariant. Fix the standard representation of $\cala_0$ as the union of two discs $\Delta'$ and $\Delta''$ with identification of the centers $0\in \Delta'$ and $0\in \Delta''$ into the nodal point of $\cala_0$, still denoted by $0$. Let $\Delta' (x, r)$ denote the subdisc of $\Delta'$ with the center $x$ and the radius $r$.

Denote by $u'_n :\Delta' \to X$ and $u''_n :\Delta'' \to X$ the corresponding “components” of the maps $u_{\alpha,n} : C_{\alpha,n} \to X$. Find the common collection of bubbling points $y^*_i$ for both sequences of maps $u'_n
:\Delta' \to X$ and $u''_n :\Delta'' \to X$. If there are no bubbling points, then we obtain a convergence of type [*B)*]{}, and the proof can be finished by induction. Otherwise, consider the first such point $y^*_1$, which lies, say, on $\Delta'$. If $y^*_1$ is distinct from the nodal point $0 \in \Delta'$, then we simply repeat all the constructions in [*Case 1$_b$)*]{}.

It remains to consider the case $y^*_1=0 \in \Delta'$. The following modifications of the argumentations are needed. Repeat the construction of the radii $r_n\lrar 0$ and the points $x_n \lrar y^*_1=0$ from [*Case 1$_b$*]{}. Then $\{x_n \}$ is a sequence in the half-disk $\delta^{\prime+}
\deff \{ z\in \Delta' : \im z \ge0 \}$. Set $\ti x_n \deff \re(x_n)$, $\rho_n
\deff \im(x_n)$ and $R_n \deff |x_n|$. Thus, $x_n= \ti x_n +\isl\rho_n$, $R_n$ is the distance from $x_n$ to the point $0=y^*_1 \in \Delta'$, whereas $\rho_n$ is the distance from $x_n$ to the interval $]-1,1[ \subset \Delta'$, the set $\tau$-invariant points of $\Delta'$. Thus $\rho_n \le R_n$. Fix $\varrho>0$ such that the disc $\Delta'(0,\varrho)$ contains no bubble points $y^*_i\not=0 \in \Delta'$.

Depending on the behavior of the sequences $r_n$, $\rho_n$ and $R_n$, we consider four subcases. [*Subcase 3$'_b$): The sequence $\bigl\{{ R_n \over r_n }\bigr\}$ is bounded.*]{} Then the sequences $\bigl\{{ {\rho_n\over r_n} }\bigr\}$ and $\bigl
\{ {\ti x_n\over r_n} \bigr\}$ are also bounded. Passing to a subsequence, we may assume that the corresponding limits exist. Let $b$ be some upper bound for the sequence $\bigl\{{ R_n \over r_n }\bigr\}$. Consider the maps $\ti
v_n: \Delta(0, {\varrho \over 2r_n}-b) \to X$ defined by $\ti v_n(z) \deff
u_n(\ti x_n+ {z\over r_n})$. Then $\ti v_n$ are $\tau$-invariant, $\ti v_n
\scirc \tau= \ti v_n$, $\ti v_n$ converge to a nonconstant map $\ti v_\infty:
\cc \to X$ on compact subsets of $\cc$, and $\ti v_\infty$ extends to a map $\ti v_\infty: S^2 \to X$.

Since $\ti v_\infty$ is nonconstant, $\norm{ d\ti v _\infty} ^2 _{L^2( S^2)}
= \area( \ti v_\infty( S^2) )\ge 3\eps_b$. Choose $b>0$ in such a way that $$\norm{d\ti v_\infty}_{L^2(\Delta (0,b))}^2 \ge 2\eps_b
\eqno(A3.4.2)$$ and $b \ge 2\lim {R_n \over r_n}+2$. Due to [*Corollary A3.2.2*]{}, for $n>\!>1$ we obtain the estimate $$\norm{du'_n}_{L^2(\Delta' (\ti x_n, br_n))}^2 =
\norm{ d\ti v_n}_{L^2(\Delta (0,b))}^2 \ge \eps_b.
\eqno(A3.4.3)$$ Note that $0\in \Delta' (\ti x_n, (b-1)r_n))$ for $n>\!>1$ by the choice of $b$.

Define the coverings of $\cala_0$ by four sets $$\mathsurround=0pt
\matrix\format\l\ \ &\l\\
W^{(n)}_1 \deff  \Delta'
\bs \barr\Delta' (0, {\textstyle{\varrho\over2}}),
&
W^{(n)}_2 \deff \Delta' (0, \varrho)
\bs \barr \Delta' (\ti x_n, br_n),
\cr
\noalign{\vskip4.5pt}
W^{(n)}_3 \deff \Delta' (\ti x_n, 2br_n)
\bs \barr\Delta'(0,{\textstyle {r_n \over 2}}),
&
W^{(n)}_4 \deff \Delta' (0, r_n) \cup
\Delta'',
\endmatrix$$ and lift them to $V_\alpha$ by putting $V^{(n)}_{\alpha,i} \deff \sigma\inv
_{\alpha,n}(W^{(n)}_i)$. Choose $n_0 >\!> 0$ such that $|x_n| < (b-1)r_n$ and the relation (A3.4.3) holds for all $n \ge n_0$. Set $V_{\alpha,i} \deff
%%%%%%%%%%%%%%%%%%%question%%%%%%%%%%%%%%%%%%%%%%%%%
V^{( n_0)} _{\alpha, i}$. Fix diffeomorphisms $\psi_n: V_\alpha \to V_\alpha$ such that $\psi_n: V_{\alpha,1} \to V^{(n)} _{\alpha,1}$ is the identity map, $\psi_n: V_{\alpha,2} \to V^{(n)}_{\alpha,2}$ and $\psi_n: V_{\alpha,3} \to
V^{(n)} _{\alpha,3}$ are diffeomorphisms, and $\psi_n: V_{\alpha,4} \to
V^{(n)} _{\alpha,4}$ correspond to isomorphisms of nodes $W^{(n)}_4 \cong
\cala_0$. Set $\sigma'_n \deff \sigma_n \scirc \psi_n$. The choice above can be made in such a way that the refined covering $\{ V_{\alpha,i} \}$ of $V_\alpha$ and parameterization maps $\sigma'_n: V_\alpha \to C_{\alpha,n}$ have the properties of [*Lemma A3.4.3*]{}. Relation (A3.4.3) implies the estimate $\area(u_n (\sigma'_n(V_{\alpha,i})) \le (N-1)\,\eps$. This provides the inductive conclusion for [*Subcase 3$'_b$)*]{}.

[*Subcase 3$''_b$): The sequence $\bigl\{{R_n \over r_n }\bigr\}$ increases infinitely but $\bigl\{{\rho_n \over r_n }\bigr\}$ remains bounded*]{}. Note that in this subcase we still have the relation $R_n \lrar
0$, or equivalently, $x_n \lrar 0$. On the other hand, $\lim {\rho_n \over
R_n} =0$. This implies that for $\ti R_n \deff |\ti x_n|$ we have $\lim
{\ti R_n \over R_n} =1$ since $R_n^2 = \ti R_n^2 + \rho_n^2$.

We proceed as follows. Define the maps $\ti v_n: \Delta(0, {\varrho \over
2r_n}-b) \to X$ setting $\ti v_n(z) \deff u'_n(\ti x_n + {z\over r_n})$. Then the $\ti v_n$ have the same properties as in [*Subcase 3$'_b$)*]{}. Choose $b>0$ obeying the relation (A3.4.2). Then for $n>\!>0$ the property (A3.4.3) follows.

For $n>\!>0$ define the coverings of $\cala_0$ by six sets $$\mathsurround=0pt
\matrix\format\l\ \ &\l\\
W^{(n)}_1 \deff  \Delta'
\bs \barr\Delta' (0, {\textstyle{\varrho\over2}}),
&
W^{(n)}_2 \deff \Delta' (0, \varrho)
\bs \barr \Delta' (\ti x_n, 2\ti R_n),
\cr\noalign{\vskip5pt}
W^{(n)}_3 \deff  \Delta' (\ti x_n, 4\ti R_n)
\bs \bigl(\barr \Delta' (\ti x_n, {\ti R_n \over 6})
\cup \barr \Delta' (0, {\ti R_n \over 6}) \bigr)
&
W^{(n)}_4 \deff \Delta' (0, {\ti R_n \over 3}) \cup \Delta'',
\cr\noalign{\vskip5pt}
W^{(n)}_5 \deff \Delta' (\ti x_n, {\textstyle{\ti R_n \over 3}})
\bs \barr\Delta'(\ti x_n, br_n),
&
W^{(n)}_6 \deff \Delta' (0, 2br_n ),
\endmatrix$$ and lift them to $V_\alpha$ by putting $V^{(n)}_{\alpha,i} \deff \sigma\inv
_{\alpha,n}(W^{(n)}_i)$. Choose $n_0 >\!> 0$ such that $R_{n_0} >\!> b
r_{n_0} $, and set $V_{\alpha,i} \deff V^{(n_0)} _{\alpha, i}$. Choose diffeomorphisms $\psi_n: V_\alpha \to V_\alpha$ such that $\psi_n: V
_{\alpha, 1} \to V^{(n)} _{\alpha,1}$ is the identity map, $\psi_n: V
_{\alpha, 2} \to V^{(n)}_{\alpha,2}$, $\psi_n: V_{\alpha,4} \to V^{(n)}
_{\alpha,4}$ and $\psi_n: V_{\alpha,5} \to V^{(n)}_{\alpha,5}$ are diffeomorphisms, and finally, $\psi_n: V_{\alpha,6} \to V^{(n)}_{\alpha,6}$ corresponds to isomorphisms of nodes $W^{(n)}_6 \cong \cala_0$. Set $\sigma'_n \deff \sigma_n \scirc \psi_n$. Note that the choices can be made in such a way that $\{ V_{\alpha,i} \}$ and parameterization maps $\sigma'_n:
V_\alpha \to C_{\alpha,n}$ have the properties of [*Lemma A3.4.3*]{}. As above, we get the estimate $\area(u_n (\sigma'_n(V_{\alpha,i} )) \le (N-1)\, \eps$ due to (A3.4.3). Thus we obtain the inductive conclusion for [*Subcase 3$''_b$)*]{} and can proceed further.

[*Subcase 3$'''_b$): The sequence $\bigl\{{\rho_n \over r_n} \bigr\}$ increases infinitely, but $\bigl\{{ R_n \over \rho_n} \bigr\}$ remains bounded*]{}. Then $\bigl\{{R_n \over r_n} \bigr\}$ also increases infinitely, but both sequences $\{R_n \}$ and $\bigl\{{ \rho_n }\bigr\}$ converge to 0. We may also assume that $\bigl\{{ \rho_n \over R_n }\bigr\}$ and $\bigl\{{ \ti x_n \over R_n }
\bigr\}$ also converge. Set $a_1 \deff \lim {\ti x_n \over R_n}$, $a_2 \deff
\lim{ \rho_n \over R_n}$, $a\deff a_1 + \isl a_2$ and $\barr a \deff a_1
- \isl a_2$. Note that $0 < a_2 \le 1$ and the involutions $\tau_n$ in $C_{\alpha, n}$ correspond to the complex conjugation $z\to \barr z$ in $\Delta'$. In particular, $\barr x_n = \tau_n(x_n)$.

Consider maps $v_n: \Delta(0, {\varrho \over 2r_n}) \to X$ defined by $v_n(z)
\deff u'_n(x_n + {z\over r_n})$. Then the sequence $\{ v_n \}$ converges on compact subsets to a nonconstant map which extends to the map $v_\infty :S^2
\to X$. Moreover, we can fix sufficiently big $b>0$ such that for $n>\!>0$ we get the property (A3.4.3).

For $n>\!>0$ define the coverings of $\cala_0$ by eight sets $$\mathsurround=0pt
\matrix\format\l\ \ \ \ \ &\l\\
W^{(n)}_1 \deff  \Delta'
\bs \barr\Delta' (0, {\textstyle{\varrho\over2}}),
&
W^{(n)}_2 \deff \Delta' (0, \varrho)
\bs \barr \Delta' (0, 2R_n),
\cr\noalign{\vskip6pt}
\rlap{$W^{(n)}_3 \deff  \Delta' (0, 4R_n)
\bs \bigl(\, \barr \Delta' ( a\, R_n , {a_2 R_n \over 4})
\cup \barr \Delta' ( \barr a\, R_n , { a_2 R_n \over 4})
\cup \barr \Delta' (0, {a_2 R_n \over 4})
\,\bigr)\qquad$}
&
\cr\noalign{\vskip6pt}
W^{(n)}_4 \deff \Delta' (0, {a_2 R_n \over 3}) \cup \Delta''.
\cr\noalign{\vskip5pt}
W^{(n)}_5 \deff \Delta' (a R_n, {a_2 R_n \over 3})
\bs \barr\Delta'(x_n, br_n),
&
W^{(n)}_6 \deff \Delta' (x_n, 2br_n ),
\cr\noalign{\vskip5pt}
W^{(n)}_7 \deff \Delta' (\barr a R_n, {a_2 R_n \over 3})
\bs \barr\Delta'(\barr x_n, br_n),
&
W^{(n)}_8 \deff \Delta' (\barr x_n, 2br_n ),
\endmatrix$$ and lift them to $V_\alpha$ by putting $V^{(n)}_{\alpha,i} \deff \sigma\inv
_{\alpha,n}(W^{(n)}_i)$. Fix sufficiently big $n_0 >\!> 0$,

and set $V_{\alpha, i} \deff V^{(n_0)} _{\alpha, i}$. Choose diffeomorphisms $\psi_n:
V_\alpha \to V_\alpha$ mapping $V _{\alpha, i}$ diffeomorphically onto $V^{(n)} _{\alpha, i}$ such that the assertions of [*Lemma A3.4.3*]{} are fulfilled. As above, we obtain the estimate $\area(u_n (\sigma'_n(V_{\alpha,i} )) \le (N-1)\, \eps$. This gives the inductive conclusion for [*Subcase 3$'''_b$)*]{}.

[*Subcase 3$''''_b$): The sequences $\bigl\{{\rho_n \over r_n} \bigr\}$ and $\bigl\{{ R_n \over \rho_n} \bigr\}$ increase infinitely*]{}. Thus $\lim {\ti R_n \over R_n} =1$. We consider the sequence of maps $\{ v_n \}$. It is defined in the same way as in the previous subcase and has the same properties. In particular, $\{ v_n \}$ converges to the map $v_\infty :S^2 \to X$ and there exists a sufficiently big $b>0$ such that for $n>\!>0$ we get the property (A3.4.3).

For $n>\!>0$ define the coverings of $\cala_0$ by ten sets $$\mathsurround=0pt
\matrix\format\l\ \ \ \ \ &\l\\
W^{(n)}_1 \deff  \Delta'
\bs \barr\Delta' (0, {\textstyle{\varrho\over2}}),
&
W^{(n)}_2 \deff \Delta' (0, \varrho)
\bs \barr \Delta' (0, 2R_n),
\cr\noalign{\vskip6pt}
\rlap{$W^{(n)}_3 \deff  \Delta' (0, 4R_n)
\bs \bigl(\, \barr \Delta' ( 0, {\ti R_n \over 4})
\cup \barr \Delta' ( \ti x_n, {\ti R_n \over 4})
\bigr),$}
&
\cr\noalign{\vskip6pt}
W^{(n)}_4 \deff \Delta' (0, {\ti R_n \over 3}) \cup \Delta'',
&
W^{(n)}_5 \deff \Delta' (\ti x_n, {\ti R_n \over 3})
\bs \barr\Delta'(\ti x_n, 2\rho_n),
\cr\noalign{\vskip6pt}
\rlap{$W^{(n)}_6 \deff \Delta' (\ti x_n, 4\rho_n )
\bs \bigl(\, \barr \Delta' ( x_n, {\rho_n \over 4})
\cup \barr \Delta' ( \barr x_n, {\rho_n \over 4})
\bigr), $}
\cr\noalign{\vskip5pt}
W^{(n)}_7 \deff \Delta' (x_n, {\rho_n \over 3})
\bs \barr\Delta'(x_n, br_n),
&
W^{(n)}_8 \deff \Delta' (x_n, 2br_n ),
\cr\noalign{\vskip5pt}
W^{(n)}_9 \deff \Delta' (\barr x_n, {\rho_n \over 3})
\bs \barr\Delta'(\barr x_n, br_n),
&
W^{(n)}_{10} \deff \Delta' (\barr x_n, 2br_n ).
\endmatrix$$ The remaining “manipulations” with $W^{(n)}_i$ are the same as in the previous subcases. As a result, we obtain the covering of $V_\alpha$ by sets $V_{\alpha,
i} \deff \sigma\inv _{\alpha, n_0}(W^{(n_0)}_i)$ with an appropriate $n_0 >\!>
0$ and refined parameterizations $\sigma'_n: V_\alpha \to C_{\alpha, n}$, for which the assertions of [*Lemma A3.4.3*]{} are fulfilled. As above, we obtain the estimate $\area(u_n (\sigma'_n(V_{\alpha,i} ))
\le (N-1)\, \eps$. This gives the inductive conclusion for [*Subcase 3$''''_b$)*]{}.

[*Case 4$_b$): $V_\alpha$ is a cylinder such that conformal radii of $(V_\alpha, \sigma_n^*j_n)$ increase infinitely*]{}. We can simply repeat the contructions made in [*Case 4)*]{} from the proof of [*Theorem 1.1*]{}. Additional attention is needed to preserve the $\tau$-invariantness.

The proof of the theorem can now be finished by induction.

Remark. Here we give some explanation of the geometric meaning of the constructions of the proof of [*Theorem 5.9*]{} and describe the picture of the bubbling. We restrict ourselves to [*Case 3$_b$)*]{} as the most complicated one; the constructions of the other cases can be treated similarly. The reflection principle allows us to reduce [*Case 3$_b$)*]{} to the consideration of $\tau$-invariant maps $u^d_n: \cala_0 \to X$ from the standard node which are $J_n$-holomorphic on $\cala^+_0$. The situation is different from those in [*Theorem 1.1*]{}, where the bubbling appears in the nodal point. In this case we must take into consideration not only the parameters $r_n$ describing the size of energy localization of the bubbled sphere, but also the additional parameters $R_n$ and $\rho_n$. These describe the position of the localization centers $x_n$ the nodal point and the set of $\tau$-invariant points of $\cala_0$. Depending on the behavior of $r_n$ $\rho_n$ and $R_n$, we can have four different types of bubbling and corrsponding [*Subcases 3$'_b$)–3$''''_b$)*]{}.

=12.3pt plus 1.5pt Turning back from a “doubled” description by $\tau$-invariant objects to the original maps $u_n: \cala^+_0 \to X$ with a totally real boundary condition, we obtain the following picture. Since every covering piece $W^{(n)}_i$ is $\tau$-invariant, for $\cala_0^+$ we obtain the covering piece $W^{(n)\,+}_i
\deff W^{(n)}_i \cap \cala_0^+$. Thus we obtain a bubbled disk represented by $W^{(n)\,+}_3$ instead of the bubbled sphere represented by $W^{(n)}_3$, the sequence of long strips $W^{(n)\,+}_2$ instead of the sequence of long cylinder $W^{(n)\,+}_2$ and so on.

However, because of the presence of the nodal point (piece $W^{(n)}_4$ on the figure), this “part inbetween” is topologically not a cylinder (an annulus) but pants. Furthermore, the complex structures on the pants are not constant. To get pants with a constant structure (piece $W^{(n)}_3$), we cut off the annuli $W^{(n)}_2$ and $W^{(n)}_5$. Since $\lim R_n =0 = \lim {r_n\over
R_n}$, the conformal radii of these annuli increase infinitely. This shows that $W^{(n)}_2$ and $W^{(n)}_5$ are sequences of long cylinders and that the sequence $W^{(n)}_3$ defines in the limit a sphere with three nodal points.

As in [*Subcase 3$'_b$)*]{} every covering piece $W^{(n)}_i$ is $\tau$-invariant, whereas $W^{(n)\,+}_i \deff W^{(n)}_i \cap \cala_0^+$ is the “half” of $W^{(n)}_n$. Thus, for a sequence of undoubled maps $u_n:
\cala^+_0 \to X$ we obtain the following bubbling picture. The limit contains two bubbled disks, represented by $W^{(n)\,+}_6$ and $W^{(n)\,+}_3$, a boundary node $W^{(n)\,+}_4$ and possibly further bubbled pieces which can appear in the limit of long strips $W^{(n)\,+}_2$ and $W^{(n)\,+}_5$. Note also that the action of the involution $\tau$ on the pants $W^{(n)}_3$ is described in Fig. 12b).

In [*Subcase 3$'''_b$)*]{} the bubbling takes place near but not at the boundary. Indeed, since ${\rho_n \over r_n}\lrar \infty$, the bubbled sphere which appears as the limit of the sequence $\{v_n \}$ is not $\tau$-invariant. To see this phenomenon, we note that for any fixed $b>0$ the covering pieces $W^{(n)}_5= \Delta' (x_n,2br_n)$ representing a sufficient big part of this sphere lie in $\cala^+_0$ for $n>\!>0$. This implies that the sequence $v_n
\scirc \tau$ converges to another bubbled sphere, which is $\tau$-symmetric to the first and represented by $W^{(n)}_7$.

Another bubbled sphere, represented by $W^{(n)}_3$, appears from pants between the first two spheres and the disk $\Delta'$. Since $\{ {R_n
\over \rho_n}\}$ remains bounded, the original nodal point remains on this latter sphere.

[A3.5. Attaching an Analytic Disk to a Lagrangian Submanifold of $\cc^n$]{}.

As in the first lecture we consider $\cc^n \approx \rr^{2n}$ together with some symplectic form $\omega $ taming the standart complex structure $J\st $. An $n$-dimensional submanifold $W\subset \cc^n$ is called $\omega $-Lagrangian if $\omega\mid_W\equiv 0$.

**Exercise 1. Prove that every Lagrangian submanifold of $\cc^n$ is totally real.**

[**2.**]{} Prove that for a Lagrangian manifold $W\subset \cc^n$ and the unit circle $\ss^1\subset \cc $ the manifold $\ss^1\times W$ is Lagrangian in $\cc^{n+1}$ with respect to $\hat\omega = {i\over 2}dz_1\wedge d\bar z_1 + 
\omega $.

A holomorphic map $u:\Delta \to \cc^n$ “sufficiently smooth” up to the boundary and such that $u(\d \Delta )\subset W$ we shall call *an analytic disk attached to $W$.*

Our goal in this paragraph is to prove the following theorem of Gromov:

Theorem A3.5.1. [*Let $W$ be a compact Lagrangian submanifold of $\cc^n$. Then there exists a non-constant analytic disk attached to $W$.* ]{}

We shall closely follow the exposition of H. Alexander, \[Al\]. Fix some point $w_0\in W$ and denote by $u_0(z)\equiv w_0$ the constant holomorphic map. Fix $p>2$ and consider the Banach manifold $L^{2,p}(\Delta , \d \Delta , 1; \cc^n , W, w_0)$ of $L^{2,p}$-maps from $\bar\Delta $ to $\cc^n$ which map $\d \Delta $ to $W$ and $1$ to $w_0$, and which are homotopic to the constant map $u_0\equiv w_0$ as $(\Delta , \d \Delta , 1)\to (\cc^n , W, w_0)$ mappings. Note that due to the Sobolev imbedding $L^{2,p}\subset C^{1,1-{2\over p}}$ our mappings are smooth up to the boundary.

Take $u\in L^{2,p}(\Delta , \d \Delta , 1; \cc^n , W, w_0)$. Denote by $E$ as usually the pull-back by $u$ of the tangent bundle of $\cc^n$. In fact $E=\Delta\times \cc^n \to \Delta $, the trivial bundle over $\Delta $. Denote by $F$ the pull-back $u^*TW$ of the tangent to $W$ bundle. $F$ is a totally real subbundle of $E$ of real dimension $n$. The tangent space to $L^{2,p}(\Delta , \d \Delta , 1; \cc^n , W, w_0)$ at $u$ is $T_uL^{2,p}(\Delta , \d \Delta , 1; \cc^n , W, w_0) = 
\{ h\in L^{2,p}(\Delta ,\cc^n): h\mid_{\d \Delta }\in F, h(1)=0\} $. In the cartesian product $L^{2,p}(\Delta , \d \Delta , 1; \cc^n , W, w_0)
\times L^{1,p}(\Delta ,\cc^n)$ consider the submanifold $\cale =\{ (u,v):
\dbar u=v\} $ with the natural projection $\pi :\cale \to L^{1,p}(\Delta ,\cc^n)$.

**Exercise 1. In $\rr^{2n+2}$ consider the operator given by the matrix**

$$J_v = \left(
\matrix
0 & -1 & ......  & v_2       & -v_1 \cr
1 &  0 & ......  & -v_1      & -v_2 \cr
. &  . & ....... & ......    & ... \cr
0 &  0 & ....... & v_{2n}    & -v_{2n-1} \cr
0 &  0 & ....... & -v_{2n-1} & -v_{2n} \cr 
0 &  0 & ....... & 0         & -1 \cr
0 &  0 & ....... & 1         & 0 \cr
\endmatrix
\right)$$ Prove that $J_v$ defines an almost complex structure in $\rr^{2n+2}$, which for every $z\in \rr^2$ on the vertical slice $\{ z\} \times \rr^{2n}$ coincides with the standart structure $J\st$ of $\cc^n$.

[**2.**]{} Prove that the equation $\dbar u=v$ for a $C^1$-map $u:\cc \to \cc^n$ is equivalent to the $J_v$-holomorphicity of the section $\hat u :z\to (z,u(z))$ of the fibration $(\rr^{2n+2},J_v)\to (\rr^2, J\st)$, i.e. to the equation

$${\d (z,u)\over \d x} + J_v(z,u)[{\d (z,u)\over \d y}] = 0.$$

Now we shall prove the following alternative:

Lemma A3.5.2. [*If there is no nonconstant analytic disk $u\in  L^{2,p}(\Delta , \d \Delta , 1;$ $ \cc^n , W, w_0)$, then the projection $\pi :\cale \to L^{1,p}(\Delta ,\cc^n)$ is surjective.* ]{} Proof.

Suppose that a nonconstant analytic disk $u\in L^{2,p}(\Delta , \d \Delta , 1; \cc^n , W, w_0)$ doesn’t exists. We are going to prove that in this case $\pi $ is surjective.

[*Step 1. *$\pi $ is a proper mapping, i.e. for the converging sequence $v_k\to v_0$ in $L^{1,p}(\Delta ,\cc^n)$ and for the sequence $u_k$ with $(u_k,v_k)\in \cale $, there is a converging subsequence $u_{k_n}$.** ]{}

Note that $\dbar u_k=v_k$. According to the Exercise above this means that the sections $\hat u_k:=(z,u_k)$ are $J_{v_k}$-holomorphic with $J_{v_k}$ converging to $J_{v_0}$ in $C^0$-sence. Note also that boundaries of our disks are on the Lagrangian (and thus totally real) submanifold $\hat W:=\ss^1\times W$ and they are homotopic to each other. From here we see that

$$\area(\hat u_k(\Delta )) = \int_{\hat u_k(\Delta )}\hat\omega = 
\int_{\hat u_k(\d \Delta )}\d \lambda ,$$ where $\lambda $ is some primitive of $\hat\omega $. The second integral does not depend on the homology class of $\hat u_k(\Delta )$ in $\sfh_1(\hat W, 
\rr)$, because $\hat\omega\mid_W = \d \lambda\mid_{\hat W} \equiv 0$ ($\hat W$ is $\hat\omega $-Lagrangian!).

So by the Theorem A3.4.1 either the limit of some subequence, still denoted as $u_k(\Delta )$, contains a nonconstant complex sphere (this is impossible in $\cc^n$), or the limit of $u_k(\Delta )$ contains some nonconstant analytic disk with boundary on $W$ (this is prohibited by our assumption), or $u_k$ $C^1$-converge.

[*Step 2. *$d\pi_{(u,v)} : T_{(u,v)}\cale \to L^{1,p}(\Delta ,\cc^n)$ is Fredholm of index zero for every $u\in \cale_v :=\pi^{-1}(v)$.** ]{} The fact that the boundary value problem $\dbar h=v, h\mid_{\d \Delta }\in F$ is Fredholm is classical, see \[Ga\]. Our manifold $\cale $ is connected, so the index of $d\pi_{(u,v)}$ doesn’t depend on $(u,v)$ and can be calculated at $(u_0,0)\in \cale $ where $d\pi $ is a bijection. Therefore $ind(d\pi )$ is everywhere zero.

One says that for the smooth mapping $\pi :\cale \to L^{1,p}(\Delta ,\cc^n)$ the point $(u,v)$ is regular if $d\pi_{(u,v)}$ is surjective. $v$ is a regular value is it is not an image of a nonregular point.

[*Step 3. (Smale’s theorem). *Let $\pi :\cale \to \calm $ be a proper Fredholm map (i.e. $d\pi_x $ is Fredholm for all points $x\in \cale $. Then the set of regular values is dence in $\calm $. Moreover, for every regular value $v\in \calm $ the set $\cale_v:=\pi^{-1}(v)$ is a manifold of dimension equal to the $\ind(d\pi_x) $ at $x\in \cale_v$. Moreover, for any two regular values $v_1$ and $v_2$ the manifolds $\cale_{v_1}$ and $\cale_{v_2}$ are cobordant.** ]{} See \[Sm\].

In our case $0$ is a regular value, so for a dence subset of $v$’s $\cale_{v}$ is cobordant to a point, therefore is a point intself. Properness of $\pi $ implies now that $\cale_v$ is allways a point.

It is not difficult to show that $\pi $ cannot be surjective. This will imply the existence of non constant analytic disk attached to $W$.

Lemma A3.5.3. [*The projection $\pi :\cale \to L^{1,p}(\Delta ,\cc^n)$ is not surjective.* ]{} Proof. [Othervice, for $v^C=(C,0,....,0)$ find $u^C=(u_1^C,...,u_n^C)$ with $\dbar u^C = v^C$. Therefore $\bar \d u_1^C=C$. This implies that $u_1^C=C\bar z - h^C$, where $h^C$ is a holomorphic function on $\Delta $. Since $u^C(\d \Delta )\subset W$ the family $u^C$ is uniformly bounded on $\d \Delta $ by a constant $k$ independent of $C$. Therefore $\vert \bar z - 
h^C(z)/C\vert \le k/C$ for $z\in \d \Delta $. Since $\bar z - h^C(z)/C$ is harmonic on $\Delta $, the bound holds for all $z\in \Delta $. This implies that $\bar z$ can be uniformly approximated on $\Delta $ by holomorphic functions. Contradiction. ]{}

[Chapter III. Global Properties and Moduli Spaces.]{}

Let $(X, J)$ be an almost-complex manifold of complex dimension $n$. An almost complex structure is always assumed to have smoothness of class $C^1$. Further, let $(S, J_S)$ denote a Riemann surface with complex structure $J_S$.

In Lecture 7 we show that for a $J$-complex curve $u:(S,J_S) \to (X,J_0)$ the pulled-back bundle $E\deff u^*TX$ possesses a natural [*holomorphic*]{} structure (the corresponding sheaf of holomorphic sections will be denoted as $\calo(E)$) such that the differential $du:TS\to E$ is a [*holomorphic*]{} homomorphism. This allows us to define the order of vanishing of the differential $du$ at point $s\in S$. We denote this number by ${\sl ord}_sdu$.

This also gives the following short, exact sequence: $$0\longrightarrow \calo(TS) \buildrel du \over\longrightarrow
\calo(E) \buildrel \pr \over\longrightarrow
\calo(N_0)\oplus \caln_1 \longrightarrow 0,\eqno(3.1)$$ where $\calo(N_0)$ denotes a free part of the quotient $\calo(E)/du(\calo(TS))$, and $\caln_1$ is supported on a finite set of cusps of $u$ ([**]{} points of vanishing of $du$).

On the Sobolev space $L^{1,p}(S,N_0)$ of $L^{1,p}$-smooth sections of the bundle $N_0$ the natural Gromov operator $D_N: L^{1,p}(S,N_0) \to
L^p(S,\Lambda^{0,1}S \otimes N_0)$ is defined. Put $\sfh^0_D(S, N_0):=
\ker D_N$ and $\sfh^1_D(S, N_0):=\coker D_N$.

We prove the following Theorem 3.1.

*Let $u: (S, J_S) \to (X, J_0)$ be a nonconstant irreducible and non- multiply covered holomorphic map such that $\sfh^1_D(S, N_0)=0$. Then*

in a neighborhood of $M\deff u(S)$ the Moduli space of nonparameterized $J_0$-holomorphic curves $\calm_{[\gamma], g, J_0}$ is a manifold whose tangent space is $T_M\calm_{[\gamma], g, J_0} =
\sfh^0_D(S, N_0) \oplus \sfh^0(S, \caln_1)$;

further, there is a neighborhood $V\ni J$ in the Banach manifold $\calj$ of $C^1$-smooth almost-complex structures on $X$ and a neighborhood $W$ of $M$ in $\calm_{[\gamma], g, V}:=\bigcup_{J\in V}
\calm_{[\gamma], g, J}$ such that the natural projection $\pr_\calj:
W\to V$ is a trivial Banach bundle;

if $\dim_{\rr }X=4$ and $c_1(E)[M] > \sum_{s\in S}{\sl ord}_sdu$, then $\sfh^1_D(S, N_0)=0$; thus, the conclusions and hold.

Let us explain our second result in this chapter, which will be strongly relied upon in Chapter IV and also in the proof of the Gromov non-squeezing theorem. Denote by $\calj $ some Banach manifold of $C^k$-smooth almost-complex structures on a manifold $X$, $k\ge 1$. In our applications $\calj $ can be

1\) the manifold $\calj_{\omega }$ of all a.-c. structures $J$ on $X$, which are tamed by some fixed symplectic form $\omega $; i.e., $\omega (u,Ju)>0$ for all $u\in TX\setminus \{ 0\} $.

2\) the manifold $\calj_U$ of all a.-c. structures $J$ on a complex Kähler manifold $(X,J\st ,\omega )$ tamed by $\omega $ and which are different from $J\st $ only on some relatively compact subset of $U$, i.e., $\{ x\in X:J_x\not= J\st \} \comp U$.

Denote by $\calm =\calm_{\calj ,[\gamma ],\Sigma }$ the (topological) space of all $J$-complex curves in $X$ ($J$ runs over all structures in $\calj $) representing some fixed homology class $[\gamma ]\in \sfh_2(X,\zz )$, and parameterized by some fixed compact surface $\Sigma $, see Chapter II. For $\calj $, as in 1) and 2), above the space $\calm $ is a manifold in all its non- multiply covered points, see [*Lemma 8.2.2*]{}.

Also let $h:[0,1]\to \calj $ be some smooth map (more generally one can consider a smooth map $h:Y\to \calj $, where $Y$ is a compact real manifold with boundary). By $\calm_h$ we denote the subset of $\{ (u,J_S,t)
\in \cals \times \ttt_{\Sigma }\times [0,1]\} $ which consists of $h(t)$-complex curves $u:(S,J_S)\to (X,h(t))$ modulo the natural action of the group $G$ of reparameterizations, see §7.4 and Definition 8.3.1. Roughly speaking, $\calm_h$ is a moduli space of $h(t)$-complex curves in $X$, $t\in [0,1]$. This is a closed subset of $\calm $. For a point $(M_0,t_0)$ in $\calm_h$ (for a $h(t_0)$-complex curve $M_0$) one can define a component $\calm_h(M_0,t_0)$ of $\calm_h$ containing $(M_0,t_0)$ as was shown in [*Definition 8.3.2*]{}.

Theorem 3.2. [*If the component $\calm_h(M_0,t_0)$ is not compact, then there exists a continuous path $\beta :[0,1]\to \calm_h(M_0,t_0)$, $\beta (t) =(M_t,J_t)$ starting at $(M_0,t_0)$ such that $\beta (t_n)$ is not bounded in $\calm_h(M_0,t_0)$ for some sequence $t_n\to 1$, but $J_{t_n}$ converge to some $J^*\in \calj $.* ]{}

This means, in fact, that in the limit the sequence $M_{t_n}$ breaks into several irreducible $J^*$-complex components.

Let us consider an example where such behavior is not possible.

Example.

Suppose that $X=S^2\times Y$ and $\omega =\omega_1
+ \omega_2$, where $\omega_1$ is some positive form on $S^2$ and $\omega_2$ is some symplectic form on a compact manifold $Y$. As a homology class $[\gamma ]$ take $S^2\times \{ pt\} $ and as a parameterizing surface for our curves again $\Sigma :=S^2$.

Observe that for any structure $J$ on $X$ tamed by $\omega $ no $J$-complex curve $C\in [\gamma ]$ can be decomposed as $C=C_1\cup C_2$ for some $J_k$-complex curves $C_k$, where $J_k\in \calj_{\omega }$!

Exercise. [Prove this.]{}

In this example $\calm_h(M_0,t_0)$ will be always be compact.

Theorem 3.3.

*Under the above conditions suppose that $S$ is a sphere $S^2$ and that $\calm_h(M_0,t_0)$ is compact. Then the path $h$ can be $C^1$-approximated by the paths $h_n$, all starting at $(M_0,t_0)$ such that for all points $(M,t)\in \calm^0_{h_n}$ the associated $D_N$-operator is surjective, i.e., $\sfh^1_{D_N}(S^2,N_{M_t})=0$.*

Moreover, $\calm_{h_n}$ is a trivial bordism: $\calm_{h_n(0)}\times [0,1]$. In particular for every $h_n(0)$-holomorphic sphere $M_0\in \calm_{h_n(0)}$ there exists a continuous family of $h_n(t)$-holomorphic spheres $M_{n,t}=
u_{n,t}(S^2)$ with $M_{n,0}=M_0$.

[Lecture 6]{}

[First Variation of the $\dbar_J$-equation.]{}

In this paragraph we want to introduce, following Gromov, a $D_{u,J}$-operator associated to the $J$ - complex curve $u : S \to X$ in an almost complex manifold $(X,J)$. For the convenience of the reader we now recall the notion of a connection.

*6.1. Symmetric Connections.*

Let $M$ be a real manifold, $E$ a real vector bundle over $M$. By $\Gamma (E)$ we denote the space of smooth sections of $E$ and by $\Lambda^p$, a space of smooth $p$-forms on $M$.

Definition 6.1.1.

*A connection on the real vector bundle $E\to M$ is a mapping $\nabla :\Gamma (E) \to \Lambda^1\otimes \gamma (E)$ such that*

$\nabla 1$. $\nabla $ is $\rr $-linear;

$\nabla 2$. for any smooth function $f$ on $M$ and any section $e\in \Gamma (E)$

$$\nabla (f\cdot e) = df\otimes e + f\cdot \nabla e.\eqno(6.1.1)$$

If $X\in \Gamma (TM)$ is a vector field on $M$, one denotes $\nabla_Xe :=
\nabla (e)(X)$, a covariant derivative of $e$ along $X$.

Remarks. 1. Choose a local frame $e_1,...,e_n$ of $E$ and write $\nabla (e_i) =
\Gamma_i^j\otimes e_j$, where $\Gamma_i^j$ are some $1$-forms. Then $\nabla $ is uniquely determined by $\{ \Gamma_i^j\}_{i,j=1}^n$.

2\. The dual connection $\nabla^*$ is defined on the dual bundle $E^*\to M$ by $$d\< h,e\> = \<\nabla^*h,e\> + \<h, \nabla e\>, \eqno(6.1.2)$$ where $h\in \Gamma (E^*)$.

3\. One defines an extension of $\nabla $ onto $E^{\otimes p}\otimes
E^{\otimes q}\to M$ as $$\nabla (e\otimes h) = \nabla e\otimes h + e\otimes \nabla h. 
\eqno(6.1.3)$$ In particular, since $\hom (E,E) \equiv E\otimes E^*$, $\nabla $ extends onto $\hom (E,E)$.

We shall usually denote all these extensions by the same symbol $\nabla$. One remarks, of course, that (6.1.1), (6.1.2) and (6.1.3) are nothing but the Leibnitz formula.

4\. Mapping $\nabla^2 = \nabla\scirc \nabla :\Gamma (E) \to \Lambda^2\otimes
\Gamma (E)$ is called the curvature of $\Gamma $ and is, in fact, linear over $\cala (M)$: $\nabla^2(fe)=\nabla (df\otimes e+f\nabla e)=
d^2f\otimes e-df\wedge \nabla e+df\wedge \nabla e+f\nabla^2e= f\nabla^2e$.

Definition 6.1.2. [*Let $E=TM$. A connection $\nabla $ is called symmetric if $\nabla_XY-\nabla_YX = [X,Y]$.* ]{}

Exercise. [Take $e_i={\d \over \d x_i}$ for some local coordinates $\{ x_i\} $. Write $\Gamma_i^j=\gamma_{ik}^jdx^k$, i.e., $\nabla_{e_k}e_i =
\Gamma_{ik}^je_j$. What does symmetricity of $\nabla $ mean in terms of $\{
\Gamma_{ik}^j\} $? ]{}

Exercise.

Choose a symmetric connection $\nabla $. Then the Nijenhuis tensor of an almost-complex structure $J$ is defined as

$$4N_J(X,Y) = (\nabla_XJ\scirc J)Y - (\nabla_YJ\scirc J)X + (\nabla_{JX}J)Y -
(\nabla_{JY}J)X.$$ Prove that this defintion doesn’t depend on the choice of a symmetric connection, i.e. express $N_J(X,Y)$ in the terms of Lee brackets.

*6.2. Definition of the $D_{u, J}$-operator.*

Recall that a $C^1$ map $u:S\to X$ from a Riemann surface $S$ with a complex structure $J_S$ is called holomorphic with respect to $J_S$ and $J$ if it satisfies the equation

$$du + J\scirc du\scirc J_S = 0. \eqno(6.2.1)$$

This simply means that $du\scirc J_S = J\scirc du$, [*i.e.,*]{} $du:T_pS \to
T_{u(p)}X$ is a [*complex*]{} linear map for every $p\in S$. Equation (6.2.1) is an elliptic quasi-linear PDE of order one. We are interested in the behavior of the solutions of (6.2.1), in particular, when the structures $J$ and $J_S$ change. So we need to choose appropriate functional spaces both for solutions and for the coefficient of (6.2.1). Our choice is based on the following facts:

[*a)*]{} The minimal reasonable smoothness of an almost complex structure $J$ on $X$, for which the Gromov operator $D_{u, J}$ can be defined, is $C^1$, see the explicit formula (6.2.5).

[*b)*]{} It is more convenient to operate with Banach spaces and manifolds and thus with finite regularity like $C^k$, $C^{k,\alpha}$ or $L^{k,p}$ than with $C^\infty$-smoothness defining only a Frechet-type topology.

[*c)*]{} Equation (6.2.1) is defined also for $u$ lying in Sobolev-type spaces $L^{k,p}(S, X)$ with $k\ge1$, $1\le p\le\infty$ and $kp>2$; such solutions are $C^1$-smooth and the topology on the space of a solution is, in fact, independent of the particular choice of such a Sobolev space.

[*d)*]{} For $J\in C^k$ the coefficients of the Gromov operator $D_{u, J}$ are, in general, only $C^{k-1}$-continuous, see (6.2.5), and hence solutions of the “tangential equation” $D_{u, J}v=0$ are only $L^{k,p}$-smooth, $1\le
p<\infty$; thus for obtaining a [*smooth*]{} structure on a space of (parameterized) $J$-complex curves, one should use Sobolev spaces $L^{k', p}(S, X)$ with $k'\le k$.

Fix a compact Riemann surface $S$, [*i.e.,*]{} a compact, connected, oriented, smooth manifold of real dimensions 2. Recall that the Sobolev space $L^{k, p}(S, X)$, $kp>2$, consists of those continuous maps $u: S \to X$, which are represented by $L^{k, p}$-functions in local coordinates on $X$ and $S$. This is a Banach manifold, and the tangent space $T_uL^{k, p}(S, X)$ to $L^{k, p}(S, X)$ in $u$ is the space $L^{k,p }(S, u^*(TX))$ of all $L^{k,p}$-sections of the pull-back under $u$ of the tangent bundle $TX$. One has the Sobolev imbeddings $$\matrix
L^{k,p}(S, X)& \hookrightarrow & L^{k-1, q}(S, X),&
\quad & \hbox{for $1\le p<2$}   &  \hbox{and} & 1\le q\le {2p\over2-p},
\cr
L^{k, p}(S, X)& \hookrightarrow & C^{k-1, \alpha}(S, X),&
\quad & \hbox{for $2<p\le\infty$}& \hbox{and} & 0\le\alpha\le 1-{2\over p}.
\cr
\endmatrix$$

Let $[\gamma ]$ be some homology class in $\sfh_2(X, \zz)$. Fix $p$ with $2<p<\infty$ and consider the Banach manifold $$\cals = \{u\in L^{1,p}(S, X):u(S)\in [\gamma]\}$$ of all $L^{1,p }$-smooth mappings from $S$ to $X$, representing the class $[\gamma]$. This makes sence, because $L^{1,p}\subset C^{0,1-{2\over p}}$, and therefore all $u$ from $\cals $ are continuous. Remark that the tangent space to $\cals $ at $u$ is

$$T_u\cals = L^{1,p}(S,u^*(TX))$$ the space of $L^{1,p}$-sections of the pulled-back by $u$ the tangent bundle of $X$.

Denote by $\calj$ the Banach manifold of $C^1$-smooth almost complex structures on $X$. In other words, $\calj = \{ J\in C^1
(X, \endo (TX)):J^2=-\id  \} $. The tangent space to $\calj$ at $J$ consists of $C^1$-smooth $J$-antilinear endomorphisms of $TX$, $$T_J\calj = \{ I\in C^1(X, \endo (TX)) : JI + IJ = 0\}
\equiv C^1(X, \Lambda^{0,1}X \otimes TX),$$ where $\Lambda^{0,1}X$ denote the complex bundle of $(0,1)$-form on $X$.

Denote by $\calj_S$ the Banach manifold of $C^1$-smooth complex structures on $S$. Thus, $\calj_S = \{ J_S\in C^1(S, \endo (TS)):J_S^2=-\id  \}$ and the tangent space to $\calj_S$ at $J_S$ is $$T_{J_S}\calj = \{ I\in C^1(S, \endo (TS)) : J_S I + IJ_S = 0\}
\equiv C^1(S, \Lambda^{0,1}S\otimes TS).$$

Consider also the subset $\calp\subset \cals\times \calj_S \times
\calj$ consisting of all triples $(u, J_S, J)$ with $u$ being $(J_S,J)$-holomorphic, [*i.e.,*]{}

$$\calp = \{(u, J_S, J)\in \cals\times \calj_S\times \calj: du +
J\scirc du\scirc  J_S = 0 \}. \eqno(6.2.2)$$

Lemma 6.2.1. *Let $J$ and $J_S$ be continuous almost-complex structures on $X$ and $S$, respectively, and let $u\in L^{1,p}(S, X)$. Then $$du + J\scirc du\scirc J_S \in L^p(S,\Lambda^{0, 1}S \otimes u^*(TX)).$$*

Proof. One can easily see that $du \in L^p(S,\hom_\rr(TS, u^*(TX))$. On the other hand, $$(du + J\scirc du\scirc J_S ) \scirc J_S= -J \scirc
(du + J\scirc du\scirc J_S ),$$ which means that $du + J\scirc du\scirc J_S$ is $L^p$-integrable $u^*(TX)$-valued (0,1)-form.

Consider a Banach bundle $\calt\to \cals\times \calj_S\times \calj$ with a fiber $$\calt_{(u, J_S, J)} = L^p(S,\Lambda^{0, 1}S\otimes
u^*(TX)),$$ where $TS$ and $TX$ are equipped with complex structures $J_S$ and $J$, respectively. $\calt$ has two distinguished sections:

1\) $\sigma_0 \equiv 0$, the zero section of $\calt$;

2\) $\sigma_{\dbar}(u, J_S, J) = du + J\scirc du\scirc J_S$.

By definition $\calp$ is the zero-set of $\sigma_{\dbar}$.

Let us compute the tangent space to $\calp$ at the point $(u, J_S, J)$. Let $(u_t, J_S(t),J(t))$ be a curve in $\calp$ such that $(u_0, J_S(0), J(0)) =
(u, J_S, J)$. Let $$(v,\dot J_S,\dot J) \deff
\left({du_t\over dt}|_{t=0},
{dJ_S(t)\over dt}|_{t=0},
{dJ(t)\over dt}|_{t=0}\right)$$ be the tangent vector to this curve and hence to $\calp$ at $t=0$. The condition $(u_t, J_S(t), J(t)) \in \calp$ means that $$du_t + J(u_t, t)\scirc du_t\scirc J_S(t) = 0 \eqno(6.2.3)$$ in $ L^p(S,\Lambda^{0, 1}(S)\otimes u_t^*(TX))$. Let $\nabla $ be some symmetric connection on $TX$, [*i.e.,*]{} $\nabla_YZ - \nabla_ ZY = [Y,
Z]$. The co-variant differentiation of (6.2.3), with respect to $t$, gives $$\nabla_{\partial \over \partial t}(du_t) + (\nabla_vJ)(du_t\scirc J_S)
+ J(u_t, t)\scirc \nabla_{\partial \over \partial t}(du_t)\scirc J_S +$$ $$+ J\scirc du_t\scirc \dot J_S +
\dot J\scirc du_t\scirc J_S = 0.$$ Let us show that $\nabla_{\partial \over \partial t}(du_t) = \nabla v$. Indeed, for $\xi \in TS$ one has $$(\nabla_{\partial \over \partial t}
du_t)(\xi ) = \nabla_{\partial \over \partial t}[du_t(\xi )] =
\nabla_{\partial \over \partial t}({\partial u_t\over \partial \xi }) =
\nabla_{\xi }({du_t\over dt}) = \nabla_{\xi }v.$$ So every vector $(v,\dot J_S,\dot J)$ which is tangent to $\calp$ satisfies the equation

$$\nabla v + J\scirc \nabla v\scirc J_S + (\nabla_vJ)\scirc (du\scirc
J_S) + J\scirc du\scirc \dot J_S +
\dot J\scirc du\scirc J_S = 0. \eqno(6.2.4)$$

[**Definition 6.2.1.**]{} Let $u$ be a $J$ - complex curve in $X$. Define the operator $D_{u, J}$ on $L^{1,p}$-sections $v$ of $u^*(TX)$ as $$D_{u, J}(v) = \msmall{1\over2}\bigl(\nabla v + J\scirc\nabla v\scirc J_S
+ (\nabla_vJ) \scirc (du\scirc J_S).  \bigr)
\eqno(6.2.5)$$

[**Remark.**]{} This operator plays a crucial role in studying properties of complex curves. In its definition we use the symmetric connections instead of those compatible with $J$, as is shown in \[G\]. The matter is that one can use the same connection $\nabla$ for changing almost complex structures $J$. The lemmas below justify our choice.

[**Lemma 6.2.2.** ]{}*$D_{u, J}$ does not depend on the choice of a symmetric connection $\nabla $ and is an $\rr$-linear operator from $L^{1,p}(S,
u^*(TX))$ to $L^p(S, \Lambda^{0, 1}S \otimes u^*(TX))$.*

[**Proof.**]{} Let $\widetilde \nabla $ be another symmetric connection on $TX$. Consider the bilinear tensor on $TX$, given by formula $Q(Z, Y) {:=} \nabla_ZY
- \widetilde \nabla_ZY$. It is easy to see that $Q$ is symmetric on $Z$ and $Y$. Also note that $\nabla_{\xi }v - \widetilde\nabla_{\xi }v =
\nabla_{du(\xi )}v
- \widetilde \nabla_{du(\xi )}v = Q(du(\xi), v)$ and, in addition, that $(\nabla_ZJ)(Y) - (\widetilde \nabla_ZJ)(Y) = Q(Z, JY) - JQ(Z, Y)$. From here one obtains $$2(D_{u, J}v)(\xi ) - 2(\widetilde D_{u, J}v)(\xi) =$$ $$= \nabla_{\xi }v - \widetilde \nabla_
{\xi }v + J(\nabla_{J_S\xi }v - \widetilde \nabla_{J_S\xi }v) + (\nabla_vJ -
\widetilde \nabla_vJ)du(J_S\xi ) =$$ $$= Q(du(\xi ), v) + JQ(du(J_S\xi ), v) + Q(v, JduJ_S
\xi ) - JQ(v, du(J_S\xi)) =$$ $$=Q(du(\xi ), v) + Q(JduJ_S\xi , v) = Q((du + JduJ_S)(\xi ), v) = 0.$$

Now let us show that $D_{u, J}(v)$ is $J_S$-antilinear: $$2D_{u, J}(v)[J_S\xi] = \nabla_{J_S\xi }v + J\scirc \nabla_{J^2
_S\xi }v + (\nabla_vJ)\scirc (du\scirc J^2_S)(\xi ) =$$ $$= \nabla_{J_S\xi }v -
J(\nabla_{\xi }v) - (\nabla_vJ)(du(\xi )) = -J[\nabla_{\xi }v + J(\nabla
_{J_S\xi }v) - J\nabla_vJdu(\xi )] =$$ $$= -J[\nabla_{\xi }v + J\scirc \nabla_{J_S\xi}v + (\nabla_vJ)
(du\scirc J_S(\xi )] = -2J\, D_{u, J}(v)[\xi].$$ Here we use the fact that $J\scirc \nabla_vJ + \nabla_vJ\scirc J = 0$ and $du\scirc J_S = J\scirc du $.

This lemma allows us to obtain expression (6.2.4) also by computation in local coordinates $x_1, x_2$ on $S$ and $u_1,\ldots, u_{2n}$ on $X$, choosing as connection $\nabla$ the de Rham differential $d$. Really, wright $u_t(x) = 
u(x,t) = (u_1(x_1,x_2,t),...,u_{2n}(x_1,x_2,t))$ for the $J_t$-holomorphic map $u:S\to X$ (note that $u_t$ and structure $J_t$ are both time dependent). In the local basis ${\d \over \d u_1},...,{\d \over \d u_{2n}}$ of the tangent bundle $TX$ $J_t$ is represented by a time dependent $2n\times 2n$ matrix, which we will also denote as $J(\cdot ,t)$. Differential of $u_t$ will be denoted as $du_t(x)$ or as ${du_t\over dx}$ and is $2\times 2n$ matrix $$du_t(x) =
\left( \matrix
{\d u_1\over \d x_1}(x,t)&{\d u_1\over \d x_2}(x,t)\cr
\cdot \cdot \cdot \cdot &\cdot \cdot \cdot \cdot \cr
{\d u_{2n}\over \d x_1}(x,t)&{\d u_{2n}\over \d x_2}(x,t)\cr
\endmatrix \right).
\eqno(6.2.6)$$

Complex structure $J_S(t)$ on the surface is also time dependent and in the local frame ${\d \over \d x_1},{\d \over \d x_2}$ of $TS$ is represented by a $2\times 2$ matrix, which we shall denote also as $J_S(t)$. Equation of holomorphicity (6.2.3) now reeds in matrix form as $${du_t\over dx} + J(u_t,t)\cdot {du_t\over dx}\cdot J_S(t)\equiv 0.
\eqno(6.2.7)$$ Differentiating this with respect to $t$ we get

$${d\over dx}\left[{du_t\over dt}\right] + 
{\d J\over \d t}(u_t,t)\cdot {du_t\over dx}\cdot J_S(t) + 
\left\<{\d J(u_t,t)\over \d u},{du_t\over dt}\right\>
\cdot {du_t\over dx} \cdot J_S(t) 
+$$ $$+ J(u_t,t)\cdot {d\over dx}
\left[{du_t\over dt}\right]\cdot J_S(t) + J(u_t,t)\cdot 
{du_t\over dx}\cdot {dJ_S(t)\over dt}  \equiv 0.
\eqno(6.2.8)$$ Here we put $\<{\d J(u_t,t)\over \d u},{du_t\over dt}\>:=\sum_{j=1}^{2n}{\d 
J\over \d u},{du_j(t)\over dt}$. Therefore the vector $$(v,\dot J_S,\dot J) \deff
\left({du_t\over dt}|_{t=0},
{dJ_S(t)\over dt}|_{t=0},
{dJ(t)\over dt}|_{t=0}\right)$$ is tangent to $\calp $ at $(u_0,J_S(0),J_0)$ if and only if $${dv\over dx} + \left\<{\d J(u_t,t)\over \d u},v \right\>
\cdot {du_t\over dx}\cdot J_S + 
J(u_0,0)\cdot {dv\over dx}\cdot J_S(0) +$$ $$+ \dot J\cdot {du\over dx}\cdot J_S + 
J\cdot {du\over dx}\cdot \dot J_S = O.\eqno(6.2.9)$$ Note that $\<{\d J(u_t,t)\over \d u},v\>={dJ\over dv}$ is the derivative of $J$ along the vector $v$. Therefore (6.2.9) with $\nabla =d$ gives the same expression as (6.2.4).

*6.3. $\dbar $-type Operators.*

Now we need to understand the structure of the operator $D_{u,J}$ in more detail. The problem arising here is that $D\deff D_{u,J}$ is only $\rr$-linear. So we decompose it into $J$-linear and $J$-antilinear parts. For the rest of this text we shall denote by $E$ the pulled-back $u^*TX$ by $u$ tangent bundle of $X$. For $\xi \in C^1 (S, TS)$ and $v\in L^{1,p}(S, E)$ write $D_\xi v= {1\over2}[ 
D_{\xi }v - JD_{\xi }(Jv)] + {1\over2} [D_{\xi }v + JD_\xi(Jv)] = 
\dbar_{u, J}[v](\xi) + R(v,\xi )$.

Definition 6.3.1. The operator $\dbar_{u, J}$, introduced above as the $J$-linear part of $D_{u, J}$, we shall call the [*$\dbar$-operator for a $J$-complex curve $u$*]{}.

Lemma 6.3.1. [*$\dbar_{u,J}: L^{1,p}(S,E) L^p(S,\Lambda^{0,1}S
\otimes E)$ is a first order differential operator satisfying $$\dbar_{u,J}(fv) = \dbar_{J_S}f\otimes v + f\otimes \dbar_{u,J}v .\eqno(6.3.1)$$ where $\dbar_{J_S}f={1\over 2}[df + i\cdot df\circ J_S]$ is $\dbar $-operator on $S$.* ]{}

Proof. Really, $$2D_{u,J}(fv)=\nabla (fv)  + J\circ \nabla (fv)\circ 
J_S + (\nabla_{fv}J)\circ (du\circ J_S) =$$

$$= f\nabla v + fJ\circ \nabla v\circ J_S + f\nabla_vJ\circ (du\circ J_S) +$$ $$+ df\otimes v + Jdf\otimes vJ_S = f2D_{u,J}v + 2\dbar_{J_S}f\otimes v.$$ From here we see that $2JD_{u,J}(Jfv) = 2fJD_{u,J}(Jv) + 2J\dbar f\otimes Jv$. So $$4\dbar_{u,J}(fv) = 2[D_{u,J}(fv) - JD_{u,J}(Jfv)] = 2f[D_{u,J}v-JD_{u,J}(Jv)] 
+ 4\dbar f\otimes v =$$ $$= 4f\dbar_{u,J}v + 4\dbar_{J_S}f\otimes v .$$

The following statement is well known in the smooth case. The line bundles case can be found in \[Hf-L-Sk\].

[**Lemma 6.3.2.**]{} [*Let $S$ be a Riemann surface with a complex structure $J_S$ and $E$ a $L^{1,p}$-smooth complex vector bundle of rank $r$ over $S$. Let also $\dbar_E : L^{1,p}(S, E) \to
 L^p(S,\Lambda^{(0, 1)}S\otimes E)$ be a differential operator, satisfying the condition $$\dbar_E(f\xi) = \dbar_S f \otimes \xi
 + f\cdot \dbar_E\xi,  \eqno(6.3.2)$$ where $\dbar_S$ is the Cauchy-Riemann operator, associated to $J_S$. Then the sheaf $$U\subset S \mapsto {\cal O}(E)(U) := \{\,\xi \in  L^{1,p}(U, E) \, :\,
 \dbar_E\xi=0 \,\}   \eqno(6.3.3)$$ is analytic and locally free of rank $r$. This defines the holomorphic structure on $E$, for which $\dbar_E$ is an associated Cauchy-Riemann operator.* ]{}

[**Remark.**]{} The condition $(6.3.2)$ means that $\dbar_E$ of order 1 and has the Cauchy-Riemann symbol.

[**Proof.**]{} It is easy to see that the problem is essentially local. So we may assume that $S$ is a unit disk $\Delta$ with the standard complex structure. Let $\xi=(\xi_1,\ldots,\xi_r)$ be a $L^{1,p}$-frame of $E$ over $\Delta$. Let also $\Gamma \in L^p(\Delta,\Lambda^{0, 1}\Delta
 \otimes Mat(r,\cc))$ be defined by relation $$\dbar_E \xi_i = \sum_j \Gamma_i^j \xi_j,
 \qquad \hbox{or in matrix form, } \qquad
 \dbar_E \xi =  \xi \cdot \Gamma.
 \eqno(6.3.4)$$ Then for any section $\eta=\sum g^i \xi_i$ the equation $\dbar_E
 \eta =0$ is equivalent to $$\dbar g^i + \sum_j \Gamma^i_j g^j=0,
 \qquad \hbox{or in matrix form, } \qquad
 \dbar g + \Gamma \cdot g =0.
 \eqno(6.3.5)$$

Let the map $\tau_t : \Delta \to \Delta$ be defined by formula $\tau_t(z) = t\cdot z$, $0<t<1$. One can easily check that $$\Vert \tau_t^*\Gamma \Vert_{L^p(\Delta)}
 \le t^{1-2/p} \Vert \Gamma \Vert_{L^p(\Delta)}.$$ So taking pull-backs $\tau_t^*E$ with $t$ sufficiently small we may assume that $\Vert \Gamma \Vert_{L^p(\Delta)}$ is small enough. Now consider the mapping $F$ from $L^{1,p}(\Delta,
  Gl(r,\cc))\subset $

$ \subset L^{1,p}(\Delta,
 Mat(r,\cc))$ to $L^p(\Delta,\Lambda^{0, 1}\Delta
 \otimes Mat(r,\cc))$ defined by formula $F(g):=
 \dbar g \cdot g^{-1}$. It is easy to see that the derivation of $F$ in $g\equiv \id$ equals to $\dbar$. Due to the [*Lemma 3.2.1*]{} and the implicit function theorem for any $\Gamma$ with $\Vert \Gamma \Vert_{L^p(\Delta)}$ small enough there exists $g$ in $L^{1,p}(\Delta, Gl(r,\cc))$ with $g(0)=\id$, satisfying the equality $F(g)=-\Gamma$, which is equivalent to (6.3.4). Consequently, in the neighbourhood of every point $p\in S$ there exists a frame $\eta =(\eta_1,\ldots,\eta_r)$ of $E$ consisting of sections of ${\cal O}(E)$. This implies that ${\cal O}(E)$ is analytic and locally free of rank $r$. This lemma has an interesting corollary, which will be not used in this notes. Let $E\to S$ be real vector bundle over a Riemann surface $(S,J_S)$, and let $J$ be a complex structure on $E$, i.e. $J\in 
 End(E)$ with $J^2=-\id $. Take some connection $\nabla $ on $E$ and consider a $J$-linear first order differential operator $D^{\nabla }v:={1\over 2}[\nabla v - J\nabla (Jv)]:\Gamma^{1,p}(S,E)\to 
 \Gamma^p(S,\Lambda^1_S\otimes E)$. Define

$$\dbar^{\nabla }v = {1\over 2}(D^{\nabla }v - JD^{\nabla }_{J_S\cdot }v).
 \eqno(6.3.6)$$

Lemma 6.3.3. [*Operator $\dbar^{\nabla }$ acts from $\Gamma^{1,p}(S,E)$ to $\Gamma^p(S,\Lambda^{0,1}_S\otimes E)$ and satisfies condition (6.3.2) i. e. $$\dbar^{\nabla }(fv) = \dbar_S f \otimes v
 + f\cdot \dbar^{\nabla }v , \eqno(6.3.7)$$ and therefore defines by [*Lemma 6.3.2*]{} a holomorphic structure on $E$ compatible with a given complex structure $J$.* ]{} Proof.

We need to check only the antilinearity of the form $\dbar^{\nabla }v$and relation (6.3.7). Take a $\xi \in TS$, then $$(\dbar^{\nabla }v)(J_S\xi ) = {1\over 2}[D^{\nabla }_{J_S\xi }v - 
 JD^{\nabla }_{\xi }v] = -J{1\over 2}[D^{\nabla }_{J\xi }v + 
 JD^{\nabla }_{J_S\xi }v] = -J\dbar^{\nabla }v(\xi ).$$ This shows antilinearity of $\dbar^{\nabla }v$ with respect to $TS$-variable $\xi $, and therefore $\dbar^{\nabla }$ acts from $\Gamma^{1,p}(S,E)$ to $\Gamma^p(S,\Lambda^{0,1}_S\otimes E)$. Note that $\dbar^{\nabla }$ is $J$-linear on the variable $v$ as $D^{\nabla }$ is. Further,

$$D^{\nabla }(fv) = {1\over 2}[df\otimes v + f\nabla v - Jf\nabla (Jv) - 
 Jdf\otimes Jv] =$$

$$= fD{\nabla }v + {1\over 2}(df\otimes v + df\otimes v) = fD^{\nabla }v + 
 df\otimes v.\eqno(6.3.8)$$ Therefore $$\dbar^{\nabla }(fv) = {1\over 2}[D^{\nabla }(fv) - JD^{\nabla }_{J_S\cdot }
 (fv)] = {1\over }[fD^{\nabla }v + df\otimes v - fJD^{\nabla }_{J_S\cdot }v 
 - Jdf\otimes v] =$$

$$= f\dbar^{\nabla }v + {1\over 2}[df\otimes v -Jdf\otimes v] = 
 f\dbar^{\nabla }v + \dbar _{J_S}f\otimes v .$$ This is because $$D^{\nabla }_{J_S\cdot }(fv) = {1\over 2}[\nabla_{J_S\cdot } (fv) - 
 J\nabla_{J_S\cdot }(fv)] = fD^{\nabla }_{J_S\cdot }v + {1\over 2}[df\circ J_S
 \otimes v - Jdf\circ J_S\otimes v] =$$ $$= fD^{\nabla }_{J_S\cdot }v + Jdf\circ J_S\otimes v =  
 fD^{\nabla }_{J_S\cdot }v  + \dbar_Sf\otimes v.$$ We used here the definition $if\otimes v = f\otimes Jv$ of multiplication by $i$ in the bundle $E$.

*6.4. Holomorphic Structure on the Induced Bundle.*

Relation (6.3.1) tells us that the $J$-linear operator $\dbar_{u, J} : 
L^{1,p}(S, E) \to L^p(S, \allowbreak 
\Lambda^{0, 1}S\otimes E)$ satisfies the condition (6.3.2) and therefore defines in way explained in the previous paragraph a holomorphic structure on the bundle $E$. We shall denote by ${\cal
O}(E)$ the sheaf of holomorphic sections of $E$.

Now let us turn to the antilinear part $R$ of Gromov’s operator.

**Lemma 6.4.1. *$R$ is a continuous $J$-antilinear operator from $E$ to $\Lambda^{0,1}\otimes E$ of the order zero, satisfying $$R(v,\xi ) = N(v, du(\xi )).\eqno(6.4.1)$$ and consequently $$R\scirc du \equiv 0.\eqno(6.4.2)$$ Here the second relations means that for all $\xi , \eta \in TS$ we have $R(du(\eta ),\xi )=0$.***

[**Proof.**]{} $J$ anti-linearity of $R$ is given by its definition. Compute $R(v,\xi )$ for $v\in L^{1,p}(S, E)$ and $\xi \in C^1(S, TS)$, setting $w\deff du(\xi)$ and $D\deff D_{u, J}$ to simplify the notations

$$4R(v,\xi ) = 2D[v](\xi ) + 2JD[Jv](\xi ) =$$

$$=\nabla_{\xi }v + J\nabla_{J_S\xi }v + \nabla_vJ\scirc du\scirc J_S(\xi )$$ $$+J\nabla_{\xi }(Jv) + J^2\nabla_{J_S(\xi )}(Jv) +
J \nabla_{Jv}J\scirc du\scirc J_S(\xi ) =$$

$$= \nabla_\xi v + J\nabla_{J_S\xi }v + \nabla_vJ \scirc du\scirc J_S(\xi ) +$$ $$+ J^2\nabla_{\xi }v + J(\nabla_wJ)v + J^3\nabla_{J_S\xi }v +
J^2(\nabla_{Jw}J)v + J\scirc \nabla_{Jv}J\scirc Jw.$$

Here we used the relations $\nabla_{du(\xi )}J = \nabla_wJ$, $du(J_S\xi) =Jw$ and $\nabla_{du(J_S\xi )}J = \nabla_{Jw}J$. Contracting terms, we obtain $$4R(v,\xi ) = \nabla_vJ(Jw) + J(\nabla_wJ)v - (\nabla_{Jw}J)v +
J(\nabla_{Jv}J(Jw)) =$$ $$= (\nabla_vJ\scirc J)w - (\nabla_wJ\scirc J)v - (\nabla_{Jw}J)v +
(\nabla_{Jv}J)w = 4N(v, w),$$ where $N(v, w)$ denotes the torsion tensor of the almost-complex structure $J$, see \[Li\], p.183, or \[Ko-No\], vol.II., p.123, where another normalization constant for the almost complex torsion is used. Finally we obtain $$R(v,\xi ) = N(v, du(\xi )).$$

$N$ is antisymmetric and $J$-antilinear on both arguments, so $$-JR\bigl(du(\eta ),\xi \bigr) = R\bigl(du(\eta ), J_S\xi \bigr) =
N\bigl(du(\eta ), du(J_S\xi ) \bigr)
= N\bigl(du(\eta ), du(\eta )\bigr) =0$$ if $\xi$ and $\eta$ were chosen in such a way that $J_S(\xi )=\eta $. The relation $R(du(\xi ),\xi )=0$ obviously follows from (6.4.1).

Remarks. 1. [If our structure $J$ is integrable, i.e. $N_J\equiv 0$ then by (6.4.2) $R\equiv 0$ and therefore $D_{u,J}=\dbar_{u,J}$ is $J$-linear. In fact we know that $J$ is integrable iff there exists a symmetric connection $\nabla $ on $TX$ compatible with $J$, i.e. $\nabla J=0$, see \[Ko-No\]. From (6.2.5) we see that in this case $D_{u,J}(v)={1\over 2}(\nabla v + J\circ 
\nabla v\circ J_S)$. So $D_{u,J}$ is onviously $J$-linear. ]{}

**2. The tangent bundle $TS$ to the Riemann surface $S$ carries a natural holomorphic structure. We shall denote by $\calo(TS)$ the corresponding analytic sheaf. If the structure $J$ on $X$ is integrable then the differential $du:(TS,\calo(TS))\to (S,E)$ of a holomorhpic map $u:S\to X$ is an analytic morphism of sheaves. We shall see know that this fact is stil true in nonintegrable case.**

**Lemma 6.4.2. *Let $u : (S, J_S) \to (X, J)$ be a non-constant complex curve in almost complex manifold $X$. Then $du$ defines an injective analytic morphism of analytic sheaves $$0\longrightarrow \calo(TS)
\buildrel du \over \longrightarrow \calo(E),  \eqno(6.4.3)$$***

where $E = u^*(TX)$ is equipped with a holomorphic structure defined as above by the operator $\dbar_{u, J}$.

[**Proof.**]{} Injectivity of a sheaf homomorphism is equivalent to its nondegeneracy, which is our case.

To prove holomorphicity of $du$ it is sufficient to show that for $\xi,\eta \in
C^1(S, TS)$ one has $$\bigl(\dbar_{u, J}(du(\xi ))\bigr)(\eta ) =
du\bigl((\dbar_S\xi ) (\eta )\bigr),  \eqno(6.4.4)$$

where $\dbar_S$ is the usual $\dbar $-operator on $TS$. We shall use the relation which is, in fact, the definition for $\dbar_S$:

$$(\dbar_S\xi )(\eta ) =
\msmall{1\over2}\left( \nabla_{\eta }\xi + J_S\nabla_{J_S\eta}\xi \right).
\eqno(6.4.5)$$ Here $\nabla$ is a symmetric connection on $S$, compatible with $J_S$. One has $$2\cdot (\dbar_{u, J}du(\xi ))(\eta ) = \nabla_{\eta }(du(\xi) ) +
J\nabla_{J_S\eta }(du(\xi))  + (\nabla_{du(\xi )}J)(du(J_S\eta )) =$$ $$= (\nabla_{\eta }du)(\xi ) + du(\nabla_{\eta }\xi ) +
J(\nabla_{J_S\eta}du)(\xi ) + J(du(\nabla_{J_S\eta }\xi ))
+ (\nabla_{du(\xi )}J)(du(J_S\eta )) =$$ $$= du\bigl[ \nabla_{\eta }\xi + J_S \nabla_{J_S\eta }\xi \bigr] +
\bigl[ (\nabla_{\eta }du)(\xi ) + J(\nabla_{J_S\eta}du)(\xi )
+ (\nabla_{du(\xi )}J)(du(J_S\eta )) \bigr]. \eqno(6.4.6)$$ The first term of $(6.4.6)$ is $2\cdot du(\dbar_S\xi)(\eta)$. To cancel the second one we use the identities $(\nabla_\xi du)[\eta] 
= (\nabla_\eta du)[\xi]$, $\nabla_wJ\scirc J = - J\scirc \nabla_w J$, and $(\nabla_\xi du)\scirc J_S= J\scirc(\nabla_\xi du) + \nabla_{du(\xi)}J\scirc 
du$. The last identity is obtained via co-variant differentiation of $du\scirc J_S=
J\scirc du$. Consequently, we obtain $$(\nabla_{\eta }du)(\xi ) + J(\nabla_{J_S\eta}du)(\xi ) +
(\nabla_{du(\xi )}J)(du(J_S\eta ))=$$ $$(\nabla_{\xi }du)(\eta ) + J(\nabla_\xi du)(J_S\eta ) +
(\nabla_{du(\xi )}J)(du(J_S\eta ))=$$ $$(\nabla_{\xi }du)(\eta ) +
J^2(\nabla_\xi du)(\eta ) +
J (\nabla_{du(\xi)}J)(du(\eta))
+ (\nabla_{du(\xi )}J)(du(J_S\eta) )=$$ $$=(J \scirc\nabla_{du\xi}J)(du\eta)
+ (\nabla_{du(\xi )}J\scirc J)(du(\eta ))=0.$$

Remark. We can give an alternative proof to both [*Lemmas 6.3.1*]{} and [*6.3.2*]{}, which does not use direct calculation. Fix a complex structure $J_S$ on $S$ and let $\phi_t$ be the one parameter group of diffeomorphisms of $S$, generated by a vector field $\xi$. Then ${d\over dt}|_{t=0}(\dbar\phi_t)=D_{J_S, \id }\xi=\dbar_S \xi$. Let a $J$-holomorphic map $u:S\to X$ also be fixed. Then $
{d\over dt}|_{t=0}(u\scirc \phi_t)=du(\xi)$ and consequently $$D_{J, u}(du(\xi))={d\over dt}\bigm|_{t=0}\dbar_J(u\scirc \phi_t)=
{d\over dt}\bigm|_{t=0}(du \scirc \dbar \phi_t)=
du\bigl({d\over dt}\bigm|_{t=0}(\dbar_{J_S} \phi_t)=$$ $$= du(\dbar_S\xi)$$ or equivalently $D_{J, u}\scirc du=du \scirc \dbar$. Taking the $J$-antilinear part of the last equality we obtain $R\scirc du=0$. Nevertheless we shall use the explicit form of the Gromov operator:

$$D_{u,J}(v)[\xi ] = \dbar_{u,J}(v)[\xi ] + N_J(v,du(\xi )).\eqno(6.4.6)$$

The zeroes of analytic morphism $du : \calo(TS) \to \calo
(E)$ are isolated. So we have the following

**Corollary 6.4.3. (\[Sk\]). *The set of critical points of a complex curve in almost the complex manifold $(X, J)$ is discrete, provided $J$ is of class $C^1$.***

For $C^\infty$-structures the result is due to McDuff, see \[McD-1\].

[Lecture 7]{}

[Fredholm Properties of the Gromov Operator]{}

*7.1. Generalized Normal Bundle.*

[*Lemma 6.3.2*]{} makes it possible to define the order of vanishing of the differential of $(J_S,J)$-holomorphic map $u:(S,J_S)\to (X,J)$, provided $J\in C^1$.

**Definition 7.1.1. *By the order of zero ${\sf ord}_p du$ of the differential $du$ at a point $p\in S$, we understand the order of vanishing at $p$ of the holomorphic morphism $du : \calo(TS)\to \calo(E)$.***

From (6.3.2) we obtain the following short exact sequence: $$0\longrightarrow \calo(TS) \buildrel du \over\longrightarrow \calo(E)
\longrightarrow \caln\longrightarrow 0. \eqno(7.1.1)$$

Here $\caln$ is a quotient-sheaf $\calo(E)/du(TS)$. We can decompose $\caln = \calo(N_0)\oplus \caln_1$, where $N_0$ is a holomorphic vector bundle and $\caln_1 =
\bigoplus_{i=1}^P \cc_{a_i}^{n_i}$. Here $\cc_{a_i}^{n_i}$ denotes the sheaf, supported at the critical points $a_i\in S$ of $du$ and having a stalk $\cc^{n_i}$ with $n_i = \ord_{a_i}du$, the order of zero of $du$ at $a_i$. We shall call $\caln$ a normal sheaf and $N_0$ a generalized normal bundle.

Denote by $[A]$ the divisor $\sum_{i=1}^Pn_i[a_i]$, and by $\calo([A])$ a sheaf of meromorphic functions on $S$ having poles in $a_i$ of order at most $n_i$. Then (7.1.1) gives rise to the exact sequence

$$0\longrightarrow \calo(TS)\otimes \calo([A])\buildrel{du}\over
{\longrightarrow} \calo(E)
\longrightarrow \calo(N_0)\longrightarrow 0. \eqno(7.1.2)$$

Denote by $L^p_{(0, 1)}(S, E)$ the space of $L^p$-integrable (0, 1)-forms with coefficients in $E$. Then (7.1.2) together with [*Lemma 6.3.1*]{} implies that the following diagram is commutative

$$\def\normalbaselines{\baselineskip20pt\lineskip3pt \lineskiplimit3pt }
\def\mapright#1{\smash{\mathop{\longrightarrow}\limits^{#1}}}
\def\mapdown#1{\Big\downarrow\rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
%
\matrix
0&\mapright{}&L^{1,p}(S, TS\otimes [A])&\mapright{du}&
L^{1,p}(S, E)&\mapright{\pr}&L^{1,p}(S, N_0)&\mapright{}&0\cr
& &\mapdown{\dbar_S}& &\mapdown{D_{u, J}}& &\mapdown{}& & \cr
0&\mapright{}&L^p_{(0,1)}(S, TS\otimes [A])&\mapright{du}&
L^p_{(0,1)}(S, E)&\mapright{}&
L^p_{(0,1)}(S, N_0)&\mapright{}&0. \cr
\endmatrix
\eqno(7.1.3)$$

This defines an operator $D_{u, J}^N : L^{1,p}(S, N_0) \longrightarrow
L^p_{(0,1)}(S, N_0)$ which has the form $D_{u, J}^N =
\dbar_N + R$. Here $\dbar_N$ is a usual $\dbar $-operator on $N_0$ and $R\in C^0(S, \hom_\rr(N_0,\Lambda^{0,1} \otimes N_0))$. This follows from the fact that $D_{u, J}$ has the same form.

Definition 7.1.2. Let $E$ be a holomorphic vector bundle over a compact Riemann surface $S$ and let $D:L^{1,p}(S, E)\to L^p(S,
\Lambda^{0,1}S \otimes E)$ be an operator of the form $D=\dbar + R$, where $R\in L^p\bigl(S,\,\hom_\rr(E,\,\Lambda^{0, 1}S\otimes E) \bigr)$ with $2<p<\infty$. Define $\sfh^0_D(S, E)\deff \ker D$ and $\sfh^1_D(S, E)
\deff \coker D$.

Remark. It is shown in [*Lemma 7.2.2*]{} below that given $S$, $E$ and $R\in L^p$, $2<p<\infty$, one can define $\sfh^i_D(S, E)$ as a (co)kernel of the operator $\dbar +R: L^{1,q}(S, E) \to
L^q(S,\,\Lambda^{0, 1}S\otimes E)$ for any $1<q\le p$. Thus, the definition is independent of the choice of a functional space.

By the standard lemma of homological algebra we obtain from (7.1.2) the following long exact sequence of $D$-cohomologies.

$$\def\normalbaselines{\baselineskip20pt\lineskip3pt \lineskiplimit3pt }
\def\mapright#1{\smash{\mathop{\longrightarrow}\limits^{#1}}}
\def\mapdown{\Big\downarrow}
%
\matrix
0& \mapright{}& \sfh^0(S, TS\otimes [A]) &\mapright{} & \sfh^0_D(S, E)
& \mapright{}& \sfh^0_D(S, N_0)   &\mapright{\delta} &\cr
%
\vphantom{\mapdown}&&&&&&&&\cr
%
& \mapright{}& \sfh^1(S, TS\otimes [A]) &\mapright{} & \sfh^1_D(S, E)
& \mapright{}& \sfh^1_D(S, N_0)         &\mapright{} &0.
\endmatrix
\eqno(7.1.4)$$

*7.2. Surjectivity of $D_{u, J}$.*

We shall use a result of Gromov (\[G\]) and Hofer-Lizan-Sikorav (\[Hf-L-Sk\]) about surjectivity of $D^N_{u, J}$, namely a vanishing theorem for $D$-cohomologies. First we prove some technical statements.

Lemma 7.2.1.

*Let $X$ and $Y$ be Banach spaces and $T:X\to Y$ a closed dense defined unbounded operator with the graph $\Gamma=\Gamma_T$ endowed with the graph norm $\Vert x \Vert_\Gamma= \Vert x \Vert_X + \Vert Tx
\Vert_Y$. Suppose that the natural map $\Gamma\to X$ is compact. Then*

$\ker(T)$ is finite-dimensional;

$\im(T)$ is closed;

the dual space $\bigl(Y/\im(T)\bigr)^*$ is naturally isomorphic to $\ker(T^*:Y^*\to X^*)$.

Proof. Obviously, for $x\in \ker(T)$ one has $\Vert x \Vert_X =
\Vert x \Vert_\Gamma$. Let $\{x_n\}$ be a sequence in $\ker(T)$ which is bounded in the $\Vert \cdot \Vert_X$- norm. Then it is bounded in the $\Vert \cdot
\Vert_\Gamma$- norm and hence relatively compact with respect to the $\Vert \cdot \Vert_X$- norm. Thus, the unit ball in $\ker(T)$ is compact which implies the statement of the lemma.

Due to finite-dimensionality, there exists a closed complement $X_0$ to $\ker(T)$ in $X$. Let $\{x_n\}$ be a sequence in $X$ such that $Tx_n \lrar y
\in Y$. Without losing generality we may assume that $x_n$ belong to $X_0$. Suppose that $\Vert x_n \Vert_X \lrar \infty$. Denote $\tilde x_n \deff {x_n
\over \Vert x_n \Vert_X }$. Then $\Vert \tilde x_n \Vert_\Gamma$ is bounded and hence some subsequence of $\{\tilde x_n\}$, still denoted by $\{\tilde
x_n\}$, converges in $X_0$ to some $\tilde x$. Note that $\tilde x\not=0$, because $\Vert \tilde x \Vert_X = \lim \Vert \tilde x_n \Vert_ X =1$. On the other hand, one can see that $T\tilde x_n \lrar 0\in Y$. Since $\Gamma$ is closed, $(\tilde x, 0)\in \Gamma$ and hence $\tilde x\in \ker(T) \cap X_0
=\{0\}$. The contradiction shows that the sequence $\{x_n\}$ must be bounded in $X$. Since $\{Tx_n \}$ is also bounded in $Y$, some subsequence of $\{x_n\}$, still denoted by $\{x_n\}$, converges in $X_0$ to some $x$. Due to the closeness of $\Gamma$, $Tx=y$. Thus $\im (T)$ is closed in $Y$.

Denote $Z\deff\ker(T^*:Y^*\to X^*)$ and let $h\in \bigl(Y/\im(T)
\bigr)^*$. Then $h$ defines a linear functional on $Y$,i.e., some element $h'\in Y^*$, which is identically zero on $\im(T)$. Thus, for any $x$ from the domain of the definition of $T$ one has $\langle h', Tx \rangle =0$, which implies $T^*(h')=0$. Consequently, $h'$ belongs to $Z$. Conversely, every $h'\in Z$ is a linear functional on $Y$ with $h'(Tx)= \langle T^*h', x\rangle
=0$ for every $x$ from the domain of the definition of $T$. Thus, $h'$ is identically zero on $\im(T)$ and is defined by some unique $h\in
\bigl(Y/\im(T)\bigr)^*$.

Lemma 7.2.2. [*(Serre Duality for $D$-cohomologies.)*]{} [*Let $E$ be a holomorphic vector bundle over a compact Riemann surface $S$, and let $D:L^{1,p}(S, E)\to L^p(S, \Lambda^{0, 1}S\otimes E)$ be an operator of the form $D=\dbar + R$, where $R\in L^p\bigl(S,\,
\hom_\rr(E,\Lambda^{0, 1}S\otimes E) \bigr)$ with $2<p <\infty$. Also let $K\deff \Lambda^{1, 0}S$ be the canonical holomorphic line bundle of $S$. Then there exists the naturally defined operator $$D^*=\dbar- R^* : L^{1,p}(S, E^* \otimes K)
\to L^p(S,\Lambda^{0, 1} \otimes E^* \otimes K)$$ with $R^* \in L^p\bigl(S,\,\homr(E^*\otimes K,\,
\Lambda^{0, 1}S \otimes E^*\otimes K) \bigr)$ and the natural isomorphisms $$\sfh^0_D(S,\, E)^*\cong \sfh^1_{D^*}(S,\, E^*\otimes K),$$ $$\sfh^1_D(S,\, E)^*\cong \sfh^0_{D^*}(S,\, E^*\otimes K).$$ If, in addition, $R$ is $\cc$-antilinear, then $R^*$ is also $\cc$-anti-linear.* ]{}

Proof. For any $1<q\le p$ we associate with $D$ an unbounded dense defined operator $T_q$ from $X_q\deff L^q(S, E)$ into $Y_q\deff L^q(S,\Lambda^{0, 1}S\otimes E)$ with the domain of definition $L^{1, q}(S, E)$. The elliptic regularity of $D$ (see [*Lemma 3.2.1*]{} above ) implies that $$\Vert \xi \Vert_{L^{1,q}(S, E)} \le C(q)
\left(
\Vert \dbar\xi +R\xi\Vert_{L^q(S, E)} +
\Vert \xi \Vert_{L^q(S, E)} \right).$$ Consequently, $T_q$ are closed and satisfy the hypothesis of [*Lemma 7.2.1*]{}. For $q>q_1$ we also have the natural imbedding $X_q
\hookrightarrow X_{q_1}$ and $Y_q \hookrightarrow Y_{q_1}$ which commutes with the operator $D$. Moreover, due to the regularity of $D$ this imbedding maps $\ker T_q$ [*identically*]{} onto $\ker T_{q_1}$. Thus we can identify $\sfh^0_D(S, E)$ with any $\ker T_q$.

Now note that for $q'\deff q/(q-1)$ we have the natural isomorphisms $$\eqalign{
X_q^*\equiv &(L^q(S, E))^*
\cong L^{q'}(S,\Lambda^{0, 1}S \otimes E^* \otimes K),\cr
Y_q^*\equiv &(L^q(S,\Lambda^{0, 1}S\otimes E))^*
\cong L^{q'}(S,  E^* \otimes K),\cr
}$$ induced by the pairing of $E$ with $E^*$ and by integration over $S$. One can easily check that the dual operator $T_q^*$ is induced by the differential operator $-D^*: L^{1,q'}(S, E^* \otimes K)
\to L^{q'}(S,\Lambda^{0, 1} \otimes E^* \otimes K)$ of the form $D^* =\dbar - R^*$. In fact, for $\xi \in L^{1, q}(S, E)$ and $\eta\in L^{1, q'}(S, E^*\otimes K)$ one has $$\langle T_q\xi,\,\eta \rangle=
\int_S \langle \dbar\xi + R\xi,\,\eta \rangle =
\int_S \dbar\langle \xi ,\,\eta \rangle +
\int_S \langle \xi,\, -(\dbar - R^*)\eta \rangle =
\int_S \langle \xi ,\, -D^*\eta \rangle,$$ since the integral of any $\dbar$-exact $(1, 1)$-form vanishes. From [*Lemma 7.2.1*]{} we obtain the natural isomorphisms $\sfh^1_D(S, E)^*\equiv (\coker
T_p)^*\cong \ker T_p^*$ and $\sfh^0_D(S, E)^*\equiv (\ker
T_p)^*\cong \coker T_p^*$, which yields the statement of the lemma.

Corollary 7.2.3. (\[G\], \[H-L-Sk\].) [*(Vanishing Theorem for $D$-cohomologies.)*]{} [*Let $S$ be a Riemann surface $S$ of the genus $g$. Also let $L$ be a holomorphic [*line*]{} bundle over $S$, equipped with a differential operator $D=\dbar + R$ with $R\in L^p\bigr(S,
\homr(L,\,  \Lambda^{0, 1}S\otimes L) \bigl)$, $p>2$. If $c_1(L)<0$, then $\sfh^0_D (S,\, L)=0$. If $c_1(L)>2g-2$, then $\sfh^1_D (S,\, L)=0$.* ]{}

Proof. Suppose $\xi$ is a nontrivial $L^{1,p}$-section of $L$ satisfying $D\xi=0$. Then due to [*Lemma 3.1.1*]{}, $\xi$ has only finitely many zeros $p_i\in S$ with [*positive*]{} multiplicities $\mu_i$. One can easily see that $c_1(L)=\sum \mu_i \ge 0$. Consequently $\sfh^0_D (S,\, L)$ vanishes if $c_1(L)<0$. The vanishing result for $\sfh^1_D$ is obtained via the Serre duality of [*Lemma 7.2.2*]{}.

*7.3. Tangent Space to the Moduli Space.*

Recall that in (7.1.4) we obtained the following long exact sequence

$$\def\normalbaselines{\baselineskip20pt\lineskip3pt \lineskiplimit3pt }
\def\mapright#1{\smash{\mathop{\longrightarrow}\limits^{#1}}}
\def\mapdown{\Big\downarrow}
%
\matrix
0& \mapright{}& \sfh^0(S, TS\otimes [A]) &\mapright{} & \sfh^0_D(S, E)
& \mapright{}& \sfh^0_D(S, N_0)   &\mapright{\delta} &\cr
%
\vphantom{.}&&&&&&&&\cr
%
& \mapright{}& \sfh^1(S, TS\otimes [A]) &\mapright{} & \sfh^1_D(S, E)
& \mapright{}& \sfh^1_D(S, N_0)         &\mapright{} &0.
\endmatrix$$ It is most important for us to associate a similar long exact sequence of $D$-cohomologies to the short exact sequence (7.1.1). Note that, due to [*Lemmas 6.3.1*]{} and 6.3.2, we obtain the short exact sequence of complexes $$\def\normalbaselines{\baselineskip20pt\lineskip3pt \lineskiplimit3pt }
\setbox1=\hbox{$\lrar$}
\def\mapright#1{\,\,\smash{\mathop{{\hbox to
\wd1{\hss\hbox{$\displaystyle\longrightarrow$}\hss}}}\limits^{#1}}\,\,}
%
\def\mapdown#1{\Big\downarrow\rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
%
\matrix\format\c&\c&\c&\c&\c&\c&\c&\c&\l\\
0&\mapright{}&L^{1,p}(S, TS)&\mapright{du}&
L^{1,p}(S, E)&\mapright{\barr\pr}&
L^{1,p}(S, E){\bigm/}du(L^{1,p}(S, TS))
&\mapright{}&0\cr
& &\mapdown{\dbar_S}& &\mapdown{D}& &\mapdown{\barr D}& &
(7.3.1)%\eqno(7.3.1)
\cr
%
0&\mapright{}&L^p_{(0,1)}(S, TS)&\mapright{du}&
L^p_{(0,1)}(S, E)&\mapright{}&
L^p_{(0,1)}(S, E){\bigm/}
du(L^p_{(0,1)}(S, TS))&\mapright{}&0\cr\vphantom{.}
\endmatrix
%\eqno(7.3.1)$$

where $\overline D$ is induced by $D\equiv D_{u, J}$.

[**Theorem 7.3.1.**]{} [*For $\overline D$ just defined, $\ker \overline D=\sfh^0_D(S, N_0) \oplus \sfh^0(S, \caln_1)$ and $\coker \overline D=\sfh^1_D(S, N_0)$.* ]{}

**Proof. Let $\pr_0: \calo(E) \to \calo(N_0)$ and $\pr_1: \calo(E) \to \caln_1$ denote the natural projections induced by $\pr: \calo(E) \to \caln\equiv \calo(N_0) \oplus \caln_1$. Also let $A$, as in (2.2.7), denote the support of $\caln_1$, [*i.e.,*]{} the finite set of the vanishing points of $du$. Then $\pr_0$ defines maps $$\matrix
\pr_0:L^{1,p}(S , E)  &  \longrightarrow
& L^{1,p}(S , N_0)
\cr
\vphantom{.}\cr
\pr_0:L^p_{(0, 1)}(S , E)  &  \longrightarrow
& L^p_{(0, 1)}(S, N_0).
\cr
\endmatrix
\eqno(7.3.2)$$ Furthermore, in a neighborhood of every point $p\in A$ the sequence (7.1.1) can be represented in the form $$0\longrightarrow \calo
\buildrel \alpha_p \over\longrightarrow \calo^{n-1}
\oplus \calo
\buildrel \beta_p \over\longrightarrow \calo^{n-1}
\oplus\caln_1\vert_p
\longrightarrow 0 \eqno(7.3.4)$$ with $\alpha_p(\xi)= (0, z^{\nu_p} \xi)$ and $\beta(\xi,\eta)=
(\xi, j^{(\nu-1)}_p\eta)$. Here $z$ denotes local a holomorphic coordinate on $S$ with $z(p)=0$, $\nu_p$ is the multiplicity of $du$ in $p$, $j^{(\nu-1)}_p\eta$ is a $(\nu-1)$-jet of $\eta$ in $p$ and $\caln_1
\vert_p$ is a stalk of $\caln_1$ in $p$.**

Now let $\overline\xi\in L^{1,p}(S, E)/du(L^{1,p}(S, TS))$ which satisfies $\overline D(\overline\xi)=0$. This means that $\overline\xi$ is represented by some $\xi \in L^{1,p}(S, E)$ with $D\xi= du(\eta)$ for some $\eta \in L^p_{(0, 1)}(S, TS)$. It is obvious that there exists $\zeta
\in L^{1,p}(S, TS)$ such that $\overline\partial \zeta = \eta$ in a neighborhood of $A$. Consequently, $D(\xi- du(\zeta))=0$ in a neighborhood of $A$. Denote $\xi_1 := \xi -du(\zeta)$. Due to (6.3.1), in a neighborhood of $p\in A$ the equation $D\xi_1=0$ is equivalent to $$\overline\partial \xi_1+ N(\xi_1, du)=0.$$ Due to [*Lemma 3.1.2*]{} above, $\xi_1= P(z)+o(|z|^{\nu_p} )$ with some (holomorphic) polynomial $P(z)$. This gives the possibility of defining $\varphi_0 := \pr_0(\xi) \in L^{1,p}(S, N_0)$ and $\varphi_1 :=
\pr_1(\xi- du(\zeta)) \in \sfh^0(S, \caln_1)$. Due to (7.1.2) and (7.1.3), $D_N\varphi_0=0$. If $\overline\partial \zeta' = \eta$ in a neighborhood of $A$ for some other $\zeta'\in L^{1,p}(S, TS)$, then $\zeta - \zeta'$ is holomorphic in a neighborhood of $A$, and consequently $\pr_1(du(\zeta - \zeta'))=0$. Thus, the map $\iota^0: \ker \overline D \to \sfh^0_D(S, N_0) \oplus \sfh^0(S, \caln_1)$, $\iota^0(\overline\xi)=(\varphi_0,\varphi_1)$ is well-defined.

Assume that $\iota^0(\overline\xi)=0$ for some $\overline\xi\in
\ker\overline D$ and that $\overline\xi$ is represented by $\xi\in
L^{1,p}(S, E)$ with $D\xi= du(\eta)$ for some $\eta \in
L^p_{(0, 1)}(S, TS)$. Let $\zeta \in L^{1,p}(S, TS)$ satisfy $\overline\partial \zeta = \eta$ in a neighborhood of $A$. The assumption $\iota^0(\overline\xi)=0$ implies that $\pr_1(\xi- du(\zeta))=0$ and that $\pr_0(\xi- du(\zeta))=0$ in a neighborhood of $A$. Consequently, $\xi-
du(\zeta)=du(\psi)$ for some $\psi \in L^{1,p}(S, TS)$ and $\xi \in
du(L^{1,p}(S, TS))$. This means that $\overline\xi=0\in \ker\overline D$ and $\iota^0$ is injective.

Let $\varphi_0 \in \sfh^0_D(S, N_0)$ and $\varphi_1 \in \sfh^0(S, {\cal
N}_1)$. For every $p\in A$ fix a neighborhood $U_p$ and representation of (7.1.1) in the form (7.3.4) over $U_p$. In every $U_p$ we find $\xi=(\xi_0,
\xi_1) \in L^{1,p}(U_p,\cc^{n-1} \times\cc)$ satisfying the following properties:

$D\xi=0$;

$\xi_0$ coincide with $\varphi_0\vert_{U_p}$ under the identification $\calo(N_0)\vert_{U_p} \cong \calo^{n-1}$;

$j^{(\nu-1)}_p \xi = \varphi_1\vert_p \in
\caln_1 \vert_p$.

The corresponding $\xi_1 \in L^{1,p}(U_p,\cc)$ can be constructed as follows. Let $D(\xi_0,\xi_1) = (\eta_0,\eta_1)$. From $\pr_0(D\xi)= D_N(\pr_0\xi)=0$ one has $\eta_0=0$. In the representation (7.3.4) the identity $R\scirc du =0$ of [*Lemma 6.3.1*]{} means that $D(0,\xi_1) = (0,\dbar\xi_1)$. Thus, one can find $\xi_1$ with $\dbar\xi_1
=-\eta_1$, which gives $D(\xi_0,\xi_1) =0$. Consequently, $j^{(\nu-1)}_p
\xi$ is well-defined. Adding an appropriate [*holomorphic*]{} term to $\xi_1$ one can satisfy condition [*c)*]{}. Using an appropriate partition of unity, we can construct $\xi'
\in L^{1,p}(S, E)$ such that $\xi'$ coincides with $\xi$ in a (possibly smaller) neighborhood of every $p\in A$ and such that $\pr_0 \xi'
= \varphi_0$ in $S$. Thus from $D_N\varphi_0=0$ and (2.2.8) one obtains $D\xi' \in du(L^p_{(0, 1)}(S, TS \otimes [A]))$. But $D\xi'
=0$ in a neighborhood of every $p\in A$, which means that $D\xi' \in du(
L^p_{(0, 1)}(S, TS))$. This shows surjectivity of $\iota^0: \ker
\overline D \to \sfh^0_D(S, N_0) \oplus \sfh^0(S, \caln_1)$.

Now consider the case of $\coker \overline D$. Since the operator $\pr_0$ satisfies the identities $\pr_0\scirc D_E=D_N\scirc \pr_0$ and $\pr_0\scirc du=0$, the induced map

$\displaystyle
\iota^1 : \coker \overline D\equiv L^p_{(0,1)}(S, E)\bigm/
\bigl(D_E(L^{1,p}(S, E) \oplus  du(L^p_{(0,1)}(S, TS)\bigr)
\longrightarrow $

$\displaystyle
\longrightarrow\sfh^1_D(S, N_0)\equiv
L^p_{(0,1)}(S, N_0)/ D_N(L^{1,p}(S, N_0))
\qquad$

is well-defined. Moreover, the surjectivity of $\pr_0:
L^p_{(0,1)}(S, E) \to L^p_{(0,1)}(S, N_0)$ easily yields the surjectivity of $\iota^1$.

Now assume that $\iota^1 \overline\xi =0$ for some $\overline\xi \in
\coker \overline D$ and $\overline \xi$ is represented by $\xi
\in  L^p_{(0,1)}(S, E)$. Then the condition $\iota^1
\overline\xi=0$ means that $\pr_0 \xi= D_N \eta$ for some $\eta
\in L^{1,p}(S, N_0)$. Find $\zeta \in L^{1,p}(S, E)$ such that $\pr_0 \zeta =\eta$. Then $\pr_0 (\xi - D\zeta)=0$, and, due to (7.1.3), $\xi- D\zeta=du(\varphi)$ for some $\varphi \in
L^p_{(0,1)}(S, TS\otimes [A])$. Find $\psi \in
L^{1,p}(S, TS\otimes [A])$ such that $\dbar \psi =\varphi$ in some neighborhood of $A$. Then $$\xi- D\zeta - D(du(\psi))= du(\varphi-\dbar\psi) \in
du(L^p_{(0,1)}(S, TS)).$$ Consequently, $\overline \xi \in \im\overline D$. This shows the injectivity of $\iota^1$.

Corollary 7.3.2. *The short exact sequence $(7.1.1)$ induces the long exact sequence of $D$-cohomologies $$\def\normalbaselines{\baselineskip20pt\lineskip3pt \lineskiplimit3pt }
\def\mapright#1{\,\,\smash{\mathop{\longrightarrow}\limits^{#1}}\,\,}
\def\mapdown{\Big\downarrow}
%
\matrix\format\c&\c&\c&\c&\c&\c&\c&\c&\c\\
0& \mapright{}& \sfh^0(S, TS) &\mapright{} & \sfh^0_D(S, E)
& \mapright{}& \sfh^0_D(S, N_0)\oplus \sfh^0(S, \caln_1)
&\mapright{\delta} &\cr
%
%\vphantom{.}&&&&&&&&\cr
%
& \mapright{}& \sfh^1(S, TS) &\mapright{} & \sfh^1_D(S, E)
& \mapright{}& \sfh^1_D(S, N_0)         &\mapright{} &0.
\endmatrix$$*

*7.4. Reparameterizations.*

Let $S$ be an oriented compact real surface without boundary. Consider the Teichmüller space $\ttt_g$ of marked complex structures on $S$. This is a complex manifold of a dimension $$\dimc \ttt_g = \cases  0      &\text{ if $g=0$;}  \\
1      &\text{ if $g=1$;}  \\
3g-3   &\text{ if $g\ge2$}
\endcases$$ which can be completely characterized in the following way. The product $S\times \ttt_g$ possesses a complex (holomorphic) structure $J_{S\times
\ttt}$ such that

the natural projection $\pi\mid_{\ttt }: S\times \ttt_g \to \ttt_g$ is holomorphic, so that for any $\tau\in \ttt_g$ the identification $S \cong S
\times \{\tau\}$ induces the complex structure $J_S(\tau)\deff J_{S\times
\ttt}\ogran_{S\times \{\tau\}}$ on $S$;

for any complex structure $J_S$ on $S$ there exists a uniquely defined $\tau\in\ttt_g$ and a diffeomorphism $f: S\to S$ such that $J_S =
f^*J_S(\tau)$ ($f: (S, J_S) \to (S, J_S(\tau))$ is holomorphic) and $f$ is homotopic to the identity map $\id_S: S\to S$.

Denote the automorphism group of $S\times \ttt_g$ by $\bfg$. It is known that $$\bfg=\cases {\bold PGl}(2,\cc)       &\text{ for $g=0$,} \\
{\bold Sp}(2,\;\;\zz) \ltimes T^2   &\text{ for $g=1$,} \\
{\bold Sp}(2g,\zz)                  &\text{ for $g\ge2$.}
        \endcases$$ We shall use the following information about $\ttt_g$.

For $g=0$ the surface $S$ is a Riemann sphere $S^2$ and all complex structures on $S^2$ are equivalent to the standard one when $S^2\cong
\cc\pp^1$. Thus, $\ttt_0$ consists of one point, and the group $\bfg={\bold
PGl}(2,\cc)$ is the group of biholomorphisms of $\cc\pp^1$.

For $g=1$ the surface $S$ is a torus $T^2$ and $\ttt_1$ is an upper half-plane $\cc_+=\{\,\tau\in\cc\;:\;\im(\tau)>0\,\}$. The product $S\times \ttt_1$ can be identified by the quotient $(\cc\times \cc_+)/\zz^2$ with respect to the holomorphic action $$\bigl((n,m),\, (z,\tau)\bigr) \in\zz^2 \times \cc\times \cc_+ \longmapsto
(m,n)\cdot(z,\tau)\deff (z+m +n\,\tau,\tau) \in \cc\times \cc_+.$$ In this case the subgroup $T^2\equiv \rr^2/\zz^2$ is a connected component of the identity ${\rom e}\in\bfg$, in particular $T^2$ is normal. The group $\bfg= {\bold Sp}(2,\;\; \zz) \ltimes T^2$ is a semi-direct product. The holomorphic action of $T^2$ on the quotient $(\cc\times \cc_+)/\zz^2$ is given by $$\eqalign{
\bigl([t_1,t_2],\, ([z],\tau)\bigr) &\in T^2 \times (\cc\times \cc_+)/\zz^2
\longmapsto
\cr
[t_1,t_2]\cdot([z],\tau)&\deff ([z+t_1 +t_2\tau],\tau )
\in (\cc\times \cc_+)/\zz^2.
}$$

For any $g\ge0$ the action of $\bfg$ on $S\times \ttt_g$ is effective and preserves fibers of the projection $\pi\mid_{\ttt }: S\times \ttt_g \to 
\ttt_g$. This induces the action of $\bfg$ on $\ttt_g$. Furthermore, given $\tau \in \ttt_g$ and $f\in\bfg$, there exists a unique diffeomorphism of $\hat f_\tau:S \to S$ such that $$f\cdot(x,\tau)=(\hat f_\tau(x), f\cdot\tau).
\eqno(7.4.1)$$

For any $\tau\in\ttt_g$ we have natural isomorphisms $T_\tau\ttt_g \cong
\sfh^1(S,\, \calo_\tau(TS))$ and $T_{\rom e}\bfg \cong \sfh^0(S,\,
\calo_\tau(TS))$, where $\calo_\tau(TS)$ denote the sheaf of a section of $TS$ which are holomorphic with respect to the complex structure $J_S(\tau)$.

Later on, we shall denote elements of $\ttt_g$ by $J_S$ and consider them as corresponding complex structures on $S$.

[Lecture 8 ]{}

[Transversality.]{}

*8.1. Moduli Space of Nonparameterized Curves.*

Let $(X,\omega, J\st)$ be a symplectic manifold with some distinguished $w$-tamed almost-complex structure $J\st$. In our applications $J\st $ will be integrable, providing, together with $w$, the Kähler structure on $X$. Let $U\Subset X$ be an open relatively compact subset which can coincide with $X$ if $X$ is compact. Also let $S$ be a (fixed) compact oriented surface of genus $g\ge0$, $u_0: S \to X$ a non-constant $C^1$-smooth map such that $u_0(S)\cap
U \not = \emptyset$.

Fix $2<p<\infty$. The Banach manifold $L^{1,p}(S,X)$ of all $L^{1,p}
$-smooth maps $u:S \to X$ is smooth with a tangent space $T_uL^{1,p}(S,X)$ equal to the space $L^{1,p}(S,u^*TX)$ of $L^{1,p}$-smooth sections of a pulled-back tangent bundle to $X$. Denote by $\cals_U$ a Banach manifold of those $u\in L^{1,p}(S,X) $, for which $u$ is homotopic to $u_0$ and $u(S) \cap U \not=\emptyset$. Fix an integer $k\ge1$ and denote by $\calj^k_U$ the set of $C^k$-smooth almost complex structures $J$ in $X$ such that $\{ x\in X:J(x)\not=J\st(x) \} \comp U$, and which are tamed by $\omega$. The latter means that $J$ is $\omega$-positive, i.e., $\omega(\xi, J\xi)>0$ for every $x\in X$ and every nonzero $\xi\in T_xX$.

The evaluation map $\ev: S\times \cals_U \times \ttt \times \calj^k_U \to X$, given by formula $\ev(x, u, J_S,J)\deff u(x)$, defines a bundle $E\deff \ev^*(TX)$ over $S\times \cals_U \times \ttt \times \calj^k_U$. We equip $E$ with the natural complex structure, which is equal to $J(u(x))$ in the fiber $E_{(x,
u, J_S,J)}\cong T_{u(x)}X$. We shall denote by $(E_u,J)$ the restriction of $E$ onto $S\times \{(u,J_S,J)\}$ which is isomorphic to $u^*TX$.

The bundle $E$ with the complex structure $J$ induces complex Banach bundles $\wh\cale$ and $\wh\cale'$ over a product $\cals_U \times \ttt \times \calj^k_U$ with fibers $$\eqalign{
\wh\cale_{(u,J_S,J)}  & \deff L^{1,p}(S, E_u),\cr
\wh\cale'_{(u,J_S,J)} & \deff L^p(S, E_u\otimes\Lambda^{(0,1)}S).
}$$ Here $S$ is equipped with the complex structure $J_S$, $\Lambda^{(0,1)}S$ is a complex bundle of (0,1)-forms on $S$ and $\otimes$ means the tensor product over $\cc$ of bundles with the corresponding complex structures. The bundle $\wh\cale$ is simply a pull-back of a tangent bundle $TL^{1,p}(S,X)$ with respect to the projection $(u,J_S,J)\in \cals_U \times \ttt \times \calj^k_U
\mapsto
u \in L^{1,p}(S,X)$.

On the other hand, the bundle $\wh\cale'$ is the target manifold of the $\dbar$-operator for the map $u\in L^{1,p}(S,X)$. Namely, we have a distinguished section $\sigma_\dbar$ of $\wh\cale'$, $$\sigma_\dbar(u,J_S,J)\deff \dbar_{J_S,J}u \deff
\half\bigl(du + J\scirc du \scirc J_S\bigr).
\eqno(8.1.1)$$ If $f$ is another section of $\wh\cale'$ (given by some explicit geometric construction), then one can consider a $\dbar$-equation $$\dbar_{J_S,J}u=f(u,J_S,J).$$ We shall consider only a homogeneous case where $f(u,J_S,J)\equiv
0$. The corresponding set of solutions is closed in $\cals_U \times \calj^k_U$ and will be denoted by $$\calp \deff \{\, (u,J_S, J) \in \cals_U \times \ttt \times \calj^k_U \;:\;
\dbar_{J_S,J}u=0 \,\}.  \eqno(8.1.2).$$ A map $u\in L^{1,p}(S,X)$ such that

$$\dbar_{J_S,J}u=0\eqno(8.1.3)$$ for some $J_S\in\ttt_g$ and $J\in \calj^k_U$ is called $J$-holomorphic, or $(J_S,J)$-holomorphic, and its image $M\deff u(S)$ is called a $J$-complex curve.

The regularity theory for $\dbar$-equation says that $\calp$ is a closed subset of $$\calx \deff \{\,(u,J_S,J)\in\cals_U\times \ttt \times \calj^k_U\;:\;
u\in C^1(S,X)\,\}.$$ Set $
\calx^* \deff \{\,(u,J_S,J)\in \calx \,:\,
\hbox{$u$ is an imbedding in a neighborhood of some $y\in S$ }\},
$ and let $\calp^*\deff \calp\cap \calx^*$. Then $\calx^*$ is open in $\calx$ and $\calp^*$ is open in $\calp$. A group $\bfg$ acts on $\cals_U\times \ttt \times \calj^k_U$ and on $\calx$ in a natural way by composition, $$(f,u,J_S,J) \in \bfg\times \cals_U\times \calj^k_U
\mapsto f\cdot(u,J_S,J) \deff (u\scirc \hat f\inv, f\cdot J_S,J),$$ where $\hat f:S\to S$ is a diffeomorphism induced by $f\in\bfg$ and $J_S$, see (7.4.1). As usual, the inverse $(\hat f)\inv$ is introduced to preserve the associative law $(f\cdot g)\cdot(u,J_S,J)=f\cdot(g\cdot(u,J_S,J))$. The sets $\calx^*$, $\calp$ and $\calp^*$ are invariant with respect to this action.

Let $\calm\deff\calp^*/\bfg$ be a quotient with respect to this action and $\pi_\calp: \calp^* \to \calm$ the corresponding projection. Also let $\pi_\calj: \calm\to\calj^k_U$ be a natural projection.

**Lemma 8.1.1. *The projections $\calx^* \longto \calx^*/\bfg$ and $\pi_\calp:\calp^* \longto \calm$ are principal $\bfg$-bundles.***

**Proof. We consider the case $g\ge2$ first. It is known that in this case the group $\bfg$ acts proper discontinuously on $\ttt_g$. This implies that the same is true for the action of $\bfg$ on $\calx$. Moreover, the definition of $\calx^*$ provides that $\bfg$ acts freely on this set. Consequently, the map $\calx^*\longto \calx^*/\bfg$ is simply an (unbranched) covering.**

Now we consider the case $g=0$. Note that in this case $S=S^2$ and $\ttt_0=\{J\st\}$. Fix some $(u^0,J\st,J^0)\in \calx^*$. Let $y_1$, $y_2$, $y_3$ be distinct points on $S^2$ such that $u^0$ is an imbedding in a neighborhood of every $y_i$, in particular $du^0$ is non-vanishing in $y_i$. Also let $Z_i$ be smooth submanifolds of codimension 2 in $X$, intersecting $u^0(S^2)$ transversally in each $u^0(x_i)$, respectively.

Let $V\ni(u^0,J\st,J^0)$ be an open set in $\calx^*$, $W$ its projection on $\calx^*/\bfg$, and $\bfg\cdot V\deff \{\,f\cdot(u,J\st,J):
f\in \bfg, (u,J\st,J)\in V\,\}$ its $\bfg$-saturation. Consider a set $$\calz \deff\{\, (u,J\st,J)\in \bfg\cdot V\;:\; u(y_i)\in Z_i\,\}.$$ One can easily show that if $V$ is chosen sufficiently small, then $\calz$ is a smooth closed Banach submanifold of $\bfg\cdot V$, intersecting every orbit $\bfg \cdot (u,J\st,J)$ transversally at exactly one point. Moreover, we have a $\bfg$-invariant diffeomorphism $\bfg\cdot V \cong \bfg \times \calz$, so that $\calz$ is a local slice of $\bfg$-action at $(u^0,J\st,J^0)$. This equips a quotient $\calx^*/\bfg$ with a structure of a smooth Banach manifold such that the projection $\calx^*\to\calx^*/\bfg$ is a smooth principle $\bfg$-bundle.

The case $g=1$ is a combination of the two above cases. We fix some $(u^0,J_S^0, J^0)\in \calx^*$ and a point $y$ on $S$ such that $u^0$ is an imbedding in a neighborhood of $y$; in particular $du^0$ is non-vanishing in $y$. Also let $Z$ be a smooth submanifold of codimension 2 in $X$, intersecting $u^0(S)$ transversally in $u^0(y)$. As in the case $g=0$, we fix a small open set $V\ni(u^0,J^0,J^0)$ in $X$ and consider a set $$\calz \deff\{\, (u,J_S,J)\in \bfg\cdot V\;:\; u(y)\in Z\,\},$$ where $\bfg\cdot V \deff \{\,f\cdot(u,J\st,J): f\in \bfg, (u,J\st,J)\in V\,\}$ is $\bfg$-saturation of $V$. If $V$ is chosen sufficiently small, then $\calz$ is a slice to the action of $T^2\subset \bfg$. Thus, the projection $\calx^*\longto \calx^*/T^2$ is a principle $T^2$-bundle.

Now we consider the action of ${\bold Sp}(2,\;\; \zz) =\bfg/T^2$ on $\calx^*/T^2$. The same arguments as in the case $g\ge2$ show that this action is free and proper discontinuous. This implies that the projection $\calx^*/T^2\longto \calx^*/\bfg$ is a covering. Consequently, $\calx^*\longto \calx^*/\bfg$ is a principle $\bfg$-bundle.

The natural inclusion $\calm \hookrightarrow \calx^*/\bfg$ is continuous and closed. Further, we have a natural $\bfg$-invariant homeomorphism $\calp^* \cong \calm \,\times_{\calx^*/\bfg} \calx^*$, which gives a desired structure of a principle $\bfg$-bundle on $\calp^*$ with a base $\calm$.

**Remark. One can show that if $(u,J_S,J)\in \calp^*$, then $u$ is an imbedding in all but finitely many points of $S$, in particular $J_S$ is completely defined by $u$ and $J$. Similarly, every class $\bfg\cdot
(u,J_S,J) \in \calm$ is completely defined by $J\in \calj^k_U$ and a $J$-complex curve $M\deff u(S)$. Therefore we shall denote elements of $\calm$ by $(M,J)$. Our motivation is that the object we really exploit is a complex curve $M=u(S)$ rather than its concrete parameterization $u$. We hope that the reader will not be confused by such a formal incorrectness.**

Using [*Lemma 8.1.1*]{}, one can obtain from $\bfg$-invariant objects on $\calp^*$ corresponding objects on $\calm$. For example, on $\calp$ we have a (trivial) $S$-bundle with a total space $\calp\times S$, natural projection on $\calp$, a complex structure $J_S$ on a fiber over $(u,J_S,J)$, and a map $\ev:\calp\times S \to X$ such that $\ev(u,J_S,J;y)\deff u(y)$ which is invariant with respect to the $\bfg$-action. It gives a $\bfg$-bundle $\pi_\calc:\calc\to \calm$ with the total space $\calc\deff\calp^* \times_\bfg S$ and fiber $S$. Moreover, $\calc$ is equipped with a map $\ev: \calc\to X$. We shall imagine $\calm$ as a moduli space of all complex curves in $X$ of appropriate topological properties, $\pi_\calc:\calc \to\calm$ as a corresponding universal bundle and $\ev:\calc \to X$ as an evaluation map. In particular, every fiber $\calc_{(M,J)} \deff \pi_\calc\inv(M,J)$ over $(M,J)\in \calm$ possesses a canonical complex structure $J_{\calc_{(M,J)}}
=J_S$.

In a similar way we define complex Banach bundles $\cale$ and $\cale'$ over $\calx^*/\bfg\supset \calm$. Note that we have a natural continuous $\bfg$-equivariant inclusion $\calx^*\hook \cals_U\times \calj^k_U$. Let $(u,J_S,J)\in \cals_U\times \calj^k_U$, $f\in\bfg$, and let $\hat f:S\to S$ be a diffeomorphism defined as in (1.1), so that $f\cdot(u,J_S,J)= (u\scirc\hat f\inv,f\cdot J_S,J)$. Define the operator $f_*$ by setting $$f_*:s\in\wh\cale_{(u,J_S,J)} \mapsto (\hat f\inv)^*s\in
\wh\cale_{f\cdot(u,J_S,J)}.$$ This defines a natural lift of a $\bfg$-action on $\cals_U\times \calj^k_U$ to a $\bfg$-actions on $\wh\cale$. In the same way we define the action of $\bfg$ on $\wh\cale'$. Since the projection $\calx^*\to \calx^*/\bfg$ admits a local $\bfg$-slice, there exist uniquely defined bundles $\cale$ and $\cale'$ over $\calx^*/\bfg$, whose lifts onto $\calx^*$ is $\bfg$-equivariant isomorphic to $\wh\cale$ and $\wh\cale'$, respectively. In particular, for any $(u,J_S,J)\in \calx^*$ representing $\bfg\cdot(u,J_S,J)\in \calx^*/\bfg$ we have the natural isomorphisms $$\eqalign{
\cale_{\bfg\cdot(u,J_S,J)}  \cong  \wh\cale_{(u,J_S,J)}
& = L^{1,p}(S, E_u),\cr
\cale'_{\bfg\cdot(u,J_S,J)} \cong  \wh\cale'_{(u,J_S,J)}
& = L^p(S, E_u\otimes\Lambda^{(0,1)}S).
}$$

The question we are interested in is whether we can deform a given (compact) $J_0$-holomorphic curve $M_0=u_0(S)$ into a compact complex curve $M_1$ which is holomorphic with respect to the given (integrable, for example) complex structure $J\st$ on $X$. The idea is to use the continuity method: one finds an appropriate homotopy $h(t)=J_t$, $t\in[0,1]$, of almost complex structures connecting $J_0$ with $J\st=J_1$, and shows that there exists a continuous deformation $u_t:S\to X$ of a map $u_0$ into $u_1$ such that $u_t$ is $J_t$-holomorphic for all $t\in[0,1]$. This can be successfully done by studying the linearization of the equation $\dbar u=0$.

**Lemma 8.1.2. *Let $\calx$ be a Banach manifold, $\cale\to \calx$ and $\cale'\to\calx$ $C^1$-smooth Banach bundles over $\calx$, $\nabla$ and $\nabla'$ linear connections in $\cale$ and $\cale'$, respectively, $\sigma$ a (local) $C^1$-section of $\cale$ and $D: \cale \to \cale'$ a $C^1$-smooth bundle homomorphism.***

If $\sigma(x)=0$ for some $x\in \calx$, then the map $\nabla\sigma_x:
T_x\calx \to \cale_x$ is independent of the choice of a connection $\nabla$ in $\cale$;

Set $K_x\deff \ker(D_x:\cale_x\to \cale'_x)$ and $Q_x\deff
\coker(D_x:\cale_x\to \cale'_x)$ with the corresponding imbedding $i_x:K_x \to \cale_x$ and projection $p_x:\cale'_x \to Q_x$. Let $\nabla^\hom$ be a connection in $\hom(\cale,\cale')$ induced by connections $\nabla$ and $\nabla'$. Then the map $$p_x \scirc(\nabla^\hom\!\! D_x)\scirc i_x:
T_x\calx\to \hom(K_x,Q_x)$$ is independent of the choice of connections $\nabla$ and $\nabla'$.

**Remark. Taking into account this lemma, we shall use the following notation. For $\sigma\in \Gamma(\calx, \cale)$, $D\in \Gamma(\calx,
\hom(\cale, \cale'))$ and $x\in \calx$ as in the hypothesis of the lemma, we shall denote by $\nabla\sigma_x:T_x\calx \to \cale_x$ and $\barr{\nabla\!\!D}:
T_x\calx\times \ker D_x \to \coker D_x$ the corresponding operators without pointing out which connections were used to define them.**

**Proof. Let $\wt\nabla$ be another connection in $\cale$. Then $\wt\nabla$ has a form $\wt\nabla=\nabla +A$ for some $A\in\Gamma(\calx,\hom(T\calx,\endo(\cale)))$. Thus, for $\xi\in T_x\calx$ we get $\wt\nabla_\xi\sigma -\nabla_\xi\sigma =
A(\xi,\sigma(x))=0$.**

Similarly, let $\wt\nabla'$ be another connection in $\cale'$, and let $\wt\nabla^\hom$ be a connection in $\hom(\cale,\cale')$ induced by $\wt\nabla$ and $\wt\nabla'$. Then $\wt\nabla'$ also has the form $\wt\nabla=\nabla +A'$ for some $A'\in\Gamma(\calx,\hom(T\calx,\endo(\cale')))$. Thus, for $\xi\in T_x\calx$ we obtain $\wt\nabla_\xi^\hom D -\nabla_\xi^\hom D=
A'(\xi)\scirc D_x - D_x \scirc A(\xi)$. The statement of the lemma now follows from the identities $p_x\scirc D_x=0$ and $D_x\scirc i_x=0$.

[8.2. Transversal Mappings.]{}

To have the possibility of deforming a complex curve along a given path of almost-complex structures, it is useful to know in which points $(u,J_S,J)$ the set $\calp$ of holomorphic maps is a Banach manifold. Note that by definition the set $\calp$ is essentially an intersection of the zero section and the $\sigma_\dbar$-section in the total space of $\cale' $ over $S\times \ttt \times \calj_U^k$. Thus, we are interested in which points these sections meet transversally.

Definition 8.2.1. [*Let $\calx$, $\caly$, and $\calz$ be Banach manifolds with $C^k$-smooth maps $f:\caly \to \calx$ and $g:\calz\to\calx$, $k\ge1$. Define the [*fiber product*]{} $\caly\times_\calx \calz $ by setting $\caly\times_\calx \calz \deff \{\,(y,z)\in \caly\times\calz \;:\;
f(y)=g(z)\,\}$. The map $f$ is called [*transversal to $g$*]{} at point $(y,z)\in \caly\times_\calx \calz$ with $x\deff f(y)=g(z)$, and $(y,z)$ is called a [*transversality point*]{}, a map $df_y\oplus -dg_z: T_y\caly \oplus T_z\calz \to T_x\calx$ is [*surjective*]{} and its kernel admits a closed complement.* ]{}

The set of transversality points $(y,z)\in \caly\times_\calx \calz$ we shall denote by $\caly\times^\trans_\calx \calz$, with $\trans$ symbolizing the transversality condition.

In the special case, when the map $g:\calz \to \calx$ is a closed imbedding, the fiber product $\caly\times_\calx \calz$ is simply the pre-image $f\inv\calz$ of $\calz\subset\calx$. In particular, every point $(y,z)\in\caly\times_\calx \calz$ is completely defined by the point $y\in\caly$, $z=f(y)\in\calz\subset \calx$. In this case we simply say that $f:\caly \to \calx$ is [*transversal to $\calz$ in $y\in\caly$*]{}, $(y,f(y))$ is a transversal point of $\caly\times_\calx \calz\cong
f\inv\calz$.

Lemma 8.2.1. [*The set $\caly\times^\trans_\calx \calz$ is open in $\caly\times_\calx \calz$ and is a $C^k$-smooth Banach manifold with a tangent space $$T_{(y,z)}\caly\times^\trans_\calx \calz =
\ker\bigl( df_y\oplus d(-g_z): T_y\caly \oplus T_z\calz \to T_x\calx \bigr).$$*]{}

Proof. Fix $w_0\deff(y_0,z_0)\in \caly\times^\trans_\calx \calz$ and set $K_0 \deff \ker( df_{y_0}\oplus dg_{z_0}:
T_{y_0}\caly \oplus T_{z_0}\calz \to T_x\calx \bigr)$. Let $Q_0$ be a closed complement to $K_0$. Then the map $df_{y_0}\oplus dg_{z_0}:Q_0\to T_x\calx$ is an isomorphism.

Due to the choice of $Q_0$, there exists a neighborhood $V\subset
\caly\times\calz$ of $(y_0,z_0)$ and $C^k$-smooth maps $w':V\to K_0$ and $w'':V\to Q_0$ such that $dw'_{w_0}$ (resp. $dw''_{w_0}$) is the projection from $T_{y_0}\caly \oplus T_{z_0}\calz$ onto $K_0$ (resp. onto $Q_0$), so that $(w',w'')$ are coordinates in some smaller neighborhood $V_1\subset
\caly\times\calz$ of $w_0=(y_0,z_0)$. It remains to consider the equation $f(y)=g(z)$ in new coordinates $(w',w'')$ and apply the implicit function theorem.

We have defined $\calp$ as a pre-image of a zero section $\sigma_0$ of $\cale'$ with respect to the map $\sigma_\dbar:\cals_U\times \calj_S\times \calj^k_U
\to \cale'$. Due to [*Lemma 8.2.1*]{}, the set $\calp$ is a Banach manifold in those points $(u,J_S,J)\in\calp$, where $\sigma_\dbar$ is transversal to $\sigma_0$. However, in any point $(u,J_S,J;0)$ on the zero section $\sigma_0$ of $\cale'$ we have the natural decomposition $$T_{(u,J_S,J;0)}\cale' =
d\sigma_0\bigl( T_{(u,J_S,J)}(\cals_U\times \calj_S\times \calj_U^k) \bigr)
\oplus \cale'_{(u,J_S,J)},$$ where the first component is the tangent space to the zero section of $\cale'$ and the second is the tangent space to the fiber $\cale'_{(u,J_S,J)}$. Let $p_2$ denote the projection on the second component. Then the transversality $\sigma_\dbar$ and $\sigma_0$ is equivalent to the surjectivity of the map $p_2\scirc d\sigma_\dbar: T_{(u,J_S,J)}(\cals_U\times \calj_U^k)\to
\cale'_{(u,J_S,J)}$. But, due to [*Lemma 8.1.2* ]{}, this map is a linearization of $\sigma_\dbar$ at $(u,J_S,J)$ and has a form (8.1.2).

Thus, the transversality of $\sigma_\dbar$ to $\sigma_0$ at $(u,J_S,J)\in \calp$ is equivalent to the surjectivity of the operator $$\eqalign{
\nabla\sigma_\dbar:\;&T_uL^{1,p}(S,X)\oplus T_{J_S}\ttt_g \oplus T_J\calj^k_U
\longto \wh\cale'_{(u,J_S,J)}
\cr
\nabla\sigma_\dbar:\;&(v,\dot J_S,\dot J) \longmapsto
D_{(u,J)}v + J\scirc du \scirc \dot J_S
+ \dot J\scirc du \scirc J_S.
}\eqno(8.2.1)$$ [*Definition 7.1.2*]{} provides that the quotient of $\wh\cale'_{(u,J_S,J)}=
L^p_{(0,1)}(S,E_u)$ by the image of $D_{u,J}$ is $\sfh^1_D(S,E_u)$. The induced map $\dot J_S \in T_{J_S}\ttt_g \mapsto J\scirc du \scirc \dot J_S \in
\sfh^1_D(S,E_u)$ is also easy to describe. From equality (8.2.1) and [*Corollary 7.3.2*]{} it follows that its image is equal to the image of the homomorphism $du\scirc J_S:\sfh^1(S,TS) \to \sfh^1_D(S,E_u)$ and its cokernel is $\sfh^1_D(S,N_u)$.

It remains to study the image of $T_J\calj^k_U$ in $\sfh^1_D(S,N_u)$. For $(u,J_S,J)\in \calp$ we define $\Psi=\Psi_{(u,J)}:T_J\calj^k_U \to
\wh\cale'_{(u,J_S,J)}$ by setting $\Psi_{(u,J)}(\dot J)\deff \dot J\scirc du
\scirc J_S$. Let $\barr\Psi=\barr\Psi_{(u,J)}:T_J\calj^k_U \to
\wh\cale'_{(u,J_S,J)}$ be induced by $\Psi$. Recall that if $(u,J_S,J)\in
\calp$, then $J_S$ is determined by $u$ and $J$.

Lemma 8.2.2. *(Infinitesimal Transversality). *Let $(u,J_S,J)\in
\calp^*$. Then the operator $\barr\Psi:T_J\calj^k_U \to \sfh^1_D(S, N_u)$ is surjective.**

Proof. In [*Lecture 3*]{} we proved that for $(u,J_S,J)\in
\calp^*$ mapping $u$ is an imbedding in the neighborhood of all but a finite number of points $x\in S$. Thus, there exists such a nonempty open set $V\subset S$ that $u(V)\subset U$ and $u\mid_V$ is an imbedding.

By [*Lemma 7.2.2*]{} we have an isomorphism $\sfh^0_D(S,N_u^*\otimes K_S)
\cong \sfh^1_D(S,N_u)^*$. The fact that operators of the type $D=\dbar + R: L^{1,p}(S,E)\to L^p_{(0,1)}(S,E)$ on the compact Riemann surface $(S,J_S)$ are Fredholm ensures the existence of the finite basis $\xi_1,...,\xi_l$ of the space $\sfh^0_D(S,N_u^*\otimes K_S)$. By [*Lemma 3.1.1*]{} every $\xi \in \sfh^0_D(S,N_u^*\otimes K_S)$ vanishes not more than in $c_1(N_u^*\otimes K_S)[S]$ points on $S$ (compare to the proof of [*Corollary 7.2.3*]{}). From here it follows that there exists $\psi_i\in C_c^k(V, N\otimes \Lambda^{0,1}), i=1,...,l$, generating an $\rr $-basis of the space $\sfh^1_D(S,N)$.

Take some $\psi_i\in C^k_c(V, N\otimes \Lambda^{0,1})$. It is a $\cc$-antilinear $C^k$-smooth homomorphism from $TS\ogran_V$ into $N\ogran_V$ vanishing outside some compact in $V$. Since $u\ogran_V$ is a $C^k$-imbedding and $u(V)\subset U$, $\psi_i$ can be represented in the form $\psi_i =\pr_N \scirc \dot J \scirc du \scirc J_{S}$ for some $J$-anti-linear $C^k$-smooth endomorphism $\dot J$ of $TX$ vanishing outside some compact in $U$. Thus, $\dot J\in
T_J\calj^k_U$ and $\Psi\dot J=\psi_i^{(\nu)}$.

Corollary 8.2.3. *$\calm$ and $\calp^*$ are $C^k$-smooth Banach manifolds and $\pi_\calj:\calm \to \calj^k_U$ is a Fredholm map. For $(M,J)\in\calm$ with $M=u(S)$ there exist natural isomorphisms $$\eqalign{
\ker(d\pi_\calj:T_{(M,J)}\calm \to T_J\calj^k_U)&\;\cong\; \sfh^0_D(S,
\caln_M),
\cr
\coker(d\pi_\calj:T_{(M,J)}\calm \to T_J\calj^k_U)&\;\cong\; \sfh^1_D(S,
\caln_M),
}$$ where $\caln_M=\calo(N_u)\oplus \caln\sing_u$ is a normal sheaf to $M$, $\sfh^0_D(S, \caln_M)$ denotes $\sfh^0_D(S,N_u)$ $\oplus\sfh^0(S,\caln_u\sing)$ and $\sfh^1_D(S, \caln_M)$ denotes $\sfh^1_D(S, N_u)$. The index of $\pi_\calj$ $$\ind_\rr(\pi_\calj)=\ind_\rr(\caln_M)\deff \dimr\sfh^0_D(S, \caln_M)
-\sfh^1_D(S, \caln_M)$$ is equal to $2(c_1(X)[M]+ (n-3)(1-g))$, where $n\deff\dimc X$.*

**Proof. One easily sees that the section $\sigma_{\dbar }$ is $C^k$-smooth if $J\in \calj^k_U$. For $\calp^*$ the statement follows from [*Lemmas 8.2.1*]{} and [*8.2.2*]{}. Moreover, $\calp^*$ is a $C^k$-smooth submanifold of $\calx^*$. Further, local slices of the projection $\pi_\calp:\calp^*\to \calm$ are also $C^k$-smooth. This induces the structure of $C^k$-smooth Banach manifold on $\calm$.**

Consider the natural projection $\pi: \calp^* \to \calj^k_U$. The tangent space $T_{(u,J_S,J)}\calp^*$ consists of $(v,\dot J_S,\dot J)$ with $D_{u,J}v + \half J\scirc du \scirc \dot J_S + \half\dot J\scirc du
\scirc J_S =0$, and the differential $d\pi: T_{(u,J_S,J)}\calp^* \to
T_J\calj^k_U$ has a form $(v,\dot J_S,\dot J)\in T_{(u,J_S,J)}\calp^*\mapsto
\dot J\in  T_J\calj^k_U$.

The kernel $\ker(d\pi) T_{(u,J_S,J)}\calp^*$ consists of solutions of the equation $$D_{u,J}v + \half J\scirc du \scirc \dot J_S =0
%\eqno(2.22)$$ with $v\in\cale_{(u,J_S,J)}$ and $\dot J_S\in T_{J_S}\ttt_g$. Since the map $\pi_\calp:\calp^*\to\calm$ is a principal $\bfg$-bundle, the kernel $\ker(d\pi_\calj:T_{(M,J)}\calm \to T_J\calj^k_U)$ is obtained from $\ker(d\pi)$ by taking a quotient with respect to the tangent space to a fiber $\bfg\cdot(u,J_S,J)$, which is equal to $du(\sfh^0(S,TS))$. Using relations $\sfh^0(S,TS)=
\ker(\dbar_{TS}:L^{1,p}(S,TS)\to L^p(S,TS\otimes\Lambda^{(0,1)}S)$, $T_{J_S}\ttt_g\cong\sfh^1(S,TS)=\coker(\dbar_{TS})$ and $du\scirc \dbar_{TS}= D_{(u,J)}\scirc du$, we conclude that the space $\ker(d\pi_\calj)$ is isomorphic to the quotient $$\{v\in L^{1,p}(S,E_u)\;:\; Dv=du(\phi) \hbox{ for some }\phi\in
L^p(S,TS\otimes\Lambda^{(0,1)}S)\}
\!\!\bigm/\!\!
du\bigl(L^{1,p}(S,TS)\bigr).$$ Hence, $\ker(d\pi_\calj:T_{(M,J)}\calm\to T_J\calj^k_U)\cong
\sfh^0_D(M,\caln_M)$ by [*Theorem 7.3.1*]{}. In particular, $\ker(d\pi_\calj)$ is finite dimensional.

Similarly, the image of $d\pi_\calj$ consists of those $\dot J$ for which the equation $$D_{u,J}v + \half J\scirc du \scirc \dot J_S + \half\dot J\scirc
du \scirc J_S =0$$ has a solution $(v,\dot J_S)$. Hence, $\im(d\pi_\calj)=\ker\barr\Psi$, and $\coker(d\pi) \cong \sfh^1_D(S,N_u)$. Thus, $d\pi_\calj$ is a Fredholm map. This implies the Fredholm property for the projection $\pi:\calp^* \to
\calj^k_U$.

Due to [*Corollary 7.3.2*]{}, $\ind_\rr(\caln_M)=\ind_\rr(E_u)-\ind_\rr(TS)$. Using the index and Riemann-Roch theorems, we get from $c_1(E)=c_1(X)[M]$ and $c_1(TS)=2-2g$ the needed formula $\ind_\rr(\caln)=2\bigl(c_1(X)[M] + n(1-g) - (3-3g)\bigr)=
2(c_1(X)[M] + (n-3)(1-g))$.

A straightforward application of this statement and the Sard Lemma for Fredholm maps gives the following

Corollary 8.2.4.

*If smoothness $k$ of the structures in $\calj^k$ is large enough and with $\ind_\rr (\pi_{\calj })=2(c_1(X)[\gamma ] 
+(n-3)(1-g))<0$ for the homology class $\gamma $, then the following holds:*

1\) For any $(M_0,J_0)\in \calm_{\calj^k}$ with $[M]=[\gamma ]$ there is a neighborhood $W$ fo $J_0$ in $\calj^k_{\omega }$ and a closed subset $S
\subset W$ of Hausdorff codimension close to $2$ such that for $J\in W
\setminus S$ $\calm_J$ is empty;

2\) In the manifold $C^k_{[\gamma ]}(I,\calm_{\calj^k})$ of $k$-smooth paths in $\calm_{[\gamma ]} $ there is a closed subset $R$ of Hausdorff codimension close to one such that for any path $h\in C^k(I,\calm_{\calj^k}\setminus R$ the moduli space $\calm_h$ is empty.

[8.3. Components of the Moduli Space.]{}

Before stating the next results, we introduce some new notations.

Definition 8.3.1. Let $Y$ be a $C^k$-smooth finite-dimensional manifold, possibly with $C^k$-smooth boundary $\d Y$, and $h:Y \to \calj^k_U$ a $C^k$-smooth map. Define the [*relative moduli space*]{} $$\calm_h \deff Y\times_{\calj^k_U}\calm \cong
\{\,(u,J_S,y)\in \cals_U\times \ttt_g\times Y\,:\, (u,J_S, h(y))\in \calp^*
\,\}/\bfg$$ with the natural projection $\pi_h:\calm_h \to Y$. In the special case $Y=\{J\}\hook \calj^k_U$, we obtain the moduli space of $J$-complex curves $\calm_J\deff \pi_\calj\inv(J)$. The projection $\pi_h:\calm_h \to Y$ is a fibration with a fiber $\pi_h\inv(y)=\calm_{h(y)}$. We shall denote elements of $\calm_h$ by $(M,y)$, where $M=u(S)$ with $h(y)$-holomorphic map $u:S\to X$.

**Lemma 8.3.1. *Let $Y$ be a $C^k$-smooth finite-dimensional manifold, and $h:Y \to \calj^k_U$ a $C^k$-smooth map. Suppose that for some $y_0\in Y$ and $u_0\in \calm_{h(y_0)}$ the map $\barr\Psi \scirc dh: T_{y_0}Y \to \sfh^1_D(S, N_{u_0})$ is surjective. Then $\calm_h$ is a $C^k$-smooth manifold in some neighborhood of $(M_0,y_0)\in \calm_h$ with the tangent space $$T_{(M,y)}\calm_h=\ker\bigl(D\;\oplus\; \Psi\scirc dh:
\cale_{u,h(y)}\oplus T_yY \longto \cale'_{u,h(y)} \bigr)
\bigm/ du(\sfh^0(S, TS)).
\eqno(8.3.1)$$***

**Proof. Let $y\in Y$, $(u,J_S,h(y)\in\calp^*$ and $M=u(S)$, so that $(M,y)\in \calm_h$. From the proof of [*Corollary 8.2.3*]{} it follows that the image of the map $d\pi_\calj:T_{(M,h(y))}\calm\to T_{h(y)}\calj^k_U$ is equal to $\ker\bigl(\barr\Psi_{(u,h(y))}\bigr)$, and that the cokernel of $d\pi_\calj$ is mapped by $\barr\Psi$ isomorphically onto $\sfh^1_D(S, N_u)$. The statement of the lemma now follows from [*Lemma 8.2.1*]{}.**

**Definition 8.3.2. Let $Y$ be a compact manifold, $h:Y\to \calj^k_U$ a $C^k$-smooth map, $\calm_h\subset \calm \times Y$ a corresponding moduli space and $(M_0,y_0)\in \calm_h$ a point. A [*component $\calm_h(M_0,y_0)$ of $\calm_h$ through $(M_0,y_0)$*]{} is the set of those $(M,y)\in\calm_h$, that for every open neighborhood $W$ of $h(Y)\subset
\calj^k_U$ there exists a continuous path $\gamma:[0,1]\to \calm$, with the following properties:**

[*a)*]{} $\gamma(0)=(M_0,h(y_0))$ and $\gamma(1)=(M,h(y))$, i.e., $\gamma$ connects $(M_0,y_0)$ and $(M,y)$ in $\calm$;

[*b)*]{} $J_t\deff\pi_\calj(\gamma(t))\in W\subset\calj^k_U$ for any $t\in[0,1]$, i.e., the corresponding path of almost complex structures $J_t$ lies in the given neighborhood $W$ of $h(Y)$.

**Lemma 8.3.2 *Let $h:Y\to \calj^k_U$, $(M_0,y_0)\in \calm_h$ and $\calm_h(M_0,y_0)$ be as in Definition 8.3.2. Then***

$\calm_h(M_0,y_0)$ is a closed subset of $\calm_h$,

if $\calm_h(M_0,y_0)$ is compact, then there exists a set $\calm_h^0$ containing $\calm_h(M_0,y_0)$, which is compact and open in $\calm_h$;

if $\calm_h(M_0,y_0)$ is not compact, then there exists a continuous path $\gamma:[0,1)\to \calm_h$ with the following properties:

[*a)*]{} $\gamma(0)=(M_0,h(y_0))$, i.e., $\gamma$ begins at $(M_0,y_0)$;

[*b)*]{} there exists a sequence $t_n\nearrow 1$ such that the sequence $(M_n,J_n)\deff\gamma(t_n)$ lies in $\calm_h$ and is discrete there, but the sequence $\{J_n\}$ converges to some $J^*\in \calj_U^k$.

**Proof. 1.2em Let $(M',y')\in \barr{\calm_h(M_0,y_0)}\subset \calm_h$ and $J'=h(y')$. Let $W$ be any open neighborhood of $h(Y)\subset \calj_U^k$, and let $\{(M_n,y_n)\}$ be a sequence in $\calm_h$ converging to $(M',y')$. Then there exists some ball $B\ni(M',J')$ in $\calm$ whose projection on $\calj^k_U$ lies in $W$. Since some $(M_n,J_n)$ with $J_n=h(y_n)$ lies in $B$, there exists a path $(M_t,J_t)$ in $\calm$ connecting $(M_0,h(y_0))$ with $(M',J')$ such that $J_t\in W$ for any $t\in [0,1]$. This proves the closedness of $\calm_h(M_0,y_0)$.**

1.2em Let $(M',y')\in \calm_h$ and $J'=h(y')$. Fix a finite dimensional subspace $F\subset T_{J'}\calj^k_U$ such that the map $D_{u',J'} \oplus \Psi:
\cale_{u',J'} \oplus F \to \cale'_{u',J'}$ is surjective. Let $B\ni0$ be a ball in $F$. Find a $C^k$-smooth map $H:Y\times B$ such that $H(y,0)\equiv h(y)$ and $dH_{(y',0)}: T_{(y',0)}(Y\times B) \to T_{J'}\calj^k_U$ induces isomorphism $T_0B\buildrel\cong\over \longto F\subset T_{J'}\calj^k_U$. Then $\calm_H$ contains a neighborhood $W\ni (M',y',0)$ which is $C^k$-smooth manifold such that $W\cap \calm_h$ is closed in $W$. This implies that $\calm_h$ is a locally compact topological space.

Since $\calm_h(M_0,y_0)$ is a compact subset of $\calm_h$, it has an open neighborhood $V$ whose closure $\barr V \subset\calm_h$ is also compact. Let $W_i\subset \calj^k_U$ be a fundamental system of neighborhoods of $h(Y)$, so that $\cap_i W_i =h(Y)$. Let $V_i$ denote the set of those $(M,y)\in \calm_h$ such that $(M,h(y))$ and $(M_0,h(y_0))$ can be connected by a continuous path $(M_t, J_t)$ in $\calm$ with $J_t$ lying in $W_i$. Then $\cap V_i =\calm_h(M_0,y_0)$. The same arguments as in the part of the proof show that every $V_i$ is both open and closed in $\calm_h$.

We state that there exists a number $N\in \nn$ such that $\bigl(\cap_{i=1}^N V_i\bigr) \cap\barr V \subset V$. If it is false, then for any $n\in \nn$ there would exist $(M_n,y_n)\in \bigl(\cap_{i=1}^n V_i\bigr)
\cap\barr V \bes V$. But, in this case, some subsequence of $\{(M_n,y_n)\}$ would converge to some $(M^*,y^*)\in\bigl(\cap_{i=1}^\infty V_i\bigr)
\cap\barr V \bes V$, which is impossible. For such $N\in \nn$ the set $$\def\capl{\mathop{\cap}}
\calm^0_h\deff \left(\capl_{i=1}^N V_i\right) \cap\barr V
=\left(\capl_{i=1}^N V_i\right) \cap V$$ satisfies the conditions of the lemma.

1.2em Suppose that $\calm_h(M_0,y_0)$ is not compact. Then there exists a discrete sequence $\{(M_n,y_n)\}$ in $\calm_h(M_0,y_0)$. Since $Y$ is compact, we may assume that $y_n$ converges to some $y^*$. For any $n\in \nn$ fix a path $\gamma_n:[0,1] \to \calm_h$ connecting $(M_{n-1}, h(y_{n-1}))$ with $(M_n,h(y_n))$. Set $t_n\deff
1-2^{-n}$. For $t\in [t_{n-1},t_n]$ define $\gamma(t)\deff
\gamma_n(2^n(t-t_{n-1}))$. Then $\gamma:[0,1)\to \calm_h$ and $t_n\nearrow1$ satisfy the conditions of the lemma.

Theorem 8.3.3.

*Let $(M_0,J_0)\in \calm$ and let $h:[0,1]\to \calj^k_U$ be $C^k$-smooth and with $h(0)=J_0$. Suppose that there exists a subset $\calm_h^0$ of $\calm_h$ which is compact, open and contains $(M_0,J_0)$. Suppose also that $\ind(\pi_\calj)=2(c_1(X)[M_0]+ (n-3)(1-g))$ is non-negative. Then $h$ can be $C^k$-approximated by smooth maps $h_n:[0,1] \to \calj^k_U$ such that*

every $\calm_{h_n}$ contains a component $\calm_{h_n}^0$ which is a smooth connected compact manifold of the expected dimension $\dimr(\calm_{h_n}^0)=\ind(\pi_\calj)+1$;

$\calm_{h_n(0)}^0\deff \pi_{h_n}\inv(0)\cap \calm_{h_n}^0$ and $\calm_{h_n(1)}^0\deff \pi_{h_n}\inv(1)\cap \calm_{h_n}^0$ are also smooth manifolds of the expected dimension $\ind(\pi_\calj)$ ($\calm^0_{h_n(1)}$ may be empty), and $\calm_{h_n}$ is a smooth bordism between $\calm_{h_n(0)}^0$ and $\calm_{h_n(1)}^0$;

every $\calm_{h_n(0)}^0$ is connected with $(M_0,J_0)$ by a path in $\calm$, i.e., there exists a $C^k$-path $\gamma_n: [0,1] \to \calm$ such that $\gamma_n(0)= (M_0,J_0)$ and $\gamma_n(1)\in \calm_{h_n(0)}^0$; in particular, every $\calm_{h_n(0)}^0$ is nonempty;

for every $(M,J)\in \calm_{h_n}$ one has $\dimr\sfh^1_D(S,N_M)\le1$.

Proof. Denote $\calm_h^0$ by $K$. Let $\cale_K$ and $\cale'_K$ be the pull-backs onto $K$ of the bundles $\cale\to \calm$ and $\cale'\to
\calm$, respectively. Further, let $\calt\deff h^*T\calj^k_U$ be the pull-back of the tangent bundle $T\calj^k_U$ onto $[0,1]$.

Due to [*Lemma 8.2.2*]{}, for every $(M,J)\in K$ with $M=u(S)$ we find that $n_{(M,J)}\in \nn$ and a $C^k$-smooth homomorphism $P_{(M,J)}: F_{(M,J)} \to \calt$ form a trivial vector bundle $F_{(M,J)}$ over $[0,1]$ of rank $\rank F_{(M,J)}=n_{(M,J)}$ such that the operator $$D_{(u,J)}\oplus \Psi_{(u,J)} \scirc P_{(M,J)} :
\cale_{(M,J)}\oplus F_{(M,J)} \to \cale'_{(M,J)}
\eqno(8.3.2)$$ is surjective.

Since $K$ is compact, we can take an appropriate finite collection of $(F_{\!(M,J)},$ $P_{\!(M,J)})$ and construct a homomorphism $P: F\to \calt$ from a trivial vector bundle $F\cong [0,1]\times \rr^n$ such that $D_{(u,J)}\oplus \Psi_{(u,J)} \scirc P : \cale_{(M,J)}
\oplus F \to \cale'_{(M,J)}$ is surjective for all $(M,J)\in K$.

For $\dot J$ lying in some small ball in $T_J\calj^k_U$ we set $\exp_J(\dot J)\deff J{(1-J\dot J/2)\over(1+J\dot J/2)}$. Differentiating the identity $J^2=-1$ gives the relation $J\dot J =- \dot JJ$, and consequently the equality $$\left(J{(1-J\dot J/2)\over(1+J\dot J/2)} \right)^2=-1,$$ which means that $\exp_J$ takes values in $\calj^k_U$. Further, it is easy to see that the differential of $\exp_J$ in $0\in T_J\calj^k_U$ is the identity map of $T_J\calj^k_U$. Thus, $\exp_J$ is a natural exponential map for $\calj^k_U$.

Take a sufficiently small ball $B=B(0,r)\subset \rr^n$ and define a map $H^*:[0,1]\times B  \to \calj^k_U$ by setting $H^*(t, y)\deff \exp_{h(t)}(P(t,y))$. The construction of $H^*$ provides that for all $(M,J)\in K$ with $J=h(t)$ the map $D_{(u,J)}\oplus \Psi_{(u,J)} \scirc dH^*(t,0) : \cale_{(M,J)}
\oplus T_{(t,0)}([0,1]\times B) \to \cale'_{(M,J)}$ is surjective. Making an appropriate small perturbation of $H^*$, we obtain a $C^k$-smooth function $H: [0,1]\times B  \to \calj^k_U$ with the following properties:

for $H(t,0)=h(t)$, i.e., $H$ is a deformation of $h$ with a parameter space $B$;

$
%\displaystyle
D_{(u,J)}\oplus \Psi_{(u,J)} \scirc dH(t,0) : \cale_{(M,J)}
\oplus T_0B \longto \cale'_{(M,J)}
$ is surjective for all $(M,t)\in K$ with $J=h(t)$ and $M=u(S)$;

in a neighborhood of every $(t,y)\in [0,1]\times B$ with $y\not=0$ the map $H$ is $\calj^\infty_U$-valued and $C^\infty$-smooth.

Let us identify $K$ and $\calm_h$ with the subset of $\calm_H$ using natural continuous imbedding $(M,t)\in \calm_h \mapsto (M,t,0)\in \calm_H$. Due to [*Lemma 8.2.2*]{}, there exists a neighborhood $V$ of $K$ in $\calm_H$ which is $C^k$-smooth manifold. Taking a smaller neighborhood of $K$ if it is needed, we may assume that the closure $\barr V$ of $V$ is compact and does not meet connected components of $\calm_h$ different from $\calm_h^0=K$. Let $p:V \to B$ be the natural projection such that $(u,t,y)\mapsto y$.

We state that there exists a smaller ball $B_1=B(0,r_1)\subset B$ such that for every $y\in B_1$ the set $p\inv(y)\subset V$ is compact. Suppose the contrary is true. Then there would exist a sequence $y_n\in B$ converging to $0\in B$ such that $p\inv(y_n)$ are not compact. Since $\barr V$ is compact, there would also exist such $u_n$ and $t_n$ that $(u_n,t_n,y_n)$ lies in the closure $\barr{p\inv(y_n)}\subset \barr V$ but not in $V$. Taking an appropriate subsequence we may assume that $(u_n,t_n,y_n)$ converges to $(u^*,t^*,0)$. However, in this case $(u^*,t^*,0)\in \calm_h \cap \barr V$ and hence $(u^*,t^*,0)\in V$. On the other hand $\barr V\bes V$ is compact, and thus $(u^*,t^*,0)$ must belong to $\barr V\bes V$. The obtained contradiction shows that the statement is true.

Set $V_1\deff p\inv(B_1)$. Due to the choice of $H$ the resticted projection $p:V_1\bes K\to \check B_1 \deff B_1\bes\{0\}$ is $C^\infty$-smooth. Due to the Sard lemma (see, e.g., \[Fed\], §[**3.4**]{}), there exists a dense subset $B_1^* \subset B_1$ such that for any $y\in B_1^*$ the set $p\inv(y)$ is a $C^\infty$, a smooth compact manifold. Fix a sequence $y_n\in B_1^*$, converging to $0\in B$ and set $h_n(t)\deff
H(t,y_n)$. Also set $\calm_{h_n}^0 \deff \calm_{h_n} \cap V$, so that $\calm_{h_n}^0 = p\inv(y_n)$. Then every $\calm_{h_n}^0$ is a $C^\infty$-smooth nonempty manifold, which is connected with $(M_0,J_0)$ by a path in $\calm$.

Due to [*Lemma 8.2.3*]{}, the tangent space to $V_1\subset \calm_H$ at $(u,t,y)$ is canonically isomorphic to $$\ker\!\Bigl(\!D_{u,H(t,y)}\,\oplus\, \Psi\scirc dH:
\cale_{u,H(t,y)}\,\oplus\, T_{(t,y)}\bigl([0,1]{\times} B_1\!\bigr)
\to \cale'_{u,H(t,y)} \!\Bigr)
\!\!\!\Bigm/ \!\!\!du(\sfh^0(S\!\!,\! TS)).$$ Since $p: V_1 \to B_1$ is a projection of the form $(u,t,y)\in V_1
\mapsto y\in B_1$, the differential $dp_{(u,t,y)}$ maps the tangent vector of the form $(\dot u,\dot t,\dot y)\in T_{(u,t,y)}V_1$ into $\dot y
\in T_yB_1$. This means that $dp_{(u,t,y)}$ is a restiction on $\ker(D_{u,H(t,y)}\,\oplus\, \Psi\scirc dH)$ of the linear projection $p_B:\cale_{u,H(t,y)} \oplus\, T_{(t,y)}\bigl([0,1]{\times} B_1\!\bigr)
\to T_y B_1$ such that $p_B(\dot u,\dot t,\dot y)=\dot y$. In particular, for $y=y_n$ the map $dp_{(u,t,y)}$ is surjective, which means the surjectivity of the map $$p_B: \ker \bigl(D_{u,H(t,y)}\,\oplus\, \Psi\scirc dH \bigr)
\longto T_{y_n}B_1.$$ This is equivalent to the surjectivity of $$D_{u,H(t,y_n)}\,\oplus\, \Psi\scirc dH\,\oplus\,p_B:
\cale_{u,H(t,y_n)}\,\oplus\, T_{(t,y_n)}\bigl([0,1]{\times} B_1\!\bigr)
\to \cale'_{u,H(t,y_n)} \oplus T_{y_n}B_1$$ and hence to the the surjectivity of $$D_{u,h_n(t)}\,\oplus\, \Psi\scirc dh_n:
\cale_{u,h_n(t)}\,\oplus\, T_t[0,1]
\to \cale'_{u,h_n(t)}.
\eqno(8.3.3)$$ Consequently, $\dimr\sfh^1_D(S, N_M)\le1$ for any $(M,t)\in \calm_{h_n}\cap
V_1$.

Corollary 8.3.4. *Under the conditions of Theorem 8.3.3 suppose additionally that $S$ is a sphere $S^2$. Then for all points $(M,t)\in \calm^0_{h_n}$ the associated $D_N$-operator is surjective, i.e., $\sfh^1_{D_N}(S^2,N_{M_t})=0$.*

Moreover, $\calm_{h_n}$ is a trivial bordism: $\calm_{h_n(0)}\times [0,1]$. In particular, for every $h_n(0)$-holomorphic sphere $M_0\in \calm_{h_n(0)}$ there exists a continuous family of $h_n(t)$, holomorphic spheres $M_{n,t}=
u_{n,t}(S^2)$ with $M_{n,0}=M_0$.

*Proof. Suppose that for some $(M,t)\in \calm^0_{h_n} $ we have $\sfh^1_{D_N}(M,N_M)\not= 0$. Then by (iv) of [*Theorem 8.3.3*]{} $\sfh^1_{D_N}(M,N_M)= 1$. But this contradicts [*Theorem 7.3.1*]{} for the case $S=S^2$ and $L=N_M$.*

Let for $(M, t)\in \calm^0_{h_n}$ we have $M =u(S)$ and $J=h_n(t)$. Let also $\dot J \not= 0 \in dh_n(T_t[0,1])$. Then by [*Lemma 8.2.1*]{} and [*Corollary 8.2.3*]{} the tangent space $’_{(M,t)} \calm^0_{h_n}$ is canonically isomorphic to

$$\ker\!\Bigl(\!D_{u,J}\,\oplus\, \Psi:
\cale_{u,J}\,\oplus\, \rr\,\dot J\!\bigr)
\to \cale'_{u,J} \!\Bigr)
\!\!\!\Bigm/ \!\!\!du(\sfh^0(S\!\!,\! TS)),$$ and the differential of the projection $d\pi_{h_n}: ’_{(M,t)} \calm^0_{h_n}
\to T_t[0,1] \cong \rr$ is of the form $d\pi_{h_n}[v,a\dot J] = a$. While in the case $S=S^2$ the space $\sfh^1(S, TS)$ is trivial, [*Corollary 8.3.4*]{} insures the surjectivity of the operator $D_{u,J}: \cale_{u,J} \to \cale'_{u,J}$. Thus for $a\not=0 \in \rr$ there exists such a $v\in \cale_{u,J}$ that $[v,a\dot J]\in ’_{(M,t)} \calm^0_{h_n}$. This means that for every $(M, t)\in \calm^0_{h_n}$ the projection $d\pi_{h_n}: ’_{(M,t)}
\calm^0_{h_n} \to T_t[0,1]$ is surjective. While the manifold $\calm^0_{h_n}$ is compact, there is a diffeomorphism $\calm_{h_n} \cong
\calm_{h_n(0)}\times [0,1]$.

*8.4. Moduli of Parameterized Curves.*

Sometimes it may be useful to consider the moduli spaces of parameterised complex curves. Therefore, in this paragraph $\calp $ will denote the space of parameterized complex curves on $X$. Suppose that $(u, J_S, J) \in \calp$, [*i.e.,*]{} $u$ belongs to $C^1(S, X)$ and satisfies the equation $du\scirc J_S = J \scirc
du$. Set $$T_{(u, J_S, J)}\calp\deff \bigl\{\, (v,\dot J_S,\dot J) \in T_u\cals
\times T_{J_S} \calj_S \times T_J\calj \, :\,
2D_{u, J}v  +  \dot J \scirc du \scirc J_S +
\dot J_S \scirc du \scirc J =0 \,\bigr\}.$$ Let $\pr_\calj: \cals \times \calj_S \times \calj \to \calj$ and $\pr_{(u, J_S, J)} : T_{(u, J_S, J)} \calp \to T_J\calj$ denote the natural projections.

Theorem 8.4.1.

*The map $\pr_{(u, J_S, J)} : T_{(u, J_S, J)}\calp
\to T_J\calj$ is surjective $\sfh^1_D(S, N_0)$ $=0$, and then the following hold:*

the kernel $\ker(\pr_{(u, J_S, J)})$ admits a closed complementing space;

for $(\tilde u,\tilde J_S,\tilde J)\in \calp$ close enough to $(u, J_S, J)$ the projection $\pr_{(\tilde u,\tilde J_S,\tilde J)}$ is also surjective;

for some neighborhood $U\subset \cals \times \calj_S
\times \calj$ of $(u, J_S, J)$ the set $\calp \cap U$ is a Banach submanifold of $U$ with the tangent space $T_{(u, J_S, J)}\calp$ at $(u, J_S, J)$;

there exists a $C^1$-map $f$ from some neighborhood $V$ of $J\in \calj$ into $U$ with $f(V)\subset \calp$, $f(J) = (u,J_S,J)$ and $\pr_\calj
\scirc f = \id_V$ such that $\im (df: T_J\calj \to
T_{(u, J_S, J)}\calp)$ is complementing to $\ker(\pr_{(u, J_S, J)})$.

Proof. Denote by $\tilde A$ the set of a singular point of $M=u(S)$, [*i.e.*]{}, the set of cuspidal and self-intersection points of $M$. Let $\xi \in L^p_{(0, 1)} (S, E)$. Using the local solvability of the $D$-equation (proved essentially in [*Lemma 3.2.1*]{}) and an appropriate partition of unity, we can find $\eta\in L^{1,p}(S, E)$ such that $\xi -2D\eta \in C^1_{(0, 1)}(S, E)$ and $\xi -2D\eta =0$ in a neighborhood of $\tilde A$. Then we find $\dot J\in T_J\calj \equiv
C^1_{(0, 1)}(X, TX)$ such that $\dot J \scirc du \scirc J_S =\xi-
2D\eta$. The surjectivity of $\pr_{(u, J_S, J)}$ and [*Theorem 7.3.1*]{} easily yield the identity $\sfh^1_D(S, N_0)=0$.

*From now on and until the end of the proof we suppose that $\sfh^1_D(S, N_0)=0$.*

Let $ds^2$ be some Hermitian metric on $S$ and let $\calh_{(0, 1)} \subset
C^1_{(0, 1)}(S, TS)$ be a space of $ds^2$-harmonic $TS$-valued $(0, 1)$-forms on $S$. Then the natural map $\calh_{(0, 1)} \to
\sfh^1(S, TS)$ is an isomorphism. Furthermore, due to [*Corollary 7.3.2*]{}, the map $$g\deff(du, 2D): \calh_{(0, 1)} \oplus L^{1,p}(S, E) \to
L^p_{(0, 1)}(S, E)$$ is surjective and has a finite-dimensional kernel. Consequently, there exists a closed subspace $Y\subset \calh_{(0, 1)} \oplus L^{1,p}(S, E)$ such that $g : Y \to L^p_{(0, 1)}(S, E)$ is an isomorphism. Let $$h=(h_{TS},h_E): L^p_{(0, 1)}(S, E) \to Y\subset \calh_{(0, 1)}
\oplus L^{1,p}(S, E)$$ be the inversion of $g\vert_Y$.

Take $\dot J\in T_J\calj= \{\, I\in C^1(X, {\sl End}(TX))\, :\,
JI+IJ=0\,\} \equiv C^1_{(0, 1)}(X, TX)$. Then $\dot J \scirc du
\scirc J_S$ lies in $C^0_{(0, 1)}(S, E)$. Let $h(\dot J \scirc du
\scirc J_S) =(\xi,\eta)$ with $\xi\in \calh_{(0, 1)}
\subset C^1_{(0, 1)}(S, TS)$ and $\eta\in L^{1,p}(S, E)$. Then we obtain $$2D(-\eta) + \dot J \scirc du \scirc J_S +
J \scirc du ( J_S \xi)=0,\eqno(8.4.1)$$ where we use the identity $J\scirc du \scirc J_S =-du$. Using that $T_{J_S}\calj_S =$ $ C^1_{(0, 1)}(S, TS)$, we conclude that the formula $F(\dot J)= \bigl(J_S h_{TS}(\dot J \scirc du
\scirc J_S), -h_E(\dot J \scirc du \scirc J_S),\dot J \bigr)$ defines a [*bounded*]{} linear operator $F: T_J\calj \to T_{(u, J_S, J)}\calp$ such that $\pr_{(u, J_S, J)}\scirc F = \id_{T_J\calj}$. In particular, $\sfh^1_D(S,
N_0)=0$ implies the surjectivity of $\pr_{(u, J_S, J)}$.

The image of the just defined operator $F$ is closed, because the convergence of $F(\dot J_n)$, $J_n\in T_J\calj$ obviously yields the convergence of $J_n=\pr_{(u, J_S, J)}\scirc F(\dot J_n)$. One can easily see that $\im(F)$ is a closed complementing space to $\ker(\pr_{(u, J_S, J)})$.

Further, for $(\tilde u, \tilde J_S,\tilde J) \in \calp$ close enough to $(u, J_S, J)$ the map $\tilde g \deff (d\tilde u, 2D_{\tilde u,\tilde J}): 
Y \to L^p_{(0, 1)}(S,\tilde E)$ is also an isomorphism. This implies the surjectivity $\pr_{(\tilde u,\tilde J_S,\tilde J)}$.

The statements and can easily be obtained from [*ii)*]{} and the implicit function theorem.

*8.5. Gromov Non-squeezing Theorem.*

We shall finish this chapter with the proof of the Gromov non-squeezing theorem.

Consider an infinite cylinder $\zz (\lambda ):=\Delta (\lambda )
\times \rr^{2n-2}$ in $\rr^{2n}$. Here $\Delta (\lambda )$ is a disk of radius $\lambda $ in the $x_1,y_1$-plane, and on $\rr^{2n-2}$ we fix coordinates $x_2,y_2,...,x_n,y_n$. By $\bb (R)$ we denote the ball of radius $R$ in $\rr^{2n}$. We consider on $\rr^{2n}$ the standard symplectic form $\omega =\Sigma_{i=1}^ndx_i\wedge dy_i$.

Let $D\subset \rr^{2n}$ be a domain. Recall that a smooth map $f:D\to \rr^{2n}$ is called a symplectomorphism if $f^*\omega =\omega $. In particular, symplectomorphisms preserve the Euclidean volume.

Exercise. [Find a volume preserving a linear map from $\bb (R)$ to $\zz (\lambda )$ for arbitrary $R$ and $\lambda $. ]{}

The following theorem shows how far symplectomorphisms are from volume preserving maps.

Theorem 8.5.1. [*(Gromov Non-squeezing Theorem). If there exists a symplectomorphism $f:\bb (R) \to \zz (\lambda )$, then $R\le \lambda$*]{}.

Proof.

Taking a smaller ball, if nessessary, and translating the image, we can suppose that $f(\bb (R))$ is contained in some compact $\bar\Delta(0,\lambda')\times [0,a]^{2n-2}$ of $\Delta (0,\lambda )\times
\rr^{2n-2}$. Factorizing $\rr^{2n-2}$ by the lattice $a\cdot \zz^{2n-2}$, we observe that $\overline{f(\bb (R))}\comp \Delta (0,\lambda )\times
T^{2n-2}$. Moreover, our syplectic form $\omega $ descends onto this factor and still can be decomposed as $\omega = \omega_1 + \omega_2$. Here $\omega_1 = dx_1\wedge dy_1$ and $\omega_2=\Sigma_{i=2}^ndx_i\wedge dy_i$.

Further, we compactify the disk $\Delta (0,\lambda )$ to a two-dimensional sphere $S^2$ by an $\eps $-disk $\Delta (0,\eps )$ and extend $\omega_1$ to a smooth strictly positive $(1,1)$-form, still denoted by $\omega_1$ on $S^2$, and having $$\int_{S^2}\omega_1 = \pi \lambda^2 + \eps . \eqno(8.5.1)$$ Note that now we have a symplectic imbedding $f:\overline{\bb (R)}
\to S^2\times T^{2n-2}$. Also note that our symplectic form $\omega $ on $S^2\times T^{2n-2}$ tames the standard complex structure $J\st$ on this manifold and satisfies $$\int_{[\gamma ]}\omega = \pi \lambda^2 + \eps , \eqno(8.5.2)$$ where $[\gamma ]=[S^2\times \{ pt\} ]$. We further observe that $\calm_{[\gamma ],J\st}$ consists of $\{ \cc\pp^1
\times \{ a\} :a\in T^{2n-2}\} $, i.e., this moduli space is diffeomorphic to a torus $T^{2n-2}$.

Lemma 8.5.2. *(a) For a generic almost-complex structure $J$ on $S^2\times T^{2n-2}$ tamed by $\omega $ the moduli space $\calm_{[\gamma ],J}$ is diffeomorphic to $T^{2n-2}$.*

\(b) Moreover, for any (not nessessarily generic!) $J$ and for any point $p\in S^2\times T^{2n-2}$ there exists a $J$-complex curve from $\calm_{[\gamma ],J}$ passing through $p$.

Proof.

\(a) For any $J\in \calj_{\omega }$, any $J$-complex curve $C$ representing $[\gamma ]$ obviously cannot be decomposed as $C=C_1\cup C_2$, where $C_k$ are $J_k$-complex for some $J_k\in \calj_{\omega }$. Therefore, we can apply Theorem 3.2 from the introduction to this chapter. Thus (a) is proved.

\(b) In fact, for a generic $J$ we have a little bit more. Namely, for a generic curve $h:[0,1]\to \calj_{\omega }$ with $h(0)=J\st $ consider an evaluation map $\ev :\calc_h\to S^2\times T^{2n-2}$, where $\calc_h$ is the universal $\cc\pp^1$-bundle over $\calm_{h}$. Recall that $\calm_h$ is a manifold diffeomorphic to $\calm_{h(0)}\times [0,1]$. This yields that $\calc_h$ is diffeomorphic to $\calc_{h(0)}\times [0,1]$.

Now, from the uniqueness properties of complex curves it easily follows that for any $t\in [0,1]$ the map $\ev :\calc_{h(t)}\to
S^2\times T^{2n-2}$ is surjective. Thus, we prove $\ev :\calc_{J}\to
S^2\times T^{2n-2}$ is surjective for generic $J\in \calj_{\omega }$.

Since generic structures are dense in $\calj_{\omega }$, we immediately obtain that the evaluation map $\ev :\calc_{h(t)}\to
S^2\times T^{2n-2}$ is surjective for all $J\in \calj_{\omega }$.

This completes the proof of the lemma.

Let us return to the proof of our theorem. Using [*Proposition 1.2.1*]{} from [*Lecture 1*]{}, we can extend the structure $f_*J\st $ from $f(\bb (R))$ onto $S^2\times T^{2n-2}$ to a $\omega $-tamed structure $J$. By [*Lemma 8.5.2*]{} there is a $J$-complex rational curve $C\ni f(0)$. Remark that $$\int_C\omega = \pi \lambda^2 + \eps , \eqno(8.5.3)$$ because $[C]=[S^2\times \{ pt\} ]$. While $f$ is a symplectomorphism, $$\int_{f^{-1}(C\cap f(\bb (R))}\omega\st  \le \pi \lambda^2 + \eps .
\eqno(8.5.4)$$ But $f^{-1}(C\cap f(\bb (R))$ is a complex curve in $\bb (R)\subset \cc^n$ passing through zero. By a well-known estimate due to Alexander-Taylor-Ullman, see \[A-T-U\], the area of such a curve is at least $\pi R^2$. So $\pi R^2\le
\pi \lambda^2 +\eps $.

[8.6. Exceptional Spheres in Symplectic $4$-Manifolds.]{}

[*Corollary 8.3.4*]{} enables us to prove an interesting result about exceptional spheres in symplectic $4$-manifolds. Recall that a smooth rational curve $C$ in a smooth complex surface $X$ is called exceptional if $[C]^2=-1$. Such curve can be contructed to a point, i.e., there exists a smooth complex surface $Y$ and a holomorphic map $h:X\to Y$ such that $h(C)=y$ is a point and $h\mid_{X\setminus C}:X\setminus C\to Y\setminus 
\{ y\} $ is a biholomorphism.

Consider now a symplectic $4$-manifold $(X,\omega )$ and let $M$ be a symplectic $2$-sphere imbedded to $X$, i.e., there is an imbedding $u:\ss^2\to M \hookrightarrow X$ with $u^*\omega$ nowhere zero.

Definition 8.6.1. [*Call $M$ exceptional if $[M]^2=-1$.* ]{}

Using [*Lemmas 1.4.2 and 1.4.3*]{} we can construct an almost-complex structure $J\in \calj_{\omega }$, which is moreover integrable in the neighborhood of $M$ and such that $M$ becomes $J$-complex. Now the complex analytic statement mentioned above allows us to construct this $M$ to a point.

Corollary 8.6.1. [*Take a maximal, linearly independant in $\sfh_2(X,\zz )$ system of exceptional symplectic spheres $M_1,...,M_k$ in $X$. Then there exists $J\in \calj_{\omega }$ making some symplectic spheres $\tilde 
M_1,...,\tilde M_k$, with $\tilde M_j$ being isotopic to $M_j$ for $1\le j\le k$, all $J$-complex.* ]{} Proof.

For $k=1$ this is the statement of [*Lemma 1.4.2*]{}. Suppose we prove this statement for $k-1$ exceptional spheres. Take our spheres $M_j,j=1,...,k$ and find a structure $J$ such that $\tilde M_j$ are isotopic to $M_j$ and $J$- complex, $j=1,...,k-1.$ The normal Gromov operator $D_{J,\tilde M_j}$ is surjective on all $\tilde M_j,j=1,...1k-1$ by [*Corollary 7.2.3*]{} and thus by [*Theorem 7.3.1*]{} and [*Corollary 8.2.3*]{} the projection $\pi_{\calj }:\calm \to \calj $ is a diffeomorphism in the neighborhood of $\tilde M_j$ and $J$ for all $j=1,...,k-1$.

Take a structure $J_1$ such that $M_k$ becomes $J_1$-complex. Again the projection $\pi_{\calj }$ is regular in the neighborhood of $M_k$. Using [*Corollary 8.2.4*]{}, we take a generic path $h$ starting close to $J$, say at $J´$ and ending close to $J_1$ such that all $\calm_h(\tilde M_j´,J´)$ are compact and open in corresponding moduli spaces. Here $\tilde M_j´$ are $J´$-complex spheres isotopic to $\tilde M_j$. Applying [*Corollary 8.3.4*]{}, we obtain an isotopy of $\tilde M_j´$ to a $J''$-complex spheres $\tilde M_j''$ with $J''$ close to $J_1$. It remains to take a $J''$-complex sphere $\tilde M_k''$ isotopic to $M_k$.

[Chapter IV. Envelopes of Meromorphy of Two-spheres.]{}

This chapter is devoted to the study of envelopes of meromorphy of neighborhoods of two-spheres in complex algebraic surfaces. The original question which motivated our studies was asked by A. Vitushkin.

*Let $M$ be a “small” perturbation of the complex line in $\cc\pp^2$. Does there exist a nonconstant holomorphic function in the neighborhood of such an $M$?*

It was asked as a test question on the way to searching for the solution to the Jakobian conjecture, and the answer, as expected, was negative.

Let us briefly test the possible approaches to the answer to this question. It is more or less clear that one should try to extend holomorphic (or meromorphic via expected nonexistence of holomorphic) functions onto the whole $\cc\pp^2$ and then conclude

[*1.*]{} If $M$ is a complex line itself, then nonconstant holomorphic functions do not exist in any neighborhood of $M$ for the following reason. First note that in this case $\cc\pp^2\setminus M=\cc^2$. Let $B_N$ denote the closed ball of radii $N$ in $\cc^2$. Then $V_N:=\cc\pp^2\setminus B_N$ is a fundumental system of strictly pseudoconcave neighborhoods of $M$. Any function holomorphic in $V_N$ holomorphically extends onto the ball $B_N$ by Hartogs’ theorem, and thus becomes holomorphic on the whole $\cc\pp^2$, i.e., constant.

One can try to construct such an exhaustion for any $M$. But the generic $M$ is totally real outside of three positive elliptic points (this follows from the Lai formulae and cancellation theorem of Kharlamov-Eliashberg, see Appendix IV) and one can check that a totally real disk has no small concave tubular neighborhoods.

[*2.*]{} As was just pointed out, a generic perturbation $M$ will have exactly three elliptic points with complex tangents. This makes it problematic to attempt to construct a family of complex disks (or any other Riemann surfaces) with boundary on $M$ starting from our elliptic points. Such an approach to the construction of the holomorphic envelope of $M$ seems to be perfect only for specially imbedded spheres, see \[B-G\],\[B-K\],\[E\], \[Sch\] and \[F-M\].

[*3.*]{} The method used here to study envelopes of meromorphy of spheres is based on Gromov’s theory of pseudoholomorphic curves. We remark first of all that a small perturbation of a complex sphere is symplectic:

Definition. *A $C^1$-smooth immersion $u\:S\to(X,\omega)$ of a real surface $S$ into a symplectic manifold $(X,\omega )$ is called [*symplectic*]{} if $u^*\omega$ does not vanish anywhere on $S$.*

Thus, we shall study envelopes of meromorphy of neighborhoods of two-spheres symplectically immersed in complex surfaces. Here a [*complex surface*]{} means a (Hausdorff) connected complex two-dimensional manifold $X$ countable at infinity.

The idea is to perturb the complex structure in the given neighborhood of $M$ in such a way that $M$ becomes complex; see [*Lemma 1.4.2*]{}, where an appropriate family $J_t$ of tamed almost complex structures is constructed. Using results from Chapter III, we then construct a family $M_t$ of $J_t$-complex spheres and try to extend functions along this family. In more formal language to extend “along a family” means “onto the envelope of meromorphy”.

Let $U$ be a domain in $X$. Its envelope of meromorphy $(\widehat U,\pi)$ is the maximal domain over $X$ satisfying the following conditions:

\(i) there exists a holomorphic embedding $i\:U\to\widehat U$ with $\pi\circ i=\id_U$;

\(ii) each meromorphic function $f$ on $U$ extends to a meromorphic function $\widehat f$ on $\widehat U$, that is, $\widehat f\circ i=f$.

The envelope of meromorphy exists for each domain $U$. This can be proved, for example, by applying the Cartan–Thullen method to the sheaf of meromorphic functions on $X$, see \[Iv-1\].

In the sequel we shall restrict ourselves to Kähler complex surfaces, that is, we assume that $X$ carries a strictly positive closed $(1,1)$-form $\omega$.

The aim of the present chapter is to prove the following result.

Theorem 4.1. *Let $u\:S^2\to X$ be a symplectic immersion of the two-sphere $S^2$ in a disk-convex Kähler surface $X$ such that $M:=u(S)$ has only positive double points. Assume that $c_1(X)[M]>0$. Then the envelope of meromorphy $(\widehat U,\pi)$ of an arbitrary neighborhood $U$ of $M$ contains a rational curve $C$ with $\pi^*c_1(X)[C]>0$.*

The definition of disk-convexity is given in Lecture 10. At this point, we only observe that all compact manifolds are disk-convex. As usual, $c_1(X)$ is the first Chern class of $X$. In Appendix IV, Corollary A4.3.2 we explain why the condition $c_1(X)[M]>0$ is nessessary.

The exposition includes a construction of a complete family of holomorphic deformations of a non-compact complex curve in a complex manifold, parameterized by a finite codimension analytic subset of a Banach ball. This will be carried out in Lecture 9.

The existence of this family is used to prove a generalization of Levi’s continuity principle, which is applied to describe envelopes of meromorphy.

[*4.*]{} Another natural approach is due to S. Nemirovski. Suppose that for some neighborhood $V\supset M$ there is a nonconstant holomorphic function in $V$. Then from the result of Fujita, \[Fu\], it follows that there exists a Stein domain over $\cc\pp^2$ which contains $M$. One can imbed by the Stout theorem, \[St\], some neighborhood of $M$ into a compact algebraic surface $X$ and observe that $b_+^2(X)>1$. The Seiberg-Witten theory imposes an adjunction inequality on $M$: $$\vert c_1(X)[M]\vert  + [M]^2\le 0,$$ see \[K-M\]. But $c_1(X)[M]=c_1(\cc\pp^2)[M]=3$ and $[M]^2=1$, a contradiction.

This method seems to work only in some special algebraic surfaces like $\cc\pp^2$ or $\cc\pp^1\times \cc\pp^1$ but does not require that $M$ be symplectic! We shall give more details in Appendix V.

[Lecture 9]{}

[Deformation of Noncompact Curves.]{}

[9.1. Banach Analytic Sets.]{}

Now we shall study a process of local deformation of noncompact complex curves in complex manifolds. We shall see that the moduli have nice analytic structure. Let start from the following

Definition 9.1.1.

*By a [*Banach ball* ]{} we mean a ball in some complex linear Banach space. A subset $\calm$ of the Banach ball $B$ is said to be a Banach analytic set of the finite codimension, b.a.s.f.c., if there exists a holomorphic map $F:B\to \cc^N$ with $N<\infty$ such that $\calm =
\{ x\in B: F(x)=0\}$.*

The importance of this notion consists in the fact that contrary to the general Banach analytic sets b.a.s.f.c., their properties are similar to the finite dimensional analytic sets. Namely, the following is true.

Theorem 9.1.1. (\[Ra\]) *Let $B$ be a ball in a Banach space $\cal F$, $\calm \subset B$ — a b.a.s.f.c.  and $x_0$ a point in $\calm $. Then there exists a neighborhood $U\ni x_0$ in $B$ such that $\calm \cap U$ is a union of finite number of irreducible b.a.s.f.c.  $\calm_j$.*

Moreover, each $\calm_j$ can be represented as a finite ramified covering over a domain in some closed linear subspace ${\cal F}_j
\subset {\cal F}$ of finite codimension.

The aim of this paragraph is to prove an existence of a complete family of holomorphic deformations of a stable curve over a complex manifold $X$, which is parameterized by a b.a.s.f.c. Before stating the result, let us introduce the following definition.

Let $C$ be a nodal curve, $E$ a holomorphic vector bundle over $C$, and $C= \cup_{i=1}^l C_i$ a decomposition of $C$ into irreducible components. Suppose that $E$ extends sufficiently smoothly to the boundary $\d C$.

Definition 9.1.2. [*Let us define an [*$L^{1,p}$-section $v$*]{} of the bundle $E$ over $C$ as a couple $(v_i)_{i=1}^l$ of $v_i \in
L^{1,p}(C_i,E)$ such that at any nodal point $z \in C_i \cap C_j$ $v_i(z) = v_j(z)$. Let us define an [*$E$-valued $L^p$-integrable $(0,1)$-form $\xi$ on $C$*]{} as a couple $(\xi_i)_{i=1}^l$ of $(0,1)$-forms $\xi_i \in L^p(C_i,
E\otimes \Lambda^{(0,1)} )$.* ]{}

Let $L^{1,p}(C,E)$ denote a Banach space of $L^{1,p}$-sections of $E$ over $C$, and $L^p(C, E\otimes \Lambda ^{(0,1)} )$ a Banach space of $L^p$-integrable $(0,1)$-forms $C$. Denote by $\calh^{1,p}(C,E)$ a Banach space of [*holomorphic*]{} $L^{1,p}$-sections of $E$ over $C$.

Analogously, for a complex manifold $X$ by $L^{1,p}(C,X)$ we denote the set of couples $u=(u_i)_{i=1}^l$ such that $u_i \in L^{1,p}(C_i,
X)$, with $u_i(z) = u_j(z)$ at any nodal point $z \in C_i \cap C_j$. One can see that $L^{1,p}(C,X)$ is a Banach manifold with a tangent space $T_uL^{1,p}(C,X) = L^{1,p}(C,u^*TX)$. Further by $\calh^{1,p}(C,X)$ we shall denote the manifold of [*holomorphic*]{} $L^{1,p}$-maps from $C$ to $X$. Note that for $u\in \calh^{1,p}(C,X)$ one has $u(C) \subset u(\barr C)
\Subset X$, because $L^{1,p}\subset C^{0,1-{2\over p}}$ and therefore $u$ is continuous up to the boundary.

*9.2. Solution of a Cousin-type Problem.*

An important role plays the following result on the possibility to solve the following Cousin-type problem.

**Lemma 9.2.1. *Let $C$ be a nodal curve and $E$ a holomorhpic vector bundle over $C$, $C^1$-smooth up to the boundary. Let $\{ V_i\}_{i=1}^l$ be a finite covering of $C$ by Stein domains with piecewise smooth boundaries. Put $V_{ij} \deff V_i \cap V_j$ and suppose that all triple intersections $V_i \cap V_j \cap V_k$ á $i \not= j \not= k \not= i$ are empty.***

Then for $2 \le p< \infty$ the Čech-differential $$\matrix
\delta :& \sum_{i=1}^l \calh ^{1,p}(V_i, E)& \lrar&
\sum_{i<j} \calh ^{1,p}(V_{ij}, E)
\cr
\delta :& (v_i)_{i=1}^l        & \longmapsto & (v_i -v_j)
\endmatrix
\eqno(9.1.1)$$ possesses the following properties:

the image $\im(\delta)$ is closed and has finite codimension; moreover, $\coker(\delta) = \sfh^1(C, E) = \sfh^1(C_{\sf comp}, E)$, where $C_{\sf comp}$ denotes the union of compact irreducible components of $C$;

the kernel $\ker(\delta)$ is isomorphic to $\calh ^{1,p}(C, E)$ and admits a closed direct complement.

Proof. Before considering the Čech complex, let us look at the corresponding $\dbar$-problem. Consider the following operator $$\dbar : L^{1,p}(C,E) \lrar L^p(C, E\otimes \Lambda^{(0,1)}_C).
\eqno(9.1.2)$$ First we shall prove that this operator possesses the same properties as $(9.1.1)$, i.e., that $\ker(\dbar)$ admits a closed direct complement, $\im(\dbar)$ has finite codemension and is closed, and $\coker(\dbar) =
\sfh^1(C, E) = \sfh^1(C_{\sf comp}, E)$. Moreover, we shall construct a natural isomorphism between (co)kernels of $(9.1.1)$ and $(9.1.2)$.

Since the boundary $C$ is smooth, there exist nodal curves $C^+$ and $C^{++}$ such that $C \Subset C^+ \Subset C^{++}$ and differences $C^+ \bs \barr C$ (resp. $C^{++} \bs \barr C^+$) consist of an annulai $A^+_\alpha$ (resp. $A^{++}_\alpha$) adjacent to the corresponding components $\gamma_\alpha$ (resp. $\gamma^+ _\alpha$) of the boundary $\d C$ (resp. $\d C^+$, see. Fig. 17.). Now $E$ extends to a holomorphic vector bundle over $C^{++}$, again denoted as $E$.

Consider the following sheaves on $C^{++}$ $$\matrix \format \r\;\; &\c\; &\; \c\; &\; \l \\
L^{1,p}_\loc(\cdot ,E): &
V & \mapsto &  L^{1,p}_\loc(V,E) \vphantom{\int_3};
\cr
L^p_\loc(\cdot ,E\otimes \Lambda^{(0,1)}_{C^{++}}): &
V & \mapsto &  L^p_\loc(V,E\otimes \Lambda^{(0,1)}_{C^{++}}),
\endmatrix$$ and a sheaf homomorphism defined by the operator $$\dbar: L^{1,p}_\loc(V,E) \lrar L^p_\loc(V,E\otimes \Lambda^{(0,1)}_{C^{++}}).$$ Then sheaves $L^{1,p}_\loc(\cdot ,E)$ and $L^p_\loc(\cdot ,E\otimes \barr
\Omega_{C^{++}})$ together with a $\dbar$-homomorphism form a fine resolution of the (coherent) sheaf $\calo^E$ of holomorphic sections of $E$ over $C^{++}$. In smooth points of $C^{++}$ this fact follows from $L^p$-regularity of the (elliptic) operator $\dbar$, and in nodal points this can be seen from the following considerations.

Let $z\in C$ be a nodal point which lies on the intersection of irreducible components $C_i$ and $C_j$ of the curve $C$. Let $\xi_i$ (resp. $\xi_j$) be an $E$-valued $L^p_\loc$-integrable (0,1)-form, defined in the neighborhood of the point $z$ in $C_i$ (resp. $C_j$). And let $v_i$ (resp $v_j$) be $L^{1,p}_\loc$-solutions of the equation $\dbar v_i = \xi_i$ (resp. $\dbar v_j= \xi_j$). Adding a local [*holomorphic*]{} section of $E$ over $C_i$, we can achieve an equality $v_i(z)= v_j(z)$. Then the pair $(v_i, v_j)$ defines a section of the sheaf $L^{1,p}_\loc(\cdot ,E)$ in the neighborhood of $z$. This shows that the Dolbeault lemma is valid for the holomorphic bundle $E$ also in the neighborhood of the nodal points.

This gives the natural isomorphisms $$\matrix \format \r&\c & \c & \l &\c\\
\ker{\Bigm(}\dbar: L^{1,p}_\loc(C^{++},E) &\lrar &
L^p_\loc(C^{++},E\otimes \Lambda^{(0,1)}_{C^{++}} )
{\Bigm)} &=& \sfh^0(C^{++}, E),
\vphantom{\int_3}
\cr
%\noalign{\rlap{and}}
\coker{\Bigm(}\dbar: L^{1,p}_\loc(C^{++},E) & \lrar&
L^p_\loc(C^{++},E\otimes \Lambda^{(0,1)}_{C^{++}} )
{\Bigm)} &=& \sfh^1(C^{++}, E).
\endmatrix
\eqno(9.1.3)$$ Note that one also has the same isomorphisms for $C$ and $C^+$. Note also that there are natural isomorphisms $\sfh^1(C^{++}, E)= \sfh^1(C^+, E) =
\sfh^1(C, E)= \sfh^1(C_{\sf comp}, E)$, induced by the restrictions $L^p_\loc(C^{++},E\otimes \Lambda^{(0,1)}_{C^{++}}) \to L^p_\loc(C^+,E\otimes
\Lambda^{(0,1)}_{C^+} ) \to L^p_\loc(C,E\otimes \Lambda^{(0,1)}_{C} ) \to
L^p_\loc(C_{\sf comp},E\otimes \Lambda^{(0,1)}_C) $.

Now take some $\xi \in L^2(C^+,E \otimes \Lambda^{(0,1)}_{C^+})$, which defines a zero cohomology class in $\sfh^1(C^+, E)$. We can extend $\xi$ by zero to the element $\ti \xi\in L^2_\loc(C^{++}, E\otimes
\Lambda^{(0,1)}_{C^{++}})$. While $[\ti\xi]_\dbar= [\xi]_\dbar =0$, there exists $\ti v \in L^{1,2}_\loc(C^{++},E)$ such that $\dbar\ti v =
\ti\xi$. The restriction $v \deff \ti v\ogran_{C^+}$ satisfies $\dbar v = \xi$ and $v \in L^{1,2}(C^+,E)$. This shows that the image of a continuous (!) operator $$\dbar: L^{1,2}(C^+,E) \lrar L^2(C^+,E\otimes
\Lambda^{(0,1)}_{C^+}) \eqno(9.1.4)$$ is of finite codimension. From here and from the Banach open mapping theorem it follows that this image is closed. Furthermore, because $L^{1,2}(C^+,E)$ is a Hilbert space, the kernel of the operator $(9.1.4)$ admits a direct complement $Q \subset L^{1,2}(C^+,E)$. Moreover, operator $(9.1.4)$ maps $Q$ isomorphically onto its image. Thus, operator $(9.1.4)$ splits, i.e., there exists a continuous operator $$\def\dash{\raise.2pt\hbox{-}} T^+:L^2(C^+,E\otimes
\Lambda^{(0,1)}_{C^+}) \lrar L^{1,2}(C^+,E)  \eqno(9.1.5)$$ such that $\im(T^+)=Q$ and for any $\xi \in L^2(C^+,E\otimes
\Lambda^{(0,1)}_{C^+})$ with $[\xi]_\dbar=0 \in \sfh^1(C^+, E)$ one has $\dbar(T^+\xi)=\xi$.

Define an operator $T:L^2(C, E\otimes \Lambda^{(0,1)}_C) \lrar
L^{1,2}(C,E)$ in the following way. For $\xi \in
L^2(C, E\otimes \Lambda^{(0,1)}_C)$ extend $\xi$ by zero to $\ti \xi \in
L^2(C^+,E\otimes \Lambda^{(0,1)}_{C^+})$ and put $T(\xi) =
T^+(\ti\xi)\ogran_{C}$. $T$ is obviously continuous; moreover $$\norm{T^+(\ti\xi)}_{L^{1,2}(C^+)} \le c\cdot \norm{\xi}_{L^2(C)}$$ with constant $c$ independent of $\xi$. By the $L^p$-regularity for the elliptic $\dbar$-operator (see ex., \[Mo\]) for $2\le p<\infty$ and

$v \in L^{1,p}_\loc(C^+,E)$ there is an interior estimate $$\norm{v}_{L^{1,p}(C)} \le c'\cdot
\bigl( \norm{v}_{L^{1,2}(C^+)} + \norm{\dbar v}_{L^p(C^+)} \bigr)
\eqno(9.1.6)$$ with the constant $c'$ independent on $v$. From this it follows that for $2\le p<\infty$ and $\xi \in L^p(C, E\otimes \Lambda^{(0,1)}_C)$ with $[\xi]_\dbar=0 \in \sfh^1(C, E)$ one has $$\norm{T(\xi)}_{L^{1,p}(C)} \le c''\cdot \norm{\xi}_{L^p(C)}$$ with a constant $c''$ independent on $\xi$. This means that the operator $T$ is a splitting off the operator $(9.1.2)$. This means that operator $(9.1.2)$ possesses the properties and of [*Lemma 9.2.1*]{}.

Let us return to the Čech operator $(9.1.1)$. Fix some partition of unity $\1=\sum_{i=1}^l\phi_i$, subordinate to the covering $\{V_i\}_{i=1}^l$ of the curve $C$. Take a cocycle $w=(w_{ij}) \in \sum_{i<j}\calh ^{1,p}(V_{ij}, E)$. For $i>j$ put $w_{ij} \deff -w_{ji}$. Define $f_i \deff \sum_j \phi_j w_{ij}$. Then $f_i \in L^{1,p}(V_i, E)$ and $f_i- f_j = w_{ij}$. Consequently, $\dbar f_i \in L^p(V_i, E\otimes
\Lambda^{(0,1)}_C)$ and $\dbar f_i = \dbar f_j$ ¢ $V_{ij}$. So, $\dbar f_i = \xi\ogran_{V_i}$ for the correctly defined $\xi \in L^p(C,
E\otimes \Lambda^{(0,1)}_C)$. Moreover, $(w_{ij})$ and $\xi$ define the same cohomology class $[w_{ij}]=[\xi]$ in $\sfh^1(C,E)$.

Suppose additionally that the induced cohomology class $[w_{ij}]$ is trivial. Put $f\deff T(\xi)$ and $v_i= f_i-f$. Then $v_i \in
L^{1,p}(V_i, E)$, $v_i- v_j = w_{ij}$, and $\dbar v_i= \dbar f_i -\dbar f=0$. Thus, $v\deff(v_i) \in \sum_{i=1}^l \calh ^{1,p}(V_i, E)$ and $\delta(v)= w$. From here it follows that the formula $T_\delta : w \mapsto v$ defines an operator $T_\delta$, which is a splitting of $\delta$. The explicit construction shows that $T_\delta$ is continuous. This completes the proof of [*Lemma 9.2.1*]{}.

*9.3. Case of a Stein Curve.*

Lemma 9.3.1. *Let $C$ be a Stein nodal curve with a piecewise smooth boundary, and $X$ a complex manifold. Then*

$\calh^{1,p}(C,X)$ possesses a natural structure of a complex Banach manifold with a tangent space $T_u\calh^{1,p}(C,X) =
\calh^{1,p}(C,u^*TX)$ at $u\in \calh^{1,p}(C,X)$.

If $C'\subset C$ is again a nodal curve, then the restriction map $\calh^{1,p}(C, X) \to \calh^{1,p}(C', X)$ is holomorphic, and its differential at $u\in \calh^{1,p}(C,X)$ is again the restriction map $\calh^{1,p}(C, u^*TX) \to \calh^{1,p}(C', u^*TX)$, $v \mapsto v|_{C'}$.

The **proof will be given in several steps.**

[*Step 1.*]{} Suppose first that the image $u(C)$ lies in the complex chart $U \subset X$ with complex coordinates $w= (w_1, \ldots, w_n) : U \buildrel \cong
\over \lrar U' \subset \cc^n$. Then the set $\calh^{1,p}(C,U)$ can be naturally identified with the set $\calh^{1,p}(C,U')$, which is an open set in the Banach space $\calh^{1,p}(C,\cc^n)$. This defines on $\calh^{1,p}(C,U)$ the structure of a complex Banach manifold with the tangent space $T_u\calh^{1,p}(C,U) \cong \calh^{1,p}(C,\cc^n) \cong \calh^{1,p}(C,u^*TU)$ at the point $u\in \calh^{1,p}(C,U)$.

Note that if $u_t$, $t\in [0,1]$, is a $C^1$-curve in $\calh^{1,p}(C,U)$, then the tangent vector $v\in \calh^{1,p}(C,u^*TU)$ to $u_t$ at $u_0$ is given by $v(z)= {\d u \over\d t}(z) \in T_{u(z)}U$. This last expression does not depend on the choice of the complex coordinates $w= (w_1,\ldots, w_n) : U \to \cc^n$ in $U$. This implies two things.

First, [*the complex structure on $\calh^{1,p}(C, U)$ does not depend on the choice of the complex coordinates $w= (w_1, \ldots, w_n) : U \to \cc^n$ in $U$*]{}. Second, for $C$ property of the lemma is fulfilled.

Thus, if $u(C)$ is contained in a coordinate chart, then [*Lemma 9.3.1*]{} is proved.

[*Step 2.*]{}  Now fix $u_0 \in \calh^{1,p}(C,X)$ and suppose that a finite covering $\{V_i\}_{i=1}^l$ of the curve $C$ is chosen such that first, the conditions of [*Lemma 9.2.1*]{} are satisfied, and second, for every $V_i$ [*Lemma 9.3.1*]{} holds, e.g., every $u_0(V_i)$ is contained in some coordinate chart $U_i \subset X$.

Set $V_{ij}\deff V_i\cap V_j$. Choose balls $B_{ij} \subset\calh^{1,p}
(V_{ij}, u_0^*TX) \cong T_{u_0}\calh^{1,p}(V_{ij},X)$ such that there exists a biholomorhpism $\psi_{ij} : B_{ij} \buildrel \cong \over \lrar
B'_{ij}
\subset \calh^{1,p}(V_{ij},X)$ with $\psi_{ij}(0) = u_0|_{V_{ij}}$ and $d\psi_{ij}
(0) = \id: \calh^{1,p}(V_{ij}, u_0^*TX) \to\calh^{1,p}(V_{ij}, u_0^*TX)$. Then take an open sets $B_i \subset \calh^{1,p}(V_i, X)$ such that $u_0|_{V_i}
\in B_i$ and for every $u_i \in B_i$ one has $u_i|_{V_{ij}} \in B'_{ij}$.

Now, the holomorphic mappings $\phi_{ij} :B_i \to B_{ij} 
\subset \calh^{1,p}(V_{ij}, \allowbreak u_0^*TX)$, $\phi_{ij} : u_i
\mapsto \psi_{ij}\inv (u_i|_{V_{ij}})$ are uniquely defined. They define a holomorphic map $$\matrix
\Phi: \prod_{i=1}^l B_i & \lrar   & \sum_{i<j} \calh^{1,p}(V_{ij},u_0^*TX)
\cr
(u_i)_{i=1}^l    & \mapsto & \phi_{ij}(u_i) - \phi_{ji}(u_j).
\endmatrix$$ One can easily see that the map $\Phi$ defines the condition of compatibility of local holomorphic mappings $u_i : V_i \to X$, namely $(u_i)_{i=1}^l \in \prod_{i=1}^lB_i$ define a holomorphic map $u: C \to X$ if and only if $\Phi(u_i)=0$. Moreover, the differential $d\Phi$ at point $(u_0|_{V_i})$ coincides with the co-boundary Ĉech operator $(9.1.1)$. Since $C$ is Stein, $\sfh^1(C, u_0^*TX) =0$. From [*Lemma 9.2.1*]{} and the implicit function theorem, we see that the statements and of [*Lemma 9.3.1*]{} are fulfilled in some neighborhood of the map $u_0\in \calh^{1,p} (C, X)$.

[*Step 3.*]{}  One can show that for any Stein nodal curve $C$ and any $u \in \calh^{1,p} (C, X)$ the statements and of [*Lemma 9.3.1*]{} are fulfilled in some neighborhood of the mapping $u$, applying [*Step 2*]{} sufficiently many times. If, for example, $C$ is an annulus $A_{r,R}$, then one can cover it by the narrow annulai $A_{r_i, R_i}$, $0< R_i - r_i <\!\!< 1$, and every $A_{r_i, R_i}$ with small sectors $V_{ij} = \{ z= \rho
e^{\isl \theta} \in \cc \,:\, r_i<\rho < R_i, \alpha_j < \theta < \beta_j\}$ á $0< \beta_j -\alpha_j <\!\!< 1$. Details are left to the reader.

*9.4. Degeneration to a Node.*

One of the diffuculties in constructing [*holomorphic*]{} families of stable curves with boundaries comes from the fact that the moduli space of complex structures on the noncompact surface with boundary $\Sigma$ does not have a natural complex structure; moreover it can have an odd real dimension. If, for example, $\Sigma$ is an annulus, then it is biholomorhic to the standard annulus $A_{r,1}$ with uniquely defined $r\in (0,1)$; thus, the interval $(0,1)$ is a moduli space in this case. In the general case, if $\Sigma$ has genus $g$ and $k$ boundary components, then the real dimension of the moduli space is equal to $d= 6g -6 + 3k$, except in four cases, where $\Sigma$ is either a sphere ($g=0$, $k=0$), a torus ($g=1$, $k=0$), a disk ($g=0$, $k=1$) or an annulus ($g=0$, $k=2$), see e.g., \[Ab\]. Note that exactly in this cases the group of biholomorphic automorphisms of corresponding complex curves $(\Sigma, J)$ has positive dimension.

One can correct the situation by introducing additionally $k$ parameters, namely, fixing $k$ marked points, one on each boundary component. Let $A$ be an annulus with boundary circles $\gamma_0$ and $\gamma_1$ and $X$ be a complex manifold.

Theorem 9.4.1. *There exist complex Banach manifolds $\calm(A, X)$ and $\calc(A, X)$, a holomorphic projection $\pi_\calc: \calc(A, X)\to \calm(A,X)$ and holomorphic mappings $\ev: \calc(A, X) \to X$, $z_1 :\calc(A, X) \to \Delta$, $z_2 :\calc(A, X)
\to \Delta$ and $\lambda_\calm :\calm(A, X) \to \Delta$ with the following properties:*

for any $y\in \calm(A,X)$ the fiber of the projection $C_y\deff \pi_\calc\inv(y)$ is a nodal curve, parameterized by an annulus $A$; moreover, the mapping $(z_1, z_2) : C_y \to \Delta^2$ defines a biholomorphism onto the curve $\{ (z_1, z_2)
\in \Delta^2 \;:\; z_1 \cdot z_2 = \lambda_\calm(y) \}$; in particular, $C_y$ is either a standard node if $\lambda_\calm(y)=0$, or a holomorphic annulus $\{ |\lambda_\calm(y)| <|z_1| <1 \}$;

the diagram $$\def\normalbaselines{\baselineskip18pt\lineskip3pt \lineskiplimit3pt }
\def\mapright#1{\!\!\smash{\mathop{\hbox to50pt{\cleaders\lowminus\hfill}
\mkern-13mu\rightarrow}\limits^{#1}}\!\!}
\def\mapdown#1{\Big\downarrow\rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
%
\matrix
\calc(A, X) & \mapright{ (\ev, z_1, z_2) }& X \times \Delta^2
\cr
\mapdown{\pi_\calc} & &
\hphantom{\hbox{$X\times{.}$}} \mapdown{\lambda= z_1 \cdot z_2}
\cr
\calm(A,X)& \mapright{ \lambda_\calm } &
\hphantom{\hbox{$X\times{.}$}} \Delta
\endmatrix
\eqno(9.4.1)$$ is commutative; moreover, for every $y\in \calm(A,X)$ the restriction $\ev|_{C_y}$ is in

$\calh^{1,p} (A_a, X)$ with $a=\lambda_\calm(y)$, and mappings $\ev_1: y\in \calm(A,X) \mapsto \ev|_{C_y}(z_1^{-1}(1))$ and $\ev_2: y\in \calm(A,X) \mapsto \ev|_{C_y}(z_2^{-1}(1))$ are holomorphic;

let $C$ be an annulus or a nodal curve with smooth boundary $\d C =\gamma_1\coprod \gamma_2$, $p_i \in \gamma_i$ be marked points, and $u:C \to X$ be a holomorphic $L^{1,p}$-smooth mapping, then there exist [*a unique*]{} $y\in \calm(A,X)$ and a [*unique*]{} biholomorphism $\phi: C \to C_y$ such that $\ev \scirc \phi = u : C\to X$ ¨ $z_i \scirc
\phi (p_i) =1\in \barr \Delta$; in other words, $\calm(A,X)$ parameterizes holomorphic mappings to $X$ from annulai and nodes with marked points on the boundary;

if the commutative diagram with complex spaces $\calw$ and $\calz$ and holomorphic mappings $$\def\normalbaselines{\baselineskip18pt\lineskip3pt \lineskiplimit3pt }
\def\mapright#1{\!\!\smash{\mathop{\hbox to50pt{\cleaders\lowminus\hfill}
\mkern-13mu\rightarrow}\limits^{#1}}\!\!}
\def\mapdown#1{\Big\downarrow\rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
%
\matrix
\calz & \mapright{ (\ev^\calz, \ti z_1,  \ti z_2) }& X \times \Delta^2
\cr
\mapdown{\pi_\calz} & &
\hphantom{\hbox{$X\times{.}$}} \mapdown{\lambda= \ti z_1  \cdot \ti z_2}
\cr
\calw & \mapright{ \lambda_\calw } &
\hphantom{\hbox{$X\times{.}$}} \Delta
\endmatrix
\eqno(9.4.2)$$ possesses the properties and , in particular, the fibers $\calz_w \deff\pi_\calz\inv(w)$ should be the nodal curves with induced maps $f_w\deff\ev^\calz|_{\calz_w} \in \calh^{1,p}(\calz_w, X)$, then diagrams $(9.4.1)$ and $(9.4.2)$ uniquely fit together to form the commutative diagram $$\def\normalbaselines{\baselineskip18pt\lineskip3pt \lineskiplimit3pt }
\def\mapright#1{\!\!\smash{\mathop{\hbox to50pt{\cleaders\lowminus\hfill}
\mkern-13mu\rightarrow}\limits^{#1}}\!\!}
\def\mapdown#1{\Big\downarrow\rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
%
\matrix
\calz & \buildrel \wt F \over \lrar &
\calc(A, X) & \mapright{ (\ev, z_1, z_2) }& X \times \Delta^2
\cr
\mapdown{\pi_\calz} & &
\mapdown{\pi_\calc} & &
\hphantom{\hbox{$X\times{.}$}} \mapdown{\lambda= z_1 \cdot z_2}
\cr
\calw & \buildrel  F \over \lrar &
\calm(A,X)& \mapright{ \lambda_\calm } &
\hphantom{\hbox{$X\times{.}$}} \Delta
\endmatrix
\eqno(9.4.3)$$ here $\lambda_\calm \scirc F = \lambda_\calw$ and $(\ev, z_1, z_2) \scirc
\wt F= (\ev^\calz, \ti z_1, \ti z_2)$;

differential $d\lambda_\calm: T_y\calm(A,X) \to T_{\lambda_\calm(y)}
\Delta \cong \cc$ is not degenerate in any point $y \in \calm(A,X)$, and for any $a \in \Delta$ the fiber $\lambda_\calm\inv(a)$ is naturally isomorphic to the manifold $\calh^{1,p} (A_a, X)$, where $A_a$ is a curve $\{ (z_1, z_2)\in
\Delta^2 : z_1 \cdot z_2 = a \}$; in particular, for any $y \in \calm(A,X)$ there is a biholomorphism $C_y \cong A_{\lambda_\calm(y)}$ and a natural exact sequence $$0 \lrar
\calh^{1,p} (C_y, u^*TX) \buildrel \iota_y \over\lrar T_y \calm(A,X)
\buildrel d\lambda_\calm(y) \over\lrar \cc \lrar 0.$$

Proof. Let $(A, p_1, p_2)$ be a smooth annulus with marked points, one on each boundary component $\gamma_i \cong S^1$, and let $J$ be a complex structure on $A$. It is known that $(A, J)$ is biholomorphic to some annulus $A_{r,1}= \{ r < |z| <1 \}$. It is easy to see that there is only one isomorphism $\psi: (A, J) \to A_{r,1}$ which smoothly extends to the diffeomorphism $\psi: \barr A \to \barr A_{r,1}$ with $\phi(p_1) =1$. Put $a \deff \phi(p_2)$. Now it is obvious that there is a unique biholomorphism $\phi: (A, J) \to A_a \deff \{ (z_1, z_2)\in
\Delta^2 \;:\; z_1 \cdot z_2 = a \}$ such that $\phi(p_1) =1$ ¨ $\phi(p_2)
=a$.

Thus, the mapping $\lambda: \Delta^2 \to \Delta$, $\lambda(z_1, z_2)
= z_1 \cdot z_2$ with a fiber $A_a$ over $a\in \Delta$ represents a holomorphic moduli space of annulai with marked points on the boundaries, completed at $a=0$ by a standard node. If $a\not =0$ the coordinates $z_i$, $i=1, 2$, realize an imbedding of each $A_a$ as an annulus into $\cc$ in such a way that the circle $\gamma_i$ becomes an outer unit circle. When $a \lrar 0$, the annulus $A_a$ degenerates into a standard node, and each $z_i$ becomes a standard coordinate on the corresponding component of the node.

Remark. In the sequel we denote by $A_a$ an annulus (or corr. a node) [*with fixed points $a$ and $1$ on the boundary $\d A_a$*]{}, and also together with uniquely determined coordinates $z_1$ and $z_2$.

Fix $r$ with $0 <r< 1$. For $|a|<r$ define a mappings $\zeta^a_1,
\zeta^a_2: A_{r,1} \to A_a$ as $\zeta^a_1(z) \deff z$ and $\zeta^a_2(z)
\deff a/z$, so that $\zeta^a_i$ are the inverse of the coordinates $z_i$. Consider a mapping $$\matrix
\Psi_r: \coprod_{|a|<r} \calh^{1,p}(A_a, X) & \lrar &
\calh^{1,p}(A_{r,1}, X) \times \calh^{1,p}(A_{r,1}, X) \times \Delta(r)
\cr
u \in \calh^{1,p}(A_a, X) & \mapsto & (u \scirc \zeta^a_1, u \scirc
\zeta^a_2, a).
\endmatrix$$ One easily sees that $\Psi_r$ is holomorphic on each $\calh^{1,p}(A_a, X)$ and the image of $\Psi_r$ consists of such triples $(u_1,u_2,a)$ that each $u_i \in \calh^{1,p}(A_{r,1}, X)$ extends to $u_i \in \calh^{1,p}
(A_{|a|,1}, X)$ and $u_2(z) = u_1(a/z)$. Thus $\Psi_r$ is injective and has closed image. On the disjoint union $\calm(A, X) \deff \coprod_{a\in
\Delta}\calh^{1,p} (A_a, X)$ define the topology induced by the mappings $\Psi_r$. Clearly it agrees with the topologies on each slice $\calh^{1,p}(A_a, X)$.

Our task now is to construct an appropriate holomorphic structure on $\calm(A, X)$, which agrees with the above introduced holomorhpic structures on each slice

$\calh^{1,p}(A_a, X)$, and also with the topology on $\calm(A, X)$, defined above.

*Case 1. Let us start with the special case $X= \cc^n$. For $a\not =0$ every function $f\in \calh^{1,p} (A_a, \cc^n)$ in a unique way can be decomposed into a Laurent series $f(z_1) =\sum_{i= -\infty} ^\infty c_i
z_1^i$. Set $f^+(z_1) \deff \sum_{i=0}^\infty c_i z_1^i$ and $f^-(z_1)
\deff \sum_{i=-\infty} ^0 c_i z_1^i$. It is convenient to consider the function $f^-$ as a function of the variable $z_2 = a/z_1$; then $f^-(z_2)= \sum_{i=0}^\infty c_{-i} (z_2/a)^i$. One has $f^+ \in \calh^{1,p}(\{ |z_1| <1\}, \cc^n)$, $f^-\in \calh^{1,p}(\{ |z_2| <
1\}, \cc^n)$, $f^+(0) = f^-(z_2 =0) = c_0$, and $f(z_1) = f^+(z_1) +
f^-(a/z_1) - c_0$, so that a pair $(f^+, f^-)$ defines the holomorhpic function $\hat f \in \calh^{1,p}(A_0, \cc^n)$. The canonical isomorphisms $\calh^{1,p}(A_a, \cc^n)\cong \calh^{1,p}(A_0, \cc^n)$ obtained in this was define on $\coprod_{|a|<1} \calh^{1,p} (A_a, \cc^n)$ the structure of the trivial Banach bundle with base $\{ |a| <1\}$ and fiber $\calh^{1,p}(A_0, \cc^n)$, and thus a structure of a Banach manifold.*

One can see that the mapping $\Psi_r : \coprod_{|a|<r} \calh^{1,p}(A_a,
\cc^n) \to
\calh^{1,p}(A_{r,1}, \cc^n)^2 \times \Delta(r)$ is holomorphic.

*Case 2. Let $X = U \subset \cc^n$ be an open subset in $\cc^n$. Then $\calm(A, U)$ is also open in $\calm(A, \cc^n)$ and consequently inherits a holomorphic structure. The natural projection $\lambda_\calm : \calm(A, U) \to \Delta$ becomes holomorphic. One easily sees that the differential $d\lambda_\calm$ is not degenerate. Thus, we can define [*a universal family of curves*]{} $\calc(A, U)$ as a fiber product $\calm(A, U) {\times}_\Delta
\Delta^2$ with respect to the mappings $\lambda_\calm : \calm(A, U) \to
\Delta$ and $\lambda: \Delta^2 \to \Delta$, which is a complex Banach manifold due to the nondegeneracy of $d\lambda_\calm$.*

Denote by $\pi_\calc : \calc(A, U) \to \calm(A, U)$ the natural projection. Then the fiber $C_y$ over $y \in \calm(A, U)$ is biholomorphic to $A_a$ with $a=\lambda_\calm(y)$. The natural projection from $\calc(A, U)$ onto $\Delta^2$ induces holomorphic functions $z_1$ and $z_2$ on $\calc(A, U)$ taking values in $\Delta$ such that the property of [*Theorem 9.4.1*]{} is satisfied.

Now let $a\not=0 \in \Delta$ and $f\in \calh^{1,p} (A_a, \cc^n)$. Represent $f$ in the form $f(z_1) = f^+(z_1) + f^-(a/z_1) - f_0$, where $f^\pm \in \calh^{1,p} (\Delta, \cc^n)$ with $f^+(0) = f^-(0) = f_0$. Analogously, for $f\in \calh^{1,p} (A_0, \cc^n)$ we have $f= (f^+, f^-)$, where again $f^\pm \in \calh^{1,p} (\Delta, \cc^n)$ á $f^+(0) = f^-(0) = f_0$. Consider a holomorphic function $\ti f(z_1, z_2) \deff f^+(z_1) + f^-(z_2)-
f_0$, $\ti f\in \calh^{1,p} (\Delta^2, \cc^n)$. Define mappings $\ti\ev_a:\calh^{1,p} (A_a, \cc^n) \times \Delta^2 \lrar \cc^n$ by the formula $\ti\ev (f, z_1, z_2) \deff \ti f(z_1, z_2)$. One easily sees that $\ti\ev_a$ define a holomorphic map $\ti\ev :\calm(A, \cc^n) \times \Delta^2
\to \cc^n$. Let $\ev$ denote the restriction of this map onto $\calc(A, \cc^n) \subset \calm(A, \cc^n) \times \Delta^2$. It is straightforward to check that $\calm(A, \cc^n)$, $\calc(A, \cc^n)$, $\ev: \calc(A, \cc^n) \to \cc^n$ and $z_{1,2} : \calc(A, \cc^n) \to \Delta$ satisfy the statements of [*Theorem 9.4.1*]{}.

Thus, for $U \subset \cc^n$ we can define $\ev: \calc(A, U) \to U$ as a restriction $\ev: \calc(A, \cc^n) \to \cc^n$. Statements of [*Theorem 9.4.1*]{} again remain true. In particular, if $G:  U \to U'\subset \cc^n$ is biholomorphic, then the natural bijections $\calm(A,U) \buildrel \cong \over\lrar \calm(A, U')$ and $
\calc(A, U) \buildrel\cong \over \lrar \calc(A, U')$ are also biholomorphisms. This means that the complex structure in $\calm(A, U)$ does not depend on the imbedding $U \subset \cc^n$.

Let $C= A_{r,1}$ be an annulus and $u: C \to X$ be a holomorphic imbedding. Then $du : TC \to u^*TX$ is an imbedding of holomorphic bundles over $C$, and this defines a holomorphic normal bundle as a factor-bundle $N_C \deff u^*TX/ TC$. Since $C$ is Stein, the bundle $N_C$ is holomorphically trivial. Fix a holomorphic frame $\sigma_1, \ldots, \sigma_{n-1} \in
\calh^{1,p} (C, N_C)$, $n= \dimc X$, and also its lift $\ti\sigma_1, \ldots,
\ti \sigma_{n-1} \in \calh^{1,p} (C, u^*TX)$. Denote by $B^{n-1}(r)$ the ball of radius $r$ in $\cc^{n-1}$ with coordinates $w= (w_1, \ldots, w_{n-1})$. As follows from [*Lemma 9.3.1*]{}, there is a holomorphic map $\Psi : C\times B^{n-1}(r) \to X$ such that $\d \Psi / \d w_i|_{z\in C,w=0}
=\sigma_i(z)$. Thus $\Psi$ is biholomorphic in the neighborhood of $C\equiv C\times \{0\}$. In particular, for $r$ sufficiently small an image $U\deff \Psi(C \times B^{n-1}(r))$ is a coordinate chart with coordinates $(z, w_1, \ldots, w_{n-1})$.

*Step 3. Consider a general case where $C\cong A_a$ is arbitrary, and $u: C \to X$ is a holomorphic map.*

If $a=0$, then $C$ is a node; thus, there exists a neigborhood $V_0$ of the nodal point such that its image $u(\barr V_0)$ lies in some coordinate chart of $X$. If $a\not=0$ and the image $u(\barr C)$ is not contained in any chart of $X$, then $u$ is not constant. Thus, for some $|a|< r <1$ mapping $u$ will be an imbedding in some neighborhood of the circle $S^1_r \deff \{|z_1| =r\} \subset C \cong \{|a|< |z_1|<1\}$.

In all cases there is a covering $\{V_0, V_1, V_2\}$ of $C=\{(z_1,
z_2) \in \Delta^2 : z_1 \cdot z_2 =a\}$ of the form $V_1 =\{ (z_1,z_2)\in C
: r_1 <|z_1| <1 \}$, $V_2 =\{ (z_1,z_2)\in C : r_2 <|z_2| <1 \}$, $V_0 =\{ (z_1, z_2) \in C : |z_1| < R_1, |z_2|< R_2 \}$, where $0<r_1 <R_1
<1$, $0< r_2 <R_2 <1$, $r_1 \cdot R_2 > |a| <r_2 \cdot R_1$, and $V_0$ is such that the image $u(\barr V_0)$ lies in some chart $U$ of $X$.

Fix coordinate $z_1$ on $V_1$ and coordinate $z_2$ on $V_2$. On $V_0$ define new coordinates $\ti z_1 \deff z_1 /R_1$ and $\ti z_2 \deff z_2 /R_2$, and also the marked points $\ti p_1 \deff R_1$ and $\ti p_2 \deff a/R_2$. Set $\ti a \deff a/(R_1R_2)$. Then $V_0 \cong A_{\ti a}= \{(\ti z_1, \ti z_2)
\in \Delta^2 : \ti z_1\cdot \ti z_2 = \ti a\}$ with $\ti z_i(\ti p_1)=1$. It is easy to see that the change of the complex parameter $a= z_1 z_2$ on $C$, which parameterizes the holomorphic structures of annulai with marked points on the boundary, can be reduced to the change of the analogous parameter in $V_0$. Namely, let $C' \cong A_{a'} = \{ (z_1,z_2) \in \Delta^2:
z_1 \cdot z_2= a'\} $ be obtained as a result of a small deformation of $a$. Put $\ti a{}'
\deff a'/(R_1R_2)$. Look on $C'$ as the result of patching of complex curves $V_0'$, $V_1$ ¨ $V_2$, where $V_0'$, $V_1$ and $V_2$ are defined in $\Delta^2$ in the same way $V_0'= \{ (z_1,z_2) \in C': |z_1| <r_1, |z_2| <r_2\}$, $V_i= \{r_i <|z_i|<1\}$ when $i=1,2$, but now the coordinates $(z_1,z_2)$ satisfy a new relation $z_1\cdot z_2 = a'$, and $V_0'\cong A_{\ti a{}'}$.

Thus, for a sufficiently small neighborhood $\calw \subset \calm(A, X)$ of the curve $(C,u)$ over $X$ a holomorphic map is defined as $$\matrix \format \c\;&\c\;&\c\;& \c& \c& \c& \c& \c \\
\Theta :& \calw &\lrar& \calm(A,U) &\llap{$\times$}&
\calh^{1,p} (V_1, X) &\times&
\calh^{1,p} (V_2, X);
\cr
\noalign{\vskip2pt}
\Theta :&(C',u') &\mapsto& ((A_{\ti a{}'}, u|_{V_0'}), &&
u|_{V_1},&& u|_{V_2}).
\endmatrix$$ On the other hand, a couple of maps $u_0 : V_0' \to X$, $u_1 : V_1 \to X$ and $u_2 : V_2 \to X$ define a map $u': C' \to X$, exactly where $u_1$ coincides with $u_0$ on $W_1 \deff V_1 \cap V_0'$, and $u_2$ coincides with $u_0$ on $W_2 \deff V_2 \cap V_0'$. Note that domains $W_i \subset V_i$ do not change in the process of deformation of the complex structure on $C$. A construction of gluing from [*Step 2*]{} of the proof of [*Lemma 9.3.1*]{} finishes the proof of the theorem.

*9.5. Banach Analytic Structure on the Stable Neighborhood.*

We are prepared now to prove the main result of this lecture. Theorem 9.5.1.

*Let $(X,J)$ be a complex manifold, and $(C_0,u_0)$ a stable complex curve over $X$, parameterized by a real surface $\Sigma$. There exists a b.a.s.f.c. $\calm $ and $\calc$ and holomorphic maps $F: \calc \lrar X$ and $\pi: \calc \lrar \calm$ such that*

a\) for any $\lambda \in \calm $ a fiber $C_\lambda =\pi^{-1}(\lambda )$ is a nodal curve parameterized by $\Sigma$, and $C_{\lambda_0} \cong C_0$ for some $\lambda_0$;

b\) for $F_\lambda \deff F|_{C_\lambda}$ a pair $(C_\lambda ,F_\lambda )$ is a stable curve over $X$; moreover $F_{\lambda_0}=u_0$;

c\) if $(C',u')$ is a stable curve over $X$, sufficiently close to $(C_0,u_0)$ in the Gromov topology, then there exists $\lambda' \in \calm$ such that $(C',u')=(C_{\lambda '},F_{\lambda'} )$;

d\) for an appropriate $N{\in}\nn$ and for a small ball $B$ in Banach space

$\calh^{1,p}(C_0, u_0^*TX) \allowbreak \oplus \cc^N$ b.a.s.f.c.  $\calm$ is realized as a zero set of a holomorphic map $\Phi$ from $B$ to the finite dimensional space $\sfh^1(C,u_0^*TX)$.

**Proof. The proof is based on the construction of local deformations of stable curves and on the analysis of the gluing of local models.**

Let $(C_0, u_0)$ be a stable curve over a complex manifold $X$ with parameterization $\sigma_0: \Sigma
\to C_0$. Using [*Proposition 3.2*]{}, fix a covering $\{ V_\alpha,
V_{\alpha\beta}\}$ of the surface $\Sigma$, having properties [*i)*]{}–[*v$\!$i)*]{} of the above-mentioned proposition. In particular, there exist biholomorphisms $\phi^0 _{\alpha \beta}: \sigma_0 (V_{\alpha \beta}) \lrar
A_{\lambda^0 _{\alpha \beta}}$. From [*i)*]{}–it follows that for every couple $\bflamb \deff (\lambda _{\alpha \beta})$, sufficiently close to $\bflamb^0 \deff (\lambda^0 _{\alpha \beta})$, there is a nodal curve $C$ and parameterization $\sigma: \Sigma \to C$ such that properties [*i*]{}–[*v$\!$i*]{} remain true and there exist biholomorphisms $\phi _{\alpha \beta}: \sigma (V_{\alpha \beta}) \lrar A_{\lambda _{\alpha
\beta}}$. In particular, the complex structures on each $\sigma (W_{\alpha
\beta})$ do not change. Moreover, in $V_{ \alpha\beta}$ there exist holomorphic coordinates $z_1$ and $z_2$ such that $z_1 |_{W_{\alpha,\beta}}$ and $z_2|_{W_{\beta, \alpha}}$ do not change by changing $\lambda _{\alpha \beta}$, and $z_1 \cdot z_2 \equiv
\lambda _{\alpha \beta}$. This means that a disk $\Delta_{\alpha
\beta}\deff \{ \lambda _{\alpha \beta} \,:\, |\lambda^0 _{\alpha \beta}-
\lambda _{\alpha \beta} | \le \epsi \,\} $ parameterizes [*a holomorphic*]{} family of curves of the type $\sigma(V_{\alpha\beta })$.

One can illustrate the changing of the complex structure in $V_{\alpha\beta}$ by the following picture.

Let $N \in \nn$ be the number of the elements of our covering of the type $V_{\alpha \beta}=V_{\beta \alpha}$. Then for sufficiently small $\epsi>0$ the polydisk $$\Delta^N_\lambda \deff \{ \bflamb \deff (\lambda _{\alpha \beta}) \;:\;
|\lambda^0 _{\alpha \beta}- \lambda _{\alpha \beta} | \le \epsi \,\}$$ parameterizes a holomorphic family of nodal curves $\{C_\bflamb\}_{\bflamb
\in \Delta^N _\lambda }$. Every curve $C_\bflamb = \sigma(\Sigma)$ is obtained by gluing the pieces $\sigma(V_\alpha)$ and $\sigma(V_{\alpha \beta})$. Denote $\calm \deff $

$\cup_{\bflamb \in\Delta^N _\lambda} \calh^{1,p}
(C_\bflamb, X)$. Since $\sigma(V_\alpha)$ does not contain nodal points and the complex structures on $\sigma(V_\alpha)$ are constant and do not depend on $\bflamb =(\lambda _{\alpha \beta})$, we can suppose that $V_\alpha$ are complex curves. We have natural isomorphisms $\sigma(V_
{\alpha \beta}) \cong A_{\lambda _{\alpha \beta}}$. Thus, the following map is well- defined $$\matrix \format \c\;& \c \;& \c& \c& \c&\c \\
\Theta: & \calm &\lrar&
\prod_\alpha \calh^{1,p} (V_\alpha, X)&\times&
\prod_{\alpha \beta} \calm(V_{\alpha \beta}, X)
\cr
\noalign{\smallskip}
\Theta: & (C_\bflamb, u) & \mapsto & \bigl(u|_{V_\alpha}, 
(\lambda _{\alpha \beta}, u|_{V _{\alpha \beta}}) \bigr).
\endmatrix$$ It is easy to see that a couple $\bigl( u_\alpha, (\lambda_{\alpha \beta},
u_{\alpha
\beta}) \bigr)\in \prod_\alpha \calh^{1,p} (V_\alpha, X)\times \prod_{\alpha
\beta} \calm(V_{\alpha \beta}, X)$ belongs to the image of $\Theta$, exactly if for all pairs $(\alpha,\beta)$ the gluing conditions $u_\alpha
|_{W_{\alpha, \beta}} = u_{\alpha \beta}|_{W_{\alpha, \beta}} $ are fulfilled.

Let us repeat the gluing procedure from [*Step 2 of Lemma 3.6.*]{} For this take the balls $$\matrix \format \l\,& \l\,& \l\,& \l\,& \l \\
B_\alpha &\subset& \calh^{1,p} (V_\alpha, u_0^*TX)
&\cong& T_{u_0}\calh^{1,p} (V_\alpha,X)
\cr
B_{\alpha \beta} &\subset& \calh^{1,p} (V_{\alpha \beta},u_0^*TX) \oplus \cc
&\cong& T_{u_0}\calm(V_{\alpha \beta}, X)
\cr
B'_{\alpha, \beta} &\subset& \calh^{1,p} (W_{\alpha, \beta}, u_0^*TX)
&\cong& T_{u_0} \calh^{1,p}(W_{\alpha, \beta}, X)
\endmatrix$$ such that there exist biholomorphisms $$\matrix \format \l\,& \l\,& \l\,& \l\,& \l\,& \l \\
\psi_\alpha: & B_\alpha &\buildrel \cong \over \lrar &\psi_\alpha(B_\alpha)&
\subset& \calh^{1,p}(V_\alpha, X)
\cr
\psi_{\alpha \beta}:& B_{\alpha \beta}&
\buildrel \cong \over \lrar& \psi_{\alpha \beta}(B_{\alpha \beta})&
\subset& \calm(V_{\alpha \beta}, X)
\cr
\psi'_{\alpha,\beta}:& B'_{\alpha,\beta}&
\buildrel \cong \over \lrar& \psi'_{\alpha,\beta} (B'_{\alpha,\beta})&
\subset &\calh^{1,p}(W_{\alpha,\beta}, X)
\endmatrix$$ having the properties $$\matrix \format \l\,& \l\,& \l\, \hskip 7pt & \l\,& \l\,& \l \\
\psi_\alpha(0)  &=& u_0|_{V_\alpha}, &
d\psi_\alpha(0) &=&
\id : T_{u_0} \calh^{1,p}(V_\alpha, X) \to T_{u_0} \calh^{1,p}(V_\alpha, X),
\cr
\psi_{\alpha\beta}(0)  &=& u_0|_{V_{\alpha\beta}}, &
d\psi_{\alpha\beta}(0) &=&
\id : T_{(\lambda^0_{\alpha,\beta}, u_0)} \calm(V_\alpha, X) \to
T_{(\lambda^0_{\alpha,\beta}, u_0)} \calm(V_\alpha, X),
\cr
\psi'_{\alpha, \beta}(0)  &=& u_0|_{W_{\alpha, \beta}}, &
d\psi'_{\alpha, \beta}(0) &=&
\id: T_{u_0} \calh^{1,p}(W_{\alpha, \beta}, X)
\to T_{u_0} \calh^{1,p}(W_{\alpha, \beta}, X).
\endmatrix$$ Shrinking the balls $B_\alpha$ and $B_{\alpha \beta}$, if nessessary, we can suppose that for all $\xi_\alpha\in B_\alpha$ and $\xi_{\alpha\beta} \in B_{\alpha \beta}$ the restrictions $\psi_\alpha(\xi_\alpha)|_{W_{\alpha,\beta}}$ and $\psi_{\alpha \beta}(\xi_{\alpha \beta})|_{ W_{\alpha, \beta}}$ belong to the image $\psi'_{\alpha, \beta} (B'_{\alpha, \beta })$. Consider a holomorphic map $$\matrix\format \c\,& \c& \c & \c\,& \c \,& \l\\
\Psi:& \prod_\alpha B_\alpha &\times&
\prod_{\alpha <\beta} B_{\alpha \beta}
& \lrar   & \sum_{\alpha, \beta} \calh^{1,p}(W_{\alpha, \beta}, u_0^*TX)
\cr
\Psi:& (v_\alpha, && v_{\alpha \beta}) & \mapsto &
\psi^{\prime\;\;-1}_{\alpha, \beta}
\bigl(\psi_\alpha(v_\alpha)|_{W_{\alpha, \beta}}\bigr)
- \psi^{\prime\;\;-1}_{\alpha, \beta}
\bigl(\psi_{\alpha \beta}(v_{\alpha \beta})|_{W_{\alpha, \beta}}\bigr).
\endmatrix$$ As in analogous situations, which have already appeared in this paper, the map $\Psi$ gives the gluing condition of local holomorphic maps $\psi_\alpha (v_\alpha): V_\alpha \to X$ and $\psi_{\alpha \beta} (v_{\alpha \beta}): V_ {\alpha \beta}\to X$. Thus we can identify $\calm \cap \prod_\alpha B_\alpha \times \prod_{\alpha
<\beta} B_{\alpha \beta}$ with the set $\Psi\inv(0)$.

Let us study in detail the behavior of $\Psi$ in the point $y_0 \in \prod
B_\alpha \times \prod B_{\alpha \beta}$, $y_0 = \bigl(\psi_\alpha \inv (u_0
|_{V_\alpha}), \psi_{\alpha \beta} \inv (u_0| _{V_{\alpha \beta}})\bigr)$, $\Psi(y_0) =0 \in \sum_{\alpha, \beta} \calh^{1,p}(W_{\alpha, \beta},
u_0^*TX)$. One easily sees that the tangent space at $y_0$ is $$T_{y_0}\bigl( \prod B_\alpha \times \prod B_{\alpha \beta}\bigr)
=
\sum_\alpha \calh^{1,p} (V_\alpha, u_0^*TX)  \oplus
\sum_{\alpha \beta} \calh^{1,p} (V_{\alpha \beta}, u_0^*TX)
\oplus \cc^N,$$ and the differential $d\Psi(y_0)$ on the summand $\sum\calh^{1,p} (V_\alpha,
u_0^*TX)\oplus \sum \calh^{1,p} (V_{\alpha \beta},$

$ u_0^*TX )$ coincides with the Čech codifferential $(9.1.1)$ with respect to the covering $\{V_\alpha, V_{\alpha\beta}\}$ of the curve $C_0$. By [*Lemma 9.2.1*]{} we can represent $\sum
\calh^{1,p} (W_{\alpha, \beta}, u_0^*TX)$ as a direct sum $\calw \oplus
\calq$, where $\calw= \im(d\Psi(y_0))$, and $\calq$ is isomorphic to $\sfh^1(C_0, u_0^*TX)$ and of finite dimension. Let $\Psi_\calw$ and $\Psi_\calq$ be the components of $\Psi= (\Psi_\calw, \Psi_\calq)$ with respect to this decomposition, and let $\wt\calm \deff \Psi_\calw\inv(0)$. By [*Lemma 9.2.1*]{} and the implicit function theorem, $\wt\calm$ is a complex submanifold in $\prod B_\alpha \times\prod B_{\alpha\beta}$ with a tangent space at $y_0\in \wt\calm$ equal to $\calh^{1,p} (C_0, u_0^*TX) \oplus
\cc^N$, and $\calm$ is defined in $\wt\calm$ as a zero set of a holomorphic mapping $\Phi\deff \Psi_\calq
|_{\wt\calm}: \wt\calm \to \calq \cong \sfh^1(C, u_0^* TX)$. This defines on $\calm$ a structure of the Banach analytic set of finite definition.

All that remains is to construct the corresponding holomorphic family of nodal curves $\pi: \calc\to \calm$ and the holomorphic mapping $F: \calc \to X$. Note that to each ball $B_\alpha$ we can naturally associate a trivial family $\pi_\alpha: \calc_\alpha
\deff B_\alpha  \times V_\alpha \to B_\alpha$, and to each ball $B_{\alpha\beta}$ a holomorphic family $\pi_{\alpha\beta}:
\calc_{\alpha\beta} \to B_{\alpha  \beta}$, with fiber $\pi_{\alpha\beta}
\inv( v_{\alpha\beta} )$ equal to $A_{\lambda_{\alpha\beta}}$, where $\lambda_{ \alpha\beta}$ is uniquely determined by the relation $\psi_{\alpha\beta} (v_{\alpha\beta}) = (\lambda_{\alpha\beta}, 
u_{\alpha\beta}) \in \calm( V_{\alpha\beta}, X)$.

Extend these families to the families $\ti\pi_\alpha: \wt\calc_\alpha \to\prod  B_\alpha \times\prod B_{\alpha\beta}$ and $\ti\pi_{\alpha\beta}: \wt\calc_{ \alpha \beta} \to\prod B_\alpha 
\times\prod B_{\alpha\beta}$. It is obvious that $\wt\calc_\alpha$ and $\wt\calc_{\alpha\beta}$ canonically patch into a globally defined family of nodal curves $\ti\pi: \wt\calc \to
\prod B_\alpha \times\prod B_{\alpha\beta}$. $\calc$ will be a Banach manifold. Moreover, we shall obtain correctly defined holomorphic maps $F_\alpha: \wt\calc_\alpha \to X$ and $F_{\alpha\beta}: \wt\calc_{
\alpha \beta} \to X$ such that for $z\in V_\alpha$ one has $F(v_\alpha,
v_{\alpha\beta}, z) \deff \psi_\alpha(v_\alpha)[z]$ plus analogous relations for $F_{\alpha\beta}$.

Let us define $\calc$ as a restriction $\calc \deff \wt\calc|_\calm$. Note that the restriction of the [*trivial*]{} holomorphic family $\wt\calc_\alpha= V_\alpha \times \prod B_\alpha \times \prod
B_{\alpha\beta}$ ­  $\calm$ is also a trivial holomorphic family.

Thus, $\calc$ is a b.a.s.f.d. in the neighborhood of the points $y \in \calc \cap
\wt\calc_\alpha$. Analogously, every holomorphic family of curves $\wt\calc_{\alpha\beta}$ is defined in the trivial bundle $\Delta^2 \times \prod B_\alpha \times \prod B_{\alpha \beta} \buildrel\pr
\over\lrar \prod B_\alpha \times\prod B_{\alpha\beta}$ by the condition $z_1\cdot z_2- \lambda_{\alpha\beta} =0$, where $\lambda_{\alpha\beta}
:B_{\alpha \beta} \to \Delta$ is a holomorphic parameter of the deformation of the complex structure in $V_{\alpha \beta}$, and $z_1$ with $z_2$ are the standard coordinates in $\Delta^2$. So $\calc$ is a b.a.s.f.d. also in the neighborhood of points $y \in \calc \cap \wt\calc_{\alpha \beta}$. Since $\calm$ was defined by the condition of the coincidence of local mappings $F_\alpha$ and $F_{\alpha\beta}$; thus, on $\calc$ a global holomorphic map $F: \calc \to X$ is defined.

Properties [*a), b) [and]{} d) of *Theorem 9.5.1**]{} for $\calm$, $\calc$ and $F$ follow directly from the construction, and one obtains property [*c)*]{} by using [*Theorem 5.3.2*]{}.

*9.6. Drawing Families of Curves.*

In the proof of the Continuity Principle for meromorphic mappings, which we need in this paper (see [*Theorem 4.2*]{} of the next paragraph or [*Theorem 5.1.3*]{} from \[I-S\]), we use the following consequence of the main results of the present paragraph.

Let $(C_n, u_n)$ be a sequence of irreducible stable curves over $X$, converging to a stable curve $(C_\infty, u_\infty)$.

Lemma 9.6.1. *There is an index $N$ and a complex (maybe singular) surface $Z$ with holomorphic mappings $\pi_Z :Z \to \Delta$ and $F: Z \to
X$, which define together a holomorphic family of stable nodal curves over $X$, joining $(C_N, u_N)$ with $(C_\infty, u_\infty)$. More precisely, the following are true:*

$1)$ For every $\lambda\in \Delta$ the fiber $C_\lambda=\pi_Z^{-1}(\lambda)$ is a connected nodal curve with boundary $\d C_\lambda$; a pair $(C_\lambda, u_\lambda)$ with $u_\lambda \deff F|_{C_\lambda}$ is a stable curve over $X$.

$2)$ All curves $C_\lambda$, except for finite numbers, are connected and smooth.

$3)$ $(C_0,u_0)$ is equal to $(C_\infty, u_\infty)$, and there exists $\lambda_N\in \Delta$ such that $(C_{\lambda_N},u_{\lambda_N})=(C_N,u_N)$.

$4)$ There exist open sets $V_1,\ldots,V_m$ in $Z$ such that each $V_j$ is biholomorphic to $\Delta \times A_j$, where $A_j$ is an annulus. Moreover, the diagram $$\matrix
V_j&\buildrel \cong \over \lrar &\Delta\rlap{$\times A_j$}
\cr
\llap{$\pi$}\downarrow & &\downarrow\rlap{$\pi_{\Delta }$}
\cr
\Delta & = & \Delta
\endmatrix$$ is commutative; each annulus $C_\lambda \cap V_j \cong \{\lambda\}
\times A_j$ is adjacent exactly to one of the components of the boundary $\d C_\lambda$, and the number $m$ of domains $V_j$ is equal to the number of the boundary components of each curve $C_\lambda$.

Remark. This lemma was stated without proof in [*Proposition 5.1.1*]{} from \[I-S\].

Proof. Let $\pi_\calc: \calc \to \calm$ and $\ev: \calc \to
X$ define a complete family of holomorphic deformations of the stable nodal curve over $X$ $(C_\infty, u_\infty)$, constructed in [*Theorem 9.4.1*]{}. Let $\lambda^* \in \calm$ parameterize the curve $(C_\infty, u_\infty)$. Choose a sequence $\lambda_n \lrar \lambda^*$ ¢ $\calm$ such that $(C_n, u_n) \cong (\pi_\calc\inv(\lambda_n),
\ev|_{\pi_\calc\inv(\lambda_n)})$ for all sufficiently big $n$.

While $(C_\infty, u_\infty) = (\pi_\calc\inv (\lambda^*), 
\ev|_{ \pi_\calc \inv(\lambda^*)})$ lifts in the neighborhood of the boundary of $\hat U$, thus by shrinking $\calm$, if nessessary, we can suppose that this is true for all $\lambda \in \calm$.

By Theorem 9.5.1 the space $\calm$ is a b.a.s.f.d. therefore, we can apply [*Theorem 9.1.1*]{}. In particular, there are only finitely many irreducible components at $\lambda^*$ of $\calm$. Let $\calm_1$ be a component of $\calm$, which contains infinitely many $\lambda_n$. Represent $\calm_1$ in the neighborhood of $\lambda^*$ as a proper ramified covering $\pi_1: \calm_1 \to B_1$ over a Banach ball $B_1$. If $\Delta$ is an imbedded disk in $B_1$, then $\pi_1\inv(\Delta)$ is a one-dimensional analytic set, irreducible components which can be also parameterized by the disks. Thus, there exists a holomorphic mapping $\phi: \Delta \to \calm_1$, passing through $\lambda^*$ and $\lambda_N$ for some $N>\!>1$. The pre-image of the family $\pi_\calc: \calc\to \calm$ with respect to $\phi$ defines a holomorphic family of stable nodal curves over $\Delta•$ with total space $\pi_Z: Z \to\Delta$ and mapping $F:Z \to X$, containing $(C_\infty,
u_\infty)$ and $(C_N, u_N)$.

While $C_N$ is smooth, the general fiber $C_\lambda =\pi_Z\inv(\lambda)$, and again is smooth. Taking if nessessary a smaller disk, we can suppose that $C_\lambda$ are singular only for a finite number of $\lambda \in \Delta$. All other properties 1)–5) follow from the construction of the family $\pi_Z: Z\to \Delta$ and a mapping $F:Z \to X$.

[Lecture 10]{}

[Envelopes of Meromorphy of Two-Spheres.]{}

*10.1. Continuity Principles Relative to Kähler Spaces.*

Our aim in this paragraph is to prove [*Theorem 4.1*]{} and give some corollaries from it. First of all we need an appropriate form of the so- called “continuity principle” for the extension of meromorphic mappings.

For the notion of meromorphic mapping from a domain $U$ in a complex manifold into a complex manifold (or space) $Y$ we refer to \[Rm\]. We only point out here that meromorphic mappings into $Y = \cc\pp^1$ are exactly meromorphic functions on $U$, see \[Rm\].

Definition 10.1.1. *A Hermitian complex space $Y$ is called disk-convex if, for any sequence $(C_n,u_n)$ of smooth curves over $Y$ parameterized by the same surface $\Sigma $, such that*

1\) $\area(u_n(C_n))$ are uniformly bounded and

2\) $u_n$ $C^1$-converges in the neighborhood of $\d C_n $,

*there is a compact $K\subset Y$ which contains all $u_n(C_n)$.*

This definition obviously carries over to the case where $Y$ is a symplectic manifold. In this case one should consider $(C_n,u_n)$ as $J_n$-holomorphic curves, with $J_n$ converging to some $J$ (everything in $C^1$-tology) and all structures being tamed by a given symplectic form.

Let $U$ now be a domain in the complex manifold $X$, and $Y$ is a complex space .

Definition 10.1.2. *An envelope of meromorphy of $U$ relative to $Y$ is the largest domain $(\hat U_Y,\pi )$ over $X$, which contains $U$ (i.e., there exists an imbedding $i:U\to \hat U_Y$ with $\pi \scirc i={\sl Id}$) such that every meromorphic map $f:U\to Y$ extends to a meromorphic map $\hat f:\hat U_Y\to Y$.*

Using the Cartan-Thullen construction for the germs of meromorphic mappings of open subsets of $X$ into $Y$ instead of germs of holomorphic functions, one can prove the existence and uniqueness of the envelope.

**Proposition 10.1.1. *For any domain $U$ in the complex space $X$ and for any complex space $Y$ there exists a maximal domain $(\hat U_Y,\pi )$ over $X$, containing $U$ such that every meromorphic mapping $f:U\longrightarrow Y$ extends to a meromorphic mapping $\hat f:\hat U_Y\longrightarrow Y$. Such a domain is unique up to a natural isomorphism.***

See \[Iv-1\] for details.

Theorem 10.1.2. (Continuity Principle-I). *Let $X$ be a disk-convex complex surface and $Y$ a disk-convex Kähler space. Then the envelope of meromorophy $(\hat U_Y,\pi )$ of $U$ relative to $Y$ is also disk-convex with respect to the pulled-back Kähler form.*

This result can be reformulated in more familar terms as follows.

Let $\{ (C_t,u_t)\}_{t\in [0, 1]}$ be a continuous (in a Gromov topology) family of complex curves over $X$ with boundaries, parameterized by a unit interval. More precisely, for each $t\in [0, 1[$ a smooth Riemann surface with boundary $(C_t, \d C_t)$ is given together with the holomorphic mapping $u_t: C_t\longrightarrow X$, which is $C^1$-smooth up to the boundary. Note that $C_1$ is not supposed to be smooth, i.e., it can be a nodal curve!

Suppose that in the neighborhood $V$ of $u_0(C_0)$ a meromorphic map $f$ into the complex space $Y$ is given.

**Definition 10.1.3. We shall say that the map $f$ meromorphically extends along the family $(C_t,u_t)$ if for every $t\in [0, 1]$ a neighborhood $V_t$ of $u_t(C_t)$ is given, and given a meromorphic map $f_t:V_t
\longrightarrow
Y$ such that**

a\) $V_0 = V$ and $f_0= f$;

b\) if $V_{t_1}\cap V_{t_2}\not=\emptyset $ then $f_{t_1}\ogran_{V_{t_1}\cap
V_{t_2}} = f_{t_2}\ogran _{V_{t_1}\cap V_{t_2}}$.

**Theorem 10.1.3. (*Continuity Principle-II) *Let $U$ be a domain in the complex surface $X$. Let $\{ (C_t,u_t)\} _{t\in [0, 1]} $ be a continuous family of complex curves over $X$ with boundaries in $U_1$, a relatively compact subdomain in $U$. Suppose moreover that $u_0(C_0)\subset U$ and that $C_t$ for $t\in [0,1[$ are smooth. Then every meromorphic mapping $f$ from $U$ to the disk-convex Kähler space $Y$ extends meromorphically along the family $(C_t,u_t)$.****

Taking an image manifold $Y$ complex line $\cc $ or $\cc \pp ^1$ we obtain the continuity principles for holomorphic or meromorphic functions. When it is necessary to emphasize that we are considering the mappings into a certain manifold $Y$, we shall refer to the statement above as the *continuity principle relative to $Y$ or *c.p. for the meromorphic mappings into $Y$.**

The discussion above leads to the following

**Corollary 10.1.4. *If we have the domain $U$ in a complex surface $X$, a Kähler space $Y$ and a family $\{ (C_t,u_t)\} $ satisfying the conditions of the “continuity principle”, then the family $\{ (C_t,u_t)\} $ can be lifted onto $\hat
U_Y$, [*i.e.,*]{} there exists a continuous family $\{ (C_t,\hat u_t) \} $ of complex curves over $\hat U$ such that $\pi \scirc \hat u_t = u_t$ for each $t$.***

Of course the point here is that the mapping can be extended to the neighborhood of $u_1(C_1)$, which is a reducible curve having in general compact components. This makes our situation considerably more general than the classical one, i.e., when $X$ is supposed to be Stein; compare with \[Ch-St\].

Remark. Let us explain the meaning of this theorem by an example. Let $X$ be disk-convex and $U\subset X$ some domain. Furthermore, let $f:U \to Y$ be a meromorphic map and $\{(C_n, u_n)\}$ a sequence of stable complex curves over $X$, converging in the Gromov topology to $(C_\infty, u_\infty)$. Suppose that the images of the boundaries $u_n (\d C_n)$ and $u_\infty (\d C_\infty)$ are contained in $U$. Suppose also that $f$ extends along every curve $u_n (C_n)$. This means that there exists a complex surface $V_n$, containing $C_n$, and a locally biholomorphic map $u'_n:V_n \to X$ such that $u'_n|_{C_n} = u_n$ and $f$ meromorphically extend from $u'_n{}\inv U$ onto the whole $V_n$.

The latter is equivalent to the lift of the curves $(C_n, u_n)$ into the envelope $\hat U$, i.e., to the existence of holomorphic mappings $\hat u_n: C_n \to \hat U$ such that $\hat \pi \scirc \hat u_n =u_n$. In other words, one can take as $V_n$ a neighborhood of the lift of $C_n$ into $\hat U$. One easily sees that the curves $(C_n,\hat u_n)$ are stable over $\hat U$, have uniformly bounded areas and converge in the neighborhood of the boundary $\d C_n$. By [*Theorem 10.1.2*]{} and by the Gromov compactness theorem , some subsequence $(C_n,\hat u_n)$ converges to the $\hat{U}$-stable curve $(C_\infty,\hat u_\infty)$, and $\hat\pi \scirc \hat u_\infty
=u_\infty$. This means that $f$ extends along the curve $u_\infty (
C_\infty)$. Thus, [*Theorem 4.2*]{} is a generalization of the E. Levi continuity principle.

[*10.2. Proof of the Continuity Principle.*]{}

Suppose that there is a subsequnce of our sequence, which we still denote by $(C_n,u_n)$ and which is not contained in any compact subset of the envelope $(\hat U_Y, \pi )$. Put $v_n=\pi \scirc u_n$ and consider our sequence as a sequence $(C_n,v_n)$ of stable curves over $X$. While the areas are bounded, we can suppose that this sequence converges in the Gromov topology by the disk-convexity of $X$. Denote by $(C_0,v_0)$ its limit. For $N$ sufficiently large take a holomorphic family $(\calc ,\pi ,\calv )$ joining $(C_N,v_N)$ with $(C_0,v_0)$ as in [*Lemma 9.6.1.*]{}

Let $f:\hat U_Y \to Y$ be some meromorphic mapping of our envelope into a disk-convex Kähler space $Y$. Composing $f$ with $\calv$, we obtain a meromorphic map $h:= f\scirc \calv :
\bigcup U_j\cup \pi^{-1}(V(\lambda_N))\to Y$. Here $V(\lambda_N)$ is a suffuciently small neighborhood of $\lambda_N \in \Delta $.

Denote by $W$ the maximal connected open subset of $\Delta $ containing $V(\lambda_N)$ such that $h$ meromorphically extends onto $\bigcup U_j\cup \pi^{-1}(W)$. We want to prove that $W=\Delta
$. Denote by $\Gamma_{f_{\lambda }}$ the graph of the restriction $f\mid_{C_{\lambda }}$. Fix some Hermitian mertic on $\calc $.

Lemma 10.2.1. *For any compact $K\subset \Delta $ there is a constant $M_K$ such that ${\sl area}(\Gamma_{f_{\lambda }})\le M_K$ for all $\lambda
\in W\cap K$.*

*Proof. Shrinking $\calc $ if nessessary, we can suppose that $f$ has only a finite number of points of indeterminancy in $\bigcup_jU_j$. Denote by $S_W$ the discrete subset in $W$, which consists of points $s\in W$ such that either the fiber $C_s$ is singular or $s$ is the projection of the indeterminancy points of $f\mid_{\calc \mid_W}$. Fix a point $\lambda_1\in W\setminus S_W$. Take a path $\gamma :[0,1]\to
W\setminus
S_W$ connecting $\lambda_1$ with some $\lambda_2\in W\setminus S_W$. Take some relatively compact in $W\setminus S_W$ neighborhood $V$ of $\gamma ([0,1])$.*

Recall that a Kähler metric on a complex space $Y$ consists of a locally finite covering $\{ V_{\alpha }\} $ of $Y$ and strictly plurisubharmonic functions $\phi_{\alpha }$ on $V_{\alpha }$ such that $\phi_{\alpha }-\phi_{\beta }$ are pluriharmonic on $V_{\alpha }\cap V_{\beta }$. The sets $\{ f^{-1}(V_{\alpha })\} $ form a covering of $\pi^{-1}(V)$ and $dd^cf^*\phi_{\alpha }=w$ is a correctly- defined semi-positive closed form on $\pi^{-1}(V)$. We have $${\sl area}(\Gamma_{f_{\lambda }})={\sl area}(C_{\lambda })+
\int_{C_{\lambda }}w.$$ By the Stokes formula $$\int_{C_{\gamma (0)}}w-
\int_{C_{\gamma (1)}}w =
\int_{\cup_tC_{\gamma (t)}}dw -
\int_{\cup_t{\d C_{\gamma (t)}}}w=$$ $$=
\int_{\cup_t{\d C_{\gamma (t)}}}w.$$ This is obviously uniformly bounded on $\lambda_2\in K\setminus S_W$, which implies the uniform bound in $K$ by the Gromov (or Bishop, in this case) compactness theorem.

Lemma 10.2.2. *Mapping $h$ meromorphically extends onto $\calc $.*

*Proof. Let us prove first that $h$ meromorhpically extends onto $\calc $ minus the singular fiber $C_0$, i.e., that $W\supset \Delta \setminus \{ 0\} $.*

The same arguments as in \[Iv-2\] (the only difference being that in \[Iv-2\] curves $C_
{\lambda }$ were disks) show that $(\Delta \cap \d W)\setminus (\{ 0\})
$ is an analytic variety. Now, using a Thullen-type extension theorem of Siu, see \[Si-1\], we can extend $h$ onto $\calc \setminus C_0$.

Denote by $\hat C_0$ the union of compact components of $C_0$. We shall show that $\hat C_0$ contracts to a finite number of normal points. To prove this we can suppose that $\hat C_0$ is connected. Otherwise, we can apply the same arguments as below to the connected components. We shall now prove that $\hat C_0$ contracts to a normal point. Denote by $E_1,...,E_n$ the irreducible components of $\hat C_0$. All we must prove is that the matrix $(E_i,E_j)$ is negatively defined, see \[Gra\]. Denote by $l_i$ the multiplicity of $E_i$. Thus $\hat C_0=\Sigma_{i=1}^nl_i\cdot E_i$. Denote by $M$ the $\zz $-module generated by $E_1,...,E_n$ with scalar product defined by intersection of divisors. Put $D=\hat C_0$. Then we have

1\) $E_i\cdot E_j\ge 0$ for $i\not= j$;

2\) $D\cdot E_i\le 0$ for all $i$, because this number is not more than the intersection of $E_i$ with the nonsingular fiber $C_{\lambda }$.

By Proposition 3 from \[Sh\], v.1, Appendix I, we have $A\cdot A\le 0$ for all $A\in M$ and $A\cdot A=0$ iff $A$ is proportional to $D$. But $D\cdot D <\hat C_0\cdot C_{\lambda }=0$, where $C_{\lambda }$ is a smooth fiber. This proves that $(E_i\cdot E_j)$ is negatively defined.

Therefore, all that is left to prove is that the normal point is a removable singularity for the meromorphic mappings into a disk-convex Kähler space.

Let $(\calc ,s)$ be a germ of two-dimensional variety with an isolated normal singularity $s$. Let a meromorphic map $h:\calc \setminus \{ s\} \to Y$ be given. Realize $\calc $ as a finite proper analytic cover over a bi-disk $\Delta^2$ with $s$ being the only point over zero. Denote by $p:\calc \to \Delta^2$ this covering.

The composition $h\scirc p^{-1}$ is a multivalued meromorphic map from $\Delta^2 \setminus \{ 0\} $ to $Y$. This can be extended to the origin, see \[Si-1\]. Consider the following analytic set in $\calc\times Y$: $$\Gamma = \{ (x,y)\in \calc \times Y: (p(x),y)\in \Gamma_{h\scirc p^{-1}} \} .$$ The irreducible component of $\Gamma$, which contains $\Gamma_h$, will be the graph of the extension of $h$ onto $\calc $.

Let us turn to the proof of the theorem. We have a holomorphic family $\calc \to \Delta $ of stable curves $(C_{\lambda },
v_{\lambda })$ over $X$ such that:

1\) $(C_{\lambda_0},v_{\lambda_0})=(C_0,v_0)$;

2\) $h=f\scirc \calv $ extends meromorphically onto $\calc $, where $\calv :\calc \to X$ is an evaluation map.

We want to lift this family to the envelope $(\hat U_Y, \pi )$. Denote by $E_1,...,E_l$ the set of all irreducible curves in $\calc $ which are contracted by $\calv :\calc \to X$ to the points. It follows that $\calc \setminus
\bigcup_{j=1}^lE_j$ lifts to the envelope by the Cartan-Thullen construction. Note further that $E_j$ do not intersect $\bigcup_{j=1}^mU_j$. That is why either $E_j\subset \hat C_0$ is a compact component of singular fiber or projects surjectively on $\Delta $ under the projection $\pi :\calc \to \Delta $. In the second case it intersects $C_0$ and the point $u(C_0 \cap E_j)$ lifts to the envelope. This proves that $\calc \setminus \hat C_0$ lifts to the envelope. More precisely, we have shown that $\calc \setminus
\hat C_0'$ lift to the envelope, where $\hat C_0'$ is a union of all compact components of the singular fiber mapped by $\calv $ into the points.

Denote by $\hat C$ some connected subset of $\hat C_0'$ and choose its neighborhood $V$ not to intersect other components of $\hat C_0'$. Then $\calv $ maps $V$ onto the neighborhood of the point $\calv (\hat C)$ in $X$. Now it is clear that this point lifts to the envelope.

Thus, we have proved that $(C_0,v_0)$ lifts to the envelope $(\hat U_Y,\pi )$ of $U$. This implies that $(C_n,u_n)$ should lie in a compct subset of $\hat U_Y$. This is the desired contradiction.

Let us give one corollary of the Continuity Principle just proved.

Corollary 10.2.3. *Let $X$ be a complex surface with one singular normal point $p$. Let $D$ be a domain in $X$, $\d D\ni p$. Suppose there is a sequence $(C_n,u_n)$ of stable curves over $X$ converging to $(C_0,u_0)$ in a Gromov topology and such that*

a\) there is a compact $K\subset D$ with $u_n(\d C_n)\subset K$ for all $n$;

b\) $p\in u_0(C_0)$.

Then every meromorphic function from $D$ extends to the neighborhood of $p$.

To our knowledge this statement is new also for holomorphic functions.

*10.3. Construction of Envelopes - I.*

We are now ready to give the proof of [*Theorem 4.1*]{}, first under some technical assumptions to clarify the main idea. Then in the next paragraph the general case will be considered.

Let a Kähler surface $(X,\omega )$ be fixed and some symplectic immersion $u:\ss^2\to X$, with only positive double intersections be given. Our technical assumptions for the moment will be the following:

1\. We suppose that $c_1(X)[M]$ is not only positive but moreover bigger than the sum $\delta $ of double points of $M:=u(\ss^2)$.

2\. $X$ is supposed to be “positive” in the sense that for any almost-complex structure $J\in \calj_{\omega }$ and $J$-complex sphere $C$ satisfies $[C]^2\ge 0$.

As we have already mentioned, those conditions will be eliminated in the next paragraph.

Now let $f$ be a meromorphic function in some neighborhood $U$ of $M$. Fix some relatively compact $U_1\Subset U$ containing $M$. Let $\{ J_t\}_{t\in [0,1]}$ be a family of $C^1$-smooth almost complex structures on the $4$-manifold $X$ constructed in [*Lemma 1.4.2*]{}. We lift the structures $J_t$ onto the envelope $(\hat U,\pi )$ in such a way that those liftings are standard on $\hat U\setminus i(U_1)$, where $i:U\to \hat U$ is a canonical imbedding.

For $t=1$ our $J_1$-holomorphic sphere $M=M_1$ is given. By assumption $c_1(X)[M]$ $\delta_1 =p \ge1$. Here $\delta_1=\delta $ is the geometric self-intersection of $M_1$, i.e., the number of double points. From [*Theorem 8.4.1*]{} we get a family $M_t=f(J_t)$ of $J_t$-holomorphic curves for $t$ sufficiently close to $1$, where $f:V\subset \calj \to
\cals \times \calj_S \times \calj$ is a local section of $\pr_\calj$ constructed in [*Theorem 8.4.1*]{}.

Definition 10.3.1. *We say that the family $\{M_t\}_{t\in(\hat t,1]}$ of (possibly reducible) $J_t$-holomorphic curves is semi-continuous if there exist a (maybe infinite) partition of the  interval $(\hat t,1]$ of the form $1=t_0 >t_1>\ldots>t_n>\ldots$ with $t_n\searrow \hat t$, natural numbers $1=N_0 \le N_1 \le\ldots \le N_n \le\ldots$, $J_t$-holomorphic maps $u_t: \bigsqcup_{j=1}^{N_i} S^j_i\to X$ for $(t_{i+1}, t_i]$ with $M_t\deff u_t(\bigsqcup_{j=1}^{N_i} S^j_i)$ such that*

$u_t: (t_{i+1}, t_i] \times \bigsqcup_{j=1}^{N_i} S^j_i\to X$ is a continuous map;

$\area(M_t)$ are uniformly bounded from above;

$\calh$-$\lim_{t\searrow t_{i+1}} M_t \deff \overline
M_{t_{i+1}} \supset M_{t_{i+1}}$.

The inclusion $\overline M_{t_{i+1}} \supset M_{t_{i+1}}$ means that $M_{t_{i+1}}$ has no other components than those of $\overline M_{t_{i+1}}$.

We say that $\{M_t\}$ is a family of spheres if all $S^j_i$ are spheres.

Let $T$ be the infimum of such $\hat t$, for which there is a semi-continuous family $\{M_t\}_{t\in (\hat t, 1]}$ of spheres such that for all irreducible components $M^1_t,\ldots M^{N_i}_t$ of $M_t$, $t\in (t_{i+1}, t_i]$, one has

[*a)*]{} $c_1(X)[M^j_t] -\delta_t^j -\varkappa_t^j=p_t^j \ge1$;

[*b)*]{} $\sum_{j=1}^{N_i} p_t^j \ge p$.

Here $\varkappa_t^j$ is the sum of the Milnor numbers of cusp points of $M_t^j$, see Appendix 2. We allow the existence of multiple components, [*i.e.,*]{} that some of $M_t^j$ can coincide.

The set $T$ is open by [*Theorem 8.4.1*]{}. To prove the closeness of $T$, we note that since all $J_t$ are tamed by the same form $\omega$, the areas of a $J_t$-holomorphic curve are uniformly in $t$ estimated from above and below by $\int_{M_t} \omega$. Moreover, since $\overline M_{t_{i+1}} \supset
M_{t_{i+1}}$, we obtain $\int_{M_{t_i}} \omega \le \int_{M_{t_{i+1}}}
\omega$. In particular, this implies that the sequence $\{N_i\}$ is bounded from above, and hence stabilizes for $i$ big enough. From Gromov’s compactness theorem and disk-convexity of the envelope (Theorem 2.2.2)we obtain that for every $j={1,\ldots,N_i}$ the sequence $\{M^j_i\}_{i=1}^\infty$ has a subsequence, still denoted in the same way, which converges to a $J_{\hat t}$-holomorphic curve $\overline
M^j\subset \hat U_Y$.

To simplify the notations, we drop the upper index $j$ from now on and write $\overline M$ instead $\overline M^j$, having in mind that we can do the same constructions for all $j=1,\ldots,N_i$. Note that all irreducible components of $\overline M$ are $J_{\hat t}$-holomorphic spheres. We write $\overline M = \bigcup_{k=1}^d m_k\cdot M_k$, where $M_k$ are distinct irreducible components of $\overline M$ with the multiplicities $m_j\ge1$. The genus formula for $\overline M$ now has (because of multiplicities!) the following form:

$$0=\msmall{[\overline M]^2 - c_1(X)[\overline M] \over2}
+ \sum_{k=1}^d m_k - \sum_{k=1}^d m_k(\delta_k +\varkappa_k)
-\sum_{k < l}m_k\cdot m_l [M_k]\cdot [M_l]$$ $$-\sum_{k=1}^d \msmall{m_k(m_k -1) \over2 } [M_k]^2.
\eqno(10.3.1)$$

The formula can be obtained by taking the sum of the genus formulas for each $M_k$ and then completing the sum $\sum_{k=1}^d m_k[M_k]^2$ to $[\overline M]^2$. Here $\delta_k$ and $\varkappa_k$ denote self-intersection and Milnor numbers of $M_k$. We also have $$\varkappa^j_i + \delta^j_i = \msmall{ [M^j_i]^2 - c_1(X)[M^j_i] \over2} +1.
\eqno(10.3.2)$$ Note that $[\overline M]^2 = [M^j_i]^2$ and $c_1(X)[\overline M]=
c_1(X)[M^j_i]$ for $i$ sufficiently big, and that $\sum_{k<l} m_k m_l
\,[M_k][M_l] \ge \sum_{k=1}^d m_k -1$, because the system of curves $\bigcup_{k=1}^d M_k$ is connected. From (10.3.1) and (10.3.2) we obtain $$\sum_{k=1}^d m_k(\delta_k + \varkappa_k) +
\sum_{k=1}^d \msmall{m_k(m_k-1) \over2 } [M_k]^2
\le \delta^j_i + \varkappa^j_i$$ and $$\sum_{k=1}^d m_k c_1(X) [M_k] = c_1(X)[M^j_i].$$ Using positivity of $(X,\omega)$, [*i.e.,*]{} the property $[M_k]^2 \ge0$, we obtain $$\sum_{k=1}^d m_k \biggl( c_1(X) [M_k] -\delta_k -\varkappa_k\biggr)
\ge c_1(X)[M^j_i] -\delta^j_i -\varkappa^j_i =p_j.$$ Thus we can choose among $M_1,\ldots,M_d$ the subset with properties [*a)*]{} and [*b)*]{}.

Returning to the old notations, we can “correct” all $\overline M^j$, choosing an appropriate subset $M^j \subset \overline M^j$. Now applying [*Theorem 8.4.1*]{} to all $M^j$, we can extend our family $\{M_t\}$ in a  neighborhood of $\hat t$, and hence define it for all $t\in [0,1]$.

For $t=0$ we get the next statement.

The next rigidity property of symplectic imbeddings is a straightforward corollary from this theorem.

Corollary 10.3.2. *Let $M_1$ and $M_2$ be two symplectically imbedded spheres in $\cc \pp^2$. Then any biholomorphism of a neighborhood of $M_1$ onto a neighborhood of $M_2$ is fractional linear.*

[10.4. Construction of Envelopes - II.]{}

Let us now give the proof in the general case. In fact, the proof works not only for the meromorphic functions but also for the meromorphic mappings into any disk-convex complex Kähler space $Y$.

Theorem 10.4.1.

*Let $u:\ss^2\to X$ be a symplectic immersion of the sphere $\ss^2$ into a disk-convex Kähler surface $X$ such that $M=u(\ss^2)$.*

Let $u:S^2\to X$ be a symplectic immersion of a sphere, having also positive self-intersections. Let $U$ be a relatively compact domain in $X$, which contains $M:=u(S^2)$. Denote by $(\hat U_Y,
\hat\pi_Y)$ its envelope of meromorphy relative to $Y$.

*Step 1. There exists an ($\omega$-tamed) $J_0\in \calj_U$ such that $M$ is $J_0$-holomorphic.*

This was proved in [*Lemma 1.1.2*]{} of \[I-S\]. Moreover, there exists a smooth homotopy $h :[0,1]\to \calj_U$ joining $J_0=h(0) $ and $J\st =h(1)$. Put $M_0:=M$ and denote as in [*Lemma 8.3.2*]{} by $\calm_h(M_0,J_0)$ a component of $\calm_h$ through the point $(M_0,J_0)$.

*Step 2. Suppose that the component $\calm_h(M_0,J_0)$ is not compact.*

Then by (iii) of [*Lemma 8.3.2*]{} there is a continuous path $\gamma :[0,1]\to \calm_h$ starting from $(M_0,J_0)$ with property (b), see (iii) of [*Lemma 2.5*]{}. Consider $J_n$-holomorphic spheres $M_n$ in $\calm_h(M_0,J_0)$, which are discrete there with $J_n\to J^*\in \calj_U$ as in (b) of [*Lemma 2.5*]{}.

If for some $n$ $M_n\cap U=\emptyset$, then, because $J_n=J\st $ on $\hat U\setminus U$, this $M_n$ will be the rational curve we are looking for.

If not, then some sequence, still denoted by $M_n$, will converge in the Gromov topology to a reducible curve $M^{(1)}$. If $M^{(1)}$ has an irreducible component $M^{(1)}_0$, lying outside of $U$ and such that $c_1(X)[M^{(1)}_0]>0$, then $M^{(1)}_0$ is our rational curve.

If not, there exists a component $M^{(1)}_0$ of the limit curve $M^{(1)}$ such that $c_1[M^{(1)}_0]>0$. Repeat Step 2 for $M^{(1)}_0$ instead of $M_0$. Since the area of a complex spheres is bounded from above (see \[G\]), after a finite number of steps we obtain a needed rational curve in the envelope $\hat U_Y$, or arrive at

*Step 3. $\calm_h(M_0,J_0)$ is compact.*

From of [*Theorem 8.3.3*]{} and [*Corollary 8.3.4*]{} we immediatly see that there are continuous (in fact, piecewise $C^k$) paths $(M^n_t,J^n_t)$ such that

1\) $M^n_0=M_0$ for all $n$;

2\) $J^n_0=J_0$ for all $n$;

3\) $J^n_1\to J\st $.

By the Gromov compactness theorem some subsequence from $M^n_1$ , still denoted by $M^n_1$, is converging to a $J\st $-holomorphic nodal curve $C^*$ with $\pi^*c_1(X)[C]>0$. Take some irreducible component $C$ of the curve $C^*$. Then $C$ is the rational curve in $\hat U_Y$ we are looking for.

*10.5. Examples.*

Here we discuss a few more examples concerning envelopes of meromorphy.

Example 1. Let $(X,\omega) = (\cc\pp^1 \times \cc\pp^1,\omega_{FS}
\oplus \omega_{FS} )$, where $\omega_{FS}$ denote the Fubini-Studi metric on $\cc\pp^1$. Note that $c_1(X)= 2[\omega]$. Let $J$ be an $\omega$-tame almost complex structure on $X$ and $C$ be a $J$ - complex curve on $X$. Denote by $e_1$ and $e_2$ the standard generators of $\sfh_2(X,\zz)= \zz^2$ and write $[C]=a\cdot e_1 + b\cdot e_2$. Then we get $a+b=\int_C\omega \ge 1$ and $c_1(X)[C]= 2(a+b)$. Furthermore, by the genus formula $0\le g(C) + \delta
+\varkappa = (2ab -2(a+b))/2 +1 =(a-1)(b-1)$. Thus, we conclude that both $a$ and $b$ are non-negative and $[C]^2=2ab \ge0$. So $\cc\pp^1 \times \cc\pp^1$ is non-negative in our sense.

Let $M$ be an imbedded symplectic sphere in $X$. Then $(a-1)(b-1)=0$ by the genus formula. Therefore, we can assume that $a=1$ and $b\ge0$. Now one concludes that the following holds:

Corollary 10.5.1. *Let $M$ be a symplectic sphere in $(\cc\pp^1
\times \cc\pp^1,\omega_{FS} \oplus \omega_{FS})$. Then the envelope of meromorphy of any neighborhood of $M$ contains a graph of a rational map of degree $0\le d \le b$ from $\cc\pp^1$ to $\cc\pp^1$.*

Example 2. $(X,\omega) = (\cc\pp^2,\omega_{FS})$. Let $M$ be a symplectic surface in $X$ of degree $m\deff \int_M \omega$ with [*positive*]{} self-intersections. Then obviously $c_1(X)[M]=3m$. Note that we proceed to the construction of a family $\{M_t\}$ if the condition $c_1(X)[M^j_t] > \varkappa(M^j_t)$ is satisfied for all irreducible components $M^j_t$ of $M_t$. By the genus formula one has $\varkappa(C) \le (d-1)(d-2)/2$ for every complex curve $C$ of degree $d$. So we can proceed if $3m > (m-1)(m-2)/2$, which is equivalent to $1\le m \le8$. Thus, we have the following

Corollary 10.5.2. *Let $M$ be a symplectic surface in $\cc\pp^2$ of degree $m\le 8$ with positive self-intersections. Then the envelope of meromorphy of any neighborhood of $M$ coincides with $\cc\pp^2$ itself.*

Remark. Note that the examples include all imbedded symplectic surfaces in $\cc\pp^2$ of genus $g\le 21$.

Example 3. Let $X$ be a ball in $\cc^2$ and $w$ is a standard Euclidean form. Blow up the origin in $\cc^2$ and denote by $E$ the exceptional curve. By $\hat X$ denote the blown-up ball $X$. The blow-up of $\cc^2$ is also Kähler, and we denote by $w_0$ some Kähler form there. Consider a sufficiently small $C^1$-perturbation of $E$. This will be a $w_0$-symplectic sphere in $\hat
X$, which is denoted by $M$. The Chern class of the normal bundle to $M$ is equal to that of $E$ and thus is $-1$. Therefore, $c_1(\hat X)[M]=1$ and the proof of Theorem 1 applies. One should only note that in the process of deformation $M_t$ cannot break into irreducible or multiply covered components in this special case. The only rational curve in $\hat X$ is $E$. Thus, we see that

*the envelope of meromorphy of any neighborhood of $M$ contains $E$.*

One can then blow down the picture to obtain downstairs a sphere $M_1$-image of $M$ under the blown-down map. This $M_1$ is homologous to zero, so cannot be symplectic, and for this $M_1$ our Theorem 1 cannot be applied.

**Example 4. Chirka in \[C\] proved the following “local version” of our Theorem 1, which he called “a generalized Hartogs’ lemma”. Denote by $\Gamma $ a graph of continuous function $f:\bar\Delta\to \cc $. Consider the following “generalized Hartogs’ figure” in $\cc^2$: $H_\Gamma := \d \Delta \times \bar\Delta\cup \Gamma $. Chirka showed that every holomorphic function in the neighborhood of $H_{\Gamma }$ extends holomorphically onto the unit bidisk $\Delta^2$. This is a corollary of our Theorem 4.1 (as is explained in \[C\]). Really, denote by $(z,w)$ coordinates in $\cc^2$. Any function $f$ holomorphic in the neighborhood of $\d \Delta \times \bar\Delta $ is a sum of a function holomorphic in the bidisk and a function $f_{-}$ holomorphic in $(\cc\pp^1
\setminus \bar\Delta )\times \Delta $. This $f$ is also holomorphic in the neighborhood of $\Gamma $. We need only to extend $f_{-}$. Extending $\Gamma $ “inside” $(\cc\pp^1\setminus \bar\Delta )\times \Delta $ we obtain $\tilde\Gamma $ - a sphere, homologous to the $\{ pt\} \times \cc\pp^1$ in $\cc\pp^1\times \cc\pp^1$. Rescailing the variable $w$ we can make $\tilde\Gamma $ symplectic and find ourselves in the conditions of [**Example 2**]{}.**

The proof of Chirka is roughly the same (i.e., uses the perturbation of the structure), but somewhat simpler. It uses, instead of Gromov’s techniques, the results of Vekua on so- called generalized analytic functions.

Answering the question, posed by Chirka, Rosay in \[R\] constructed an example showing that a “generalised Hartogs’ lemma” does not hold in $\cc^3$.

Example 5. This example was explained to us by E. Chirka, and it shows that our Continuity Principle is not valid when the complex dimension of the manifold $X$ is more than two.

Take as $X$ the total space of the bundle $\calo(-1)\oplus \calo (-1)$ over $\cc\pp^1$. Denote an affine coordinate on $\cc\pp^1$ by $z$, coordinates on the fibers by $\xi_1$, $\xi_2$ and $\eta_1$, $\eta_2$ such that $\eta_1 = z\xi_1$ and $\eta_2 = z\xi_2$. Identify $\cc\pp^1$ with the zero section of the bundle. Consider the meromorphic function $f= 
e^{\xi_2 /\xi_1}$. The set of essential singularities of $f$ is $\{\xi_1=0 \}$, which contains the zero section $\cc\pp^1$.

Consider the following sequence of analytic disks $C_n$ in $U\deff X \bs
\{\xi_1=0 \}$, $C_n \deff \{ \xi_2 =0, |z|\le n, \xi_1 = {z\over n} \}$. Then the function $f$ is defined in a neighborhood of every $C_n$. On the other hand, the limit curve of the sequence is $C_0 = \cc\pp^1 \cup \Delta_\infty$, where $\Delta_\infty \deff \{ \eta_2=0, z=\infty, |\eta_1| \le 1\}$. In particular, $f$ does not extend in a neighborhood of $C_0$.

[Appendix IV.]{}

[Complex Points and Stein Neighborhoods.]{}

**The condition $c_1(X)[M]>0$ (in the case of imbedded surfaces, i.e., when $\delta =0$) in Theorem 4.1 cannot be dropped. In \[N-1\] Nemirovsky, using results of Eliashberg-Kharlamov and Forstneric, showed that any imbedded complex curve $C$ with $c_1(X)[M]\le 0$ can be perturbed to an imbedded surface $M$ which has a basis of Stein neighborhoods.**

*A4.1. Complex Points of Real Surfaces in Complex Surfaces.*

Let $S\subset X$ be a real surface embedded in a complex surface. A point $p\in S$ is said to be [*complex*]{} if the tangent plane $T_p S\subset T_p X$ is a complex line.

If locally $S=\Gamma_f$ is the graph of a smooth complex valued function $f$, then a point $(z,f(z))\in S$ is complex if and only if ${\d f\over \d\bar z}(z)=0$. It is therefore not difficult to show that for generic embeddings the complex points are isolated, and near each complex point there exists a complex system of coordinates $(z,w)$ such that $S$ is given by the equation $$w=|z|^2+{\gamma\over 2}(z^2+\bar z^2)+\bar o(|z|^2),\quad\gamma\in\R.
\eqno (A4.1.1)$$ If $|\gamma|<1$, then $p=(0,0)$ is called an [*elliptic*]{} complex point, and if $|\gamma|>1$, then it is called [*hyperbolic*]{}. (The ‘parabolic’ case $|\gamma|=1$ is not generic.) Note that the property of being elliptic or hyperbolic is invariant under biholomorphic transformations.

It is easy to check that if $S$ is a graph $\Gamma_f$, then elliptic and hyperbolic complex points correspond to positive and negative zeros of ${\d f\over \d\bar z}$, respectively.

Assume now that an embedded real surface $S\subset X$ in the general position is compact and oriented. Then there are two orientations of $T_pS$ at each complex point $p\in S$. The first one is the orientation of $S$ and the second one is the canonical complex orientation of $T_pX$. A complex point is called positive if these two orientations coincide; otherwise, it is called negative.

Denote by $e_\pm=e_\pm(S)$ and $h_\pm=h_\pm(S)$ the numbers of positive and negative elliptic and hyperbolic complex points of $S$. Introduce [*Lai’s indices*]{} $I_\pm:=e_\pm-h_\pm$. It turns out that $I_\pm$ are topological invariants of the embedding.

Let $S\subset X$ be a compact oriented real surface in the general position in a complex surface $X$. Then $$\matrix
I_+ +I_-&=\chi(S)+S^2,\\
I_+ -I_-&=c_1(X)\cdot[S],
\endmatrix
\eqno(A4.1.2)$$ where $\chi(S)=2-2g(S)$ is the Euler characteristic of $S$, $S^2=[S]\cdot[S]$ is the self-intersection index of $S$ in $X$, and $c_1(X)\in \sfh^2(X, \R)$ is the first Chern class of $X$.

Proof. We shall only sketch the proof. For the first formula, choose a tangent vector field $\xi\in TS$, twist it by the complex structure $J$ on $X$ and consider the projection $\eta=\pi(J\xi)\in NS$ to the normal bundle of $S$. The zeros of $\eta$ are exactly the zeros of $\xi$ and the complex points of $S$. Careful inspection of the signs (using local equations $(*)$) yields the first formula.

To prove the second formula, pick a non-vanishing 2-form $\omega$ on $S$ and observe that for a local frame $\xi_1,\xi_2\in TS$, the expression $\omega(\xi_1,\xi_2)^{-1}\xi_1\wedge_\C\xi_2$ gives a well-defined section of the linear bundle $\Lambda^2_\C(TX)|_S$. The zeros of this section are the complex points of $S$, because at these points $\xi_1$ and $\xi_2$ are linearly dependent over $\C$. It remains to calculate the signs of these zeros and to use the definition of $c_1(X)$.

Sometimes it is convenient to write Lai’s formulae in the following way: $$I_\pm(S)={1\over 2}(\chi(S)+S^2\pm c_1(X)\cdot[S]).\eqno(A4.1.3)$$ For example, if $C\subset X$ is a non-singular compact complex curve, then after a small real perturbation we obtain an embedded real surface $S$ that obviously has only positive complex points. Thus, $I_-(S)=0$ and therefore $-\chi(C)=C^2-c_1(C)\cdot[C]$, which is precisely the adjunction formula for $C$.

One can also give the following geometric intepretation of Lai’s indices. Consider the Grassmanian $Gr(X,\R,2)$ of oriented real 2-planes in $TX$. It has two natural oriented submanifolds ${\frak I}_\pm$ formed by complex lines with the complex and anti-complex orientation, respectively. Then $I_\pm=\tau(S)\cdot {\frak I}_\pm$, where $\tau:S\to Gr(X,\R,2)$ is the tangential Gauss map. This again can be proved by calculating local signs.

*A4.2. Cancellation of Complex Points. Lai’s indices $I_\pm$ are essentially the only topological invariants of the position of embedded real surfaces with respect to the complex structure. The following theorem of Kharlamov and Eliashberg shows that it is possible to cancel an elliptic and a hyperbolic complex point with the same sign. In particular, if $I_+=I_-=0$, then the surface is isotopic to a totally real one. This can be regarded as a partial case of Gromov’s $h$-principle for $CR$-embeddings.*

Let $S\subset X$ be an embedded real surface in a general position, and assume that $e_+(S)h_+(S)>0$. Then there exists an isotopic embedded surface $S'\subset X$ such that $e_+(S')=e_+(S)-1$ and $h_+(S')=h_+(S)-1$.

Join the pair of points that we want to remove by a smooth path in $S$ and choose a finite covering of this path by coordinate neighborhoods such that in each chart $S$  there is the graph of a smooth function with natural orientation. Now to move complex points along this path and to eventually ‘cancel’ them, it suffices to prove the following local result.

Let $f:\bar\Delta\to\C$ be a smooth function in a neighborhood of the unit disk. Assume that ${\d f\over \d\bar z}$ has two ordinary zeros $a,b\in\Delta$ of opposite sign. Then there exists a function $g:\bar\Delta\to\C$ that is $C^0$ close to $f$ in the disk, coincides with $f$ near the boundary and has a non-vanishing ${\d g\over \d\bar z}$.

Conversely, if ${\d f\over \d\bar z}$ does not vanish, then $g$ can be chosen so that ${\d g\over \d\bar z}$ has two zeros of opposite sign at the given points $a,b\in\Delta$.

Proof. By obvious topological reasons there exists a smooth function $\psi$ supported in a thin neighborhood of the segment $[a,b]$ such that ${\d f\over \d\bar z}+\psi$ does not vanish in $\Delta$. Note also that we may choose $\psi$ so that $\|\psi\|_{C^0}\le 2\|{\d f\over\d\bar z}\|_{C^0}$. Hence, if the area of $\supp(\psi)$ is small enough, then the Cauchy-Green transform $T_{CG}\psi $ is $L^{1,p}$-small by Calderon-Zygmund inequality and the graph of $f+T_{CG}\psi $ is totally real. It remains to multiply $T_{CG}\psi$ by a cut-off function $\phi $ so that $\phi T_{CG}\psi \equiv 0$ near the boundary of $\Delta$. This will not spoil $\d(f+\phi T_{CG}\psi)\over \d\bar z$, because $T_{CG}\psi$ is holomorphic near the boundary and $C^0$-small.

The second part of the lemma is proved in exactly the same way.

The following statement immediately follows from Lai’s formulae and Cancellation theorem.

Corollary A4.2.2. [*Let $\ss^2\subset \cc^2$ be a symplectically imbedded two-sphere in complex projective plane, which is homologous to the projective line. Then there is an imbedded two-sphere $\tilde \ss^2\subset 
\cc^2$, which is $C^0$-close to $\ss^2$ and has precisely three points with complex tangents. Moreover, those points are positive and elliptic.* ]{}

*A4.3. Neighborhoods of Real Surfaces. Now we can give a construction of Stein neighborhoods of deformations of real surfaces satisfying certain topological conditions.*

Let $S\subset X$ be an embedded real compact oriented surface in a complex surface. If $S$ satisfies both inequalities $$2I_\pm=\chi(S)+S^2\pm c_1(X)\cdot[S]\leq 0, \eqno(A4.3.1)$$ then there exists an isotopic embedded real surface with a basis of strictly pseudoconvex Stein neighborhoods.

This result was proved in [@F] for $X=\C^2$. The case where $S$ is a smooth deformation of a complex curve was used in [@N1] and the general case in [@N2].

Proof. By the Cancellation Theorem there exists an isotopic real surface, denoted by $S$, that has only [*hyperbolic*]{} complex points.

Let $p\in S$ be a totally real or a hyperbolic complex point. It is not difficult to see from the local equations of $S$ that there exist coordinate neighborhoods $U\supset\!\supset V\ni p$ and a non-negative smooth function $\psi$ in $U$ with the following properties:

“1)” $S\cap U=\{x\in U\mid \psi(x)=0\}$;

“2)” $d\psi\equiv 0$ on $S\cap U$;

“3)” $\psi$ is strictly plurisubharmonic in $U\setminus(S\cap V)$.

Choose a finite covering of $S$ by $V_j\subset\!\subset U_j$. Let, furthermore, $\chi_j$ be smooth non-negative functions supported in $U_j$ such that

“1)” $\chi_j\equiv 1$ on $\bar{V_j}$;

“2)” $\displaystyle\sum_{|\alpha|\leq 2} |D^\alpha\chi_j|\leq A\chi_j$ everywhere in $U_j$ for some constant $A$. Such functions indeed exist: one may first pick functions satisfying the first condition and then raise them to the third power.

Now by the Leibnitz rule we have $$L(\chi_j\psi_j)=i\,\d\bar{\d}(\chi_j\psi_j)=
i\bigl(\psi_j\d\bar{\d}\chi_j+\d\psi_j\wedge\dbar\chi_j
+\d\chi_j\wedge\dbar\psi_j+\chi_j\d\bar{\p}\psi_j\bigr).$$ It readily follows that the Levi form $L(\chi_j\psi_j)$ is non-negative in some neighborhood $W_j\supset S\cap U_j$ and positive in $(W_j\setminus S)\cap V_j$.

Consider the function $\Phi:=\displaystyle\sum_j\chi_j\psi_j$. By construction

“1)” $\Phi\equiv 0$ on $S$ and $\Phi>0$ outside of $S$;

“2)” $\Phi$ is strictly plurisubharmonic in a punctured neighborhood of $S$. Thus, the sets $U_\eps=\{\Phi<\eps\}$, where $\eps>0$ is a regular value of $\Phi$, form a basis of strictly pseudoconvex Stein neighborhoods of $S$.

It is worth observing that an embedded real surface $S\subset X$ with elliptic complex points cannot have a Stein neighborhood basis. Indeed, according to Bishop for an elliptic point $p\in S$, there exists a non-trivial continuous family of holomorphic discs $f_t:(\Delta,\d\Delta)\to (X,S)$ with $f_0\equiv p$. By the continuity principle every holomorphic function in a neighborhood of $S$ extends holomorphically along this family, and therefore a sufficiently small neighborhood of $S$ cannot be Stein.

In fact, it can be proved that, at least when $S^2\geq 0$, a surface $S$ that ‘potentially’ has elliptic complex points (which means that one of the indices $I_\pm(S)$ is positive) does not admit a (not necessarily small) Stein neighborhood (see [@N2] and [@N3]).

Now, consider a complex surface $X$ and a smooth, compact complex curve $C\subset X$.

Corollary A4.3.2. [*If $c_1(X)\cdot[C]\le 0$ then a generic embedded real surface $S$ irotopic to $C$ has a basis of Stein neighborhoods. In particular such $S$ has no non trivial envelope of meromorphy.* ]{}

Proof. [If $c_1(X)\cdot[C]\le 0$, then a generic embedded real surface $S$ isotopic to $C$ satisfies $I_\pm(S)\le 0$, because of the adjunction formula and Lai’s formulae. The statement now follows from the Theorem A4.3.1. ]{}

Appendix V. Seiberg-Witten Invariants and Envelopes.

[A5.1. The Genus Estimate.]{} Throughout this Appendix $X$ will denote a compact, real, oriented manifold of dimension $4$. Fix some $c\in H^2(X, \rr )$. Let $M$ be a closed, oriented surface of genus $g(M)$ imbedded in $X$. For a riemannian metric $g$ on $X$ denote by $s_g$ the scalar curvature of $g$, i.e. $s_g=g_{ij}R^{ij}={\sl tr}R_g$-trace of the Ricci curvature. In the case when $X=M$ is a compact surface the Gauss-Bonnet theorem tells us that

$${1\over 4\pi }\int_Ms_gd\mu_g = 2-2g(M), \eqno(A5.1.1)$$ where $\mu_g$ is the area form associated to $g$.

The following statement is due to P. Kronheimer.

Theorem A5.1.1.

*Assume that $[M]^2=0$ and that for every riemannian metric $g$ on $X$ one can find a two-form $\omega $ representing $c$ such that*

$$\Vert \omega \Vert_{L^2_g}\le {1\over 4\pi }\Vert s_g\Vert_{L^2_g} + K(X),
\eqno(A5.1.2)$$ where constant $K(X)$ depends only on the topology of $X$. Then

$$\vert c[M]\Vert \le \max \bigl\{ 2g(M)-2,0\bigr\} .\eqno(A5.1.3)$$

Proof.

Take some neighborhood $U$ of $M$ which is diffeomorphic to $M\times D$, where $D$ is a disk in $\rr^2$ with coordinates $x,y$. Denote by $g_M$ the riemannian metric on $M$ of constant scalar curvature $s_g$ equal to $4\pi (2-2g(M))$.

(A5.1.1) tells us that if $M$ is not a torus, then the ${\sf area}_{g_M}M=1$. In the case of torus we just take the flat metric with total area one. For real positive $R$ consider the metrics $g_{D,R} = R(dx^2 + dy^2)$ on $D$.

Take now the following metric on $X$:

$$ g\_R = g\_Mg\_[D,R]{} & on $U$ g & away from $U$ $$

Let $\omega_R$ be the harmonic representative of $c$ with respect to the metric $g_R$. Harmonic form minimases the $L^2$-norm in its cohomology class. So by the assumption (A5.1.2) of our theorem

$$\Vert \omega_R\Vert_{L^2_{g_R}}\le (2g(M)-2)\sqrt{\pi }R + O(1). \eqno(A5.1.4)$$ Really, $\Vert \omega_R\Vert_{L^2_{g_R}}\le {1\over 4\pi }\vert s_{g_R}\Vert
+ O(1) $ $= \bigl(\int_{M\times D}[4\pi (2g-2)]^2d\mu_{g_R}\bigr)^{{1\over 2}}= 
{1\over 4\pi }\bigl(\pi R^2\cdot 4\pi (2g(M)-2)\bigr)^{{1\over 2}} + O(1) $ $ = \sqrt{\pi } \cdot R\cdot (2g(M)-2) + O(1)$.

Further

$$\Vert \omega_R\Vert_{L^2_{g_R}}\ge \vert c[M]\vert \sqrt{\pi }R.
 \eqno(A5.1.5)$$ Really, $\Vert \omega_R\Vert^2_{L^2_{g_R}}\ge \int_D\Vert \omega\mid_{M\times 
(x,y)}\Vert^2_{L^2_{g_M}}R^2dxdy$ because $\omega\mid_{M\times (x,y)}$ is only one of the coefficients of $\omega $. $\Vert \omega\mid_{M\times 
(x,y)}\Vert^2_{L^2_{g_M}}$ is bounded from below by the $L^2$-norm of the corresponding harmonic representative. Note that the harmonic representative of $\omega\mid_{M\times(x,y)}=c\mid_M$ is just $c[M]*_{g_M}1$ because Hodge $*_{g_M}$ commutes with the Laplacian: $\Delta_{g_M}: \Delta_{g_M}(*_{g_M}1)=*_{g_M}(\Delta_{g_M}1)=0$. So the $L^2$-norm of this harmonic representative and thus of $\omega\mid_{M\times (x,y)}$ is bounded from below by $c[M]$. Inequality (A5.1.5) follows.

Inequalities (A5.1.4) and (A5.1.5) clearly imply inequality (A5.1.2).

This theorem reduces the problem of minimazing the genus of an imbedded surfaces in a fixed homology class to the a priori estimate (A5.1.2). This estimate can be obtained if the Seiberg-Witten invariant $SW(X,c)$ for our cohomology class $c$ is not zero.

[A5.2. The use of Seiberg-Witten Invariant.]{}

The intersection form on the second cohomology space $H^2(X,{\Bbb R})$ of a compact $4$-manifold $X$ is symmetric, and hence it has Sylvester indices $b^\pm(X)$. By definition, the signature of $X$ is the signature $\sigma(X)=b^+ -b^-$ of its intersection form.

If $X$ is a Kähler complex surface, then the maximal positive subspace in $H^2(X,{\Bbb R})$ is spanned by the class of the Kähler form $\omega$ and by the classes of the real and imaginary parts of holomorphic $2$-forms on $X$. In particular, $b^+(X)=1+2\dim_\cc\Omega^2(X)$ is an odd positive integer.

Suppose now that for our $4$-manifold $X$ we have that $b^+(X)>1$. Let us pick a characteristic class $c\in H^2(X,{\Bbb Z})$, that is, an integral cohomology class which is congruent $\mathop{\roman{mod}} 2$ to the second Stiefel–Whitney class $w_2(X)\in H^2(X,{\Bbb Z}_2)$. (For example, the first Chern class of an almost-complex structure $J$ on $X$ is characteristic.)

In this situation one can define the [*Seiberg–Witten invariant*]{} $SW(X,c)\in{\Bbb Z}/2{\Bbb Z}$ of the smooth oriented manifold $X$ (if $\psi$ is an orientation-preserving diffeomorphism of $X$, then $SW(X,\psi^*c) =SW(X,c)$).

Roughly speaking, the invariant $SW(X,c)$ is non-zero if the moduli space of gauge equivalence classes of solutions of the Seiberg–Witten differential equations for a Riemannian metric on $X$ has non-zero homology class in the configuration space. For the precise definition of the invariant, the reader is referred to the original paper \[Wt\] and to the books \[Wg\] and \[Mr\].

Theorem of Witten - I.

*Let $X$ be a compact Kähler surface with $b^+>1$. Then the moduli space of solutions of the (suitably perturbed) Seiberg–Witten equations can be identified with the space of all effective divisors that are Poincaré dual to the cohomology class $\frac12\bigl(c-c_1(X)\bigr)$.*

In particular, $SW(X,c_1(X))=1$.

Really, if $c$ is chosen to be equal to $c_1(X)$ there is only one divisor Poincaré dual to the zero class, namely the zero divisor. Therefore the moduli space of the solutions of $SW$-equations in this case is a singleton.

Remark. [In the opposite direction, the Seiberg–Witten invariants of $X$ can be used to produce complex curves with given cohomology class, namely, if $SW(X,c)\ne 0$, then there is a complex curve dual to $\frac12\bigl(c-c_1(X)\bigr)$. This statement is due to Taubes, see \[Ta\]. ]{}

Suppose that, for a our pair $(X,c)$ the invariant $SW(X,c)$ is non-zero. Then, by definition, for every metric $g$ on $X$, there is a solution of the Seiberg–Witten equations. Using the properties of the equations, one can obtain [*a priori*]{} estimates of the solutions in terms of the metric $g$. We do not write out the equations themselves and only state the key estimate of this type, which is essentially due to Witten (see also \[Mr\] and \[Kr\]).

Theorem of Witten - II. [*If $SW(X,c)\ne 0$ then for every Riemannian metric $g$ on $X$ with there is a closed two-form $\eta$ representing this class in the de Rham cohomology and such that $$\Vert \eta \Vert_{L_g^2}\le {1\over 4\pi }\Vert s_g\Vert_{L^2_g} + K(X),
\eqno(A5.2.1)$$ where $K(X)$ is a constant depending only on the topology of $X$.* ]{} As a consequence we see that

Corollary A5.2.1. [*The number of classes $c\in H^2(X,\zz )$ with non-trivial Seiberg–Witten invariants $SW(X,c)$ is finite.* ]{} Proof.

Indeed, for a fixed metric $g$, the harmonic representatives of these integer classes are uniformly bounded.

Corollary A5.2.2 [*Let $X$ be a compact Kähler surface with $b^{+} >1$. Let $M\subset X$ be an embedded real surface with non-negative self-intersection $M^2\ge0$. Then $$\left| c_1(X)\cdot M \right|+M^2\le \max\{0,2g(M)-2\}. \eqno(A5.2.2)$$* ]{} Proof.

Let us choose an orientation on $M$ so that $c_1(X)\cdot M\le 0$. We blow up the surface $X$ at $d=M^2$ distinct points and denote the blown-up surface by $\widetilde X$, the exceptional curves by $E_j$, and the corresponding blow-down by $\pi:\tilde X\to\nomathbreak X$. Then $c_1(\tilde X)=\pi^*c_1(X)-\sum_j [E_j]$.

Let $\Sigma$ be the interior connected sum of $M$ with $\overline{E_j}$ in $\widetilde X$. Then $[\Sigma]=\pi^*[M]-\sum_j [E_j]$, and therefore $\Sigma^2=0$. Observe that the genus of $\Sigma$ is equal to that of $M$.

By Theorem A5.1.1 we have that $$\max\{0,2g(M)\} \ge\bigl|c_1(\tilde X)\cdot\Sigma\bigr|
=\left| c_1(X)\cdot M-M^2\right|=\left| c_1(X)\cdot M \right|+M^2.$$

Corollary A5.2.2, as it is stated, makes no difference between tori and spheres, i.e. $\max\{0,2g(M)\} =0$ in both cases. However for spheres it can be improved. We shall derive now the following result of Kronheimer and Mrowka \[Kr-M\] and Morgan, Szabó, and Taubes \[M-S-T\].

Corollary A5.2.3. [ *An embedded oriented real surface $M\subset X$ with non-negative self-intersection index in a compact Kähler surface with $b^+(X)>1$ satisfies the inequality $$M^2+\bigr|c_1(X)\cdot M\bigl|\le 2g(M) - 2 \eqno(A5.2.3)$$ provided that $M$ is not a $2$-sphere with trivial real homology class.* ]{} Proof.

The case $g(M)\ge 1$ is covered by (A5.2.2). So let $M$ be a sphere. (A5.2.2) tells that $M^2=0$ and $c_1(X)\cdot M=0$.

Let us blow up the surface $X$ at one point. Let $E\subset \tilde X$ be the exceptional curve. For every $n\ge 1$, we consider the interior connected sum $M_n=E\#nM$. This is an embedded two-sphere such that $M_n^2=-1$ and $c_1(\tilde X)\cdot M_n=1$.

Let us use the existence of an orientation-preserving diffeomorphism $\psi\:\tilde X\to\tilde X$ (with support in a neighbourhood of $M_n$) that maps $M_n$ into itself with opposite orientation. We have $\psi^*c_1(\tilde X)=c_1(\tilde X)+2[M_n]$. If the class $[M]\in H^2(X,\rr )$ were non-zero, this would give us an infinite set of cohomology classes on $\tilde X$ with non-zero Seiberg–Witten invariant, which contradicts Corollary A5.2.1.

Remark. [ By the usual adjunction formula, a complex curve $C$ with $C^2\ge 0$ violates our inequality if and only if $c_1(X)\cdot C>0$. This situation occurs in $\cc\pp^2$ and other complex surfaces with negative canonical class indeed. For this reason, it is impossible to drop the assumption $b^+>1$ in the above theorem. ]{}

In A5.3 we shall repeatedly use the fact that complex curves in complex surfaces with $c_1(X)>0$ violate the “general” form of the adjunction inequality.

[A5.3. The Genus Estimate on Stein Surfaces and Envelopes.]{}

We start from the following

Theorem of Stout. [ *Let $K\subset Y$ be a compact set in a Stein manifold. Then there is a neighbourhood $V$ of $K$ and biholomorphic imbedding $h$ of $V$ onto an open subset in an affine algebraic variety $X$. Moreover, if $Z\subset Y$ is a complex submanifold, then one can find such $h$ that $Z\cap V$ will be mapped onto the intersection of an algebraic subvariety in $X$ with $h(V)$.* ]{}

We shall see now that imbedded real surfaces in Stein surfaces remind real surfaces in compact Kähler surfaces with $b^+>1$.

Theorem A5.3.1.

*Let $M\subset Y$ be an imbedded oriented real surface in a Stein surface with non-negativ self-intersection. Suppose that $M$ is not a $2$-sphere with trivial homology class. Then $$S^2+\bigr|c_1(X)\cdot S\bigl|\le 2g-2. \eqno(A5.3.1)$$*

Proof.

We may always assume that the homology class of $M$ is non-zero because our inequality becomes trivial if $[M]=0$ and $M$ is not a sphere. Hence, there is a non-singular (non-compact) complex curve $L\subset Y$ such that $L\cdot M\ne 0$. Indeed, $\sfh_2(X,\zz)$ is torsion-free for any Stein surface, and hence by the Poincaré duality we can find a class $c\in H^2(Y, \zz )$ with $c\cdot M\ne 0$. Since $Y$ is a Stein surface, it follows that $c=c_1({\Cal L})$ for a holomorphic line bundle $\Cal L$. Then $L$ is the divisor of a generic holomorphic section of $\Cal L$.

Using the algebraic approximation theorem of Stout we obtain an open imbedding of a neighbourhood of $M$ into an affine part $X_a$ of a compact algebraic surface $X$ such that $L$ is mapped to an algebraic curve $C\subset X$ so that $C\cdot Y=L\cdot M\ne 0$. Therefore, the real homology class of $M$ remains non-trivial after the algebraization.

Denote by $H=X\setminus X_a$-“the curve at infinity”. We have non-zero classes $[M]$ and $[H]$ in $H^2(Y,\rr )$ such that $[M]^2\ge 0$,  $[H]^2>0$, and $[H]\cdot [M]=0$. Since the intersection form is non-degenerate, it immediately follows that $b^+(Y)>1$. Hence, the desired inequality is a consequence of Corollary A5.2.3.

Remarks.

**1. It follows from Theorem A4.3.1 that the inequality A5.3.1 is sharp, namely, an imbedded real surface satisfying this inequality is isotopic to a surface with a basis of Stein neighbourhoods.**

[**2.**]{} This theorem is also true for the surfaces $M$ with negative self-intersection, see \[N-4\].

[**3.**]{} Concerning a real surface with non-negative self-intersection index the Theorem A5.3.1 still holds for non-Stein strictly pseudoconvex domains as well. This can proved by the same argument (see \[N-2\] and \[N-3\]) by using the Lempert algebraic approximation theorem \[Le\] for such domains.

[**4.**]{} [*Immersed Surfaces*]{} We assume that a surface $S$ is immersed with $\kappa_+$ positive and $\kappa_-$ negative ordinary double points. Then, after embedding a neighbourhood of $S$ in an algebraic surface $Y$, we replace each positive double point with an embedded handle and perform a blow-up at each negative double point. This operation increases the genus by $\kappa_+$ and does not change $S^2$ and $c_1(Y)\cdot S$. Hence, Theorem A5.3.1 holds with $g$ replaced by $g+\kappa_+$.

Now we shall apply the adjunction inequalities (or Genus estimates) for imbedded real surfaces in Stein surfaces to the study of envelopes of holomorphy and meromorphy of domains in complex surfaces.

The main idea is fairly simple. Suppose that we can solve the Levi problem for domains over a complex surface $X$, that is to prove that if $D$ is a pseudoconvex domain over $X$ then $D$ is a Stein domain unless some very special case occurs. I.e. for example $D=X$ (this is the case of $X=\cc\pp^2$) or $X$ is foliated by complex curves (case of $X=\cc\pp^1\times \cc\pp^1$. Then Theorem A5.3.1 yields then topological restrictions on real surfaces in $D$. The below are due to Nemirovski, see \[N-2\].

Let $M\subset\cc\pp^2$ be an embedded oriented real surface in the complex projective plane. By definition, the degree of $M$ is the intersection index of $M$ with complex lines, $d=M\cdot [\cc\pp^1]$. We can choose the orientation of $M$ so that $d\ge 0$ and note that $M$ is homologous to zero if and only if $d=0$.

For an embedded surface of degree $d$, we have $M^2=d^2$ and $c_1(\cc\pp^2)\cdot M=3d$. In particular, if $S$ is a smooth complex curve, then the classical genus formula reads $g(M)=g_\Bbb C(d)=\frac12(d^2-3d+2)$.

Corollary A5.3.2.

*Let $M\subset \cc\pp^2$ be an imbedded oriented real surface of positive degree $d>0$ and genus $g$. If there is a non-constant holomorphic function in a neighbourhood of $M$, then $$g\ge\frac12(d^2+3d+2)=g_\Bbb C(d)+3d.$$*

Conversely, if $M$ satisfies this inequality, then it is isotopic to an embedded surface with a basis of Stein neighbourhoods.

Corollary: Vitushkin’s conjecture.

*If an embedded two-sphere in $\cc\pp^2$ is not homologous to zero, then every holomorphic function in a neighbourhood of this sphere is constant.*

Proof.

Let us consider a neighbourhood $U\supset S$. If there is a non-constant holomorphic function in $U$, then, by Theorem 1, the envelope of holomorphy $\tilde U\supset U$ is a Stein domain because it cannot coincide with $\cc\pp^2$. Hence, the desired inequality immediately follows from Theorem A5.3.1.

Conversely, if the inequality of the theorem is satisfied, then, by Theorem 4, there is an isotopic surface with a basis of Stein neighbourhoods.

Corollary A5.3.3.

*If $M\subset \cc\pp^2$ be an imbedded oriented real surface of positive degree $d>0$ and genus $g$ with*

$$g < \frac12(d^2+3d+2)=g_\Bbb C(d)+3d.$$ then the envelope of meromorphy of any neighborhood of $M$ consides with $\cc\pp^2$.

For a real oriented surface $M\subset\cc\pp^1\times\cc\pp^1$, we consider its “bidegree” $d=(d_1,d_2)$, that is, the pair of intersection indices with horizontal and vertical complex lines. Then $c_1(\cc\pp^1\times\cc\pp^1)\cdot S=2(d_1+d_2)$ and $S^2=2d_1d_2$.

A smooth complex curve $C\subset\cc\pp^1\times\cc\pp^1$ has “positive” bidegree $d\in{\Bbb Z}^2_+\setminus\{0\}$. The genus of $C$ is given by the formula $g_\cc(d)=d_1d_2-d_1-d_2+1$.

Corollary A5.3.4.

*Let $S\subset\cc\pp^1\times\cc\pp^1$ be an embedded oriented real surface of non-zero bidegree $d=(d_1,d_2)\in{\Bbb Z}^2\setminus\{0\}$ and genus $g$. If there is a non-constant holomorphic function in a neighbourhood of $S$, then either $$ gd\_1d\_2+|d\_1+d\_2|+1,d\_1d\_2&gt;0, $**$ $$ or $d_1d_2=0$ and the envelope of $S$ contains a vertical or a horizontal line.*

Conversely, if $S$ satisfies $(**)$, then it is isotopic to an embedded surface with a basis of Stein neighbourhoods.

Remark. [In particular, it is impossible to deform a complex curve in $\cc\pp^1\times\cc\pp^1$ other than a vertical or a horizontal line into a real surface with holomorphic functions in its neighbourhoods. ]{}

Proof.

By the Theorem of Fujita the envelope of holomorphy of a domain $U\subset\cc\pp^1\times\cc\pp^1$ either is a Stein domain or contains one of the fibers. We also note that if $\tilde U$ contains, say, a vertical line and intersects the other vertical lines, then $\tilde U$ coincides with the entire $\cc\pp^1\times\cc\pp^1$ by the usual Hartogs Theorem.

It remains to apply Theorems A5.3.1 and A4.3.1 in the same way as above.

Let us consider the direct product $X=\cc\pp^1\times Y$, where $Y$ is a non-compact Riemann surface. We recall that $H^2(Y,{\Bbb Z})=0$, and hence $H^2(X,{\Bbb Z})\cong H^2(\cc\pp^1,{\Bbb Z})\cong{\Bbb Z}$ by the Künneth formula. For a real oriented surface $S\subset X$, let $d=\deg\pi_{\cc\pp^1}$ be the topological degree of its projection to the $\cc\pp^1$-factor. Then $S^2=0$ and $c_1(X)\cdot S=2d$. We also note that we can orient $S$ so that $d\ge 0$.

By the classical Behnke–Stein theorem, $Y$ is a Stein manifold. Hence, Theorem A4.3.1 gives the following description of envelopes of holomorphy over $X$. The envelope of a domain $U\subset X$ either is a Stein domain or contains a fiber $\cc\pp^1\times\{y\}$. In the latter case, holomorphic functions in $U$ are constant in the $\cc\pp^1$-direction, and therefore the envelope is of the form $\cc\pp^1\times\pi_Y(U)$.

Using the same argument as above, we obtain the following result.

Corollary A5.3.5.

*Let $S\subset\cc\pp^1\times Y$ be an embedded oriented real surface of genus $g$ and positive degree $d>0$. If $g<d+1$, then every holomorphic function in a neighbourhood of $S$ extends to the set $\cc\pp^1\times\pi_Y(S)$.*

If $g\ge d+1$, then $S$ is isotopic to a surface with a Stein neighbourhood basis.

Remark. The results of this section are much more general then the [*Theorem 4.1*]{} in the case of $X=\cc\pp^2, \cc\pp^1\times \cc\pp^1$. For example, if a symplectic sphere $M\subset \cc\pp^2$ is in the homology class of the line, then the procedure of the proof of the Theorem 4.1 gives an isotopy of $M$ to a complex line. However there are knoted two-spheres in $\cc\pp^2$ in the homology class of the line (but they are of course non symplectic). From the other hand the Theorem 4.1 is valid for the spheres not nessessarily in compact Kähler surfaces, where the envelopes of holomorphy or meromorphy are not nessessarily Stein. And for a non-Stein envelope the existence of a compact curve is also a strong conclusion.

To our understanding of the subject a statement containing the both type of results is not avaiable by neither Gromov no Seiberg-Witten techniques.

[A5.4. Two-spheres in $\cc^2$.]{} In this section we apply Theorem A5.3.1 to embedded spheres in $\cc^2$ and its blow-ups. This enables us to show that an embedded 2-sphere in $\cc^2$ is homologous to zero in its envelope of holomorphy.

It follows from Theorem A5.3.1 that an embedded sphere $S$ with self-intersection index $S^2=0$ in a Stein complex surface $X$ is homologous to zero in $\sfh_2(X,\zz)$. For an embedded two-sphere $S\subset\cc^2$, the condition $S^2=0$ holds because $\sfh_2(\cc^2,\zz)=0$. Hence, $S$ is homologous to zero in every Stein domain over $\cc^2$ that contains a neighbourhood of $S$. Since the envelope of holomorphy of every domain in $\cc^2$ is a Stein domain, we obtain the following result \[N-3\].

Corollary A5.4.1. [ *An embedded two-sphere $S\subset\cc^2$ is homologous to zero in the envelope of holomorphy of every neighbourhood $U\supset S$.* ]{}

One can note that the theorem does not work for [*immersed*]{} spheres. Indeed, a simple modification of the results in Appendix IV for immersed surfaces (we must replace $S^2$ in Lai’s formulas by the Euler number of the normal bundle) shows that there is an immersed totally real 2-sphere in $\cc^2$ with a single positive double point. This sphere has a basis of Stein neighbourhoods in each of which this sphere is certainly not homologous to zero.

[A5.5. Attaching complex disks to strictly pseudoconvex domains.]{}

Let $U$ be a relatively compact domain with smooth strictly pseudoconvex boundary in a Stein surface. We say that an embedded analytic disk $D\subset X$, smooth up to the boundary, is attached to $U$ (from the outside) if $D\subset X\setminus U$ and $\partial D\subset\partial U$. We shall always assume that $D$ is transversal to $\partial U$.

For example, the disk $\{|z|\le1,\,w=0\}\subset{\Bbb C}^2$ is attached to the boundary of the $\eps$-neighbourhood of the circle $\{|z|=1+\eps,\,w=0\}$. We note that the boundary of this disk is not homologous to zero in $U$.

A more intricate example can be obtained as follows. Let us consider a totally real torus $T\subset{\Bbb C}^2$. Reversing the proof of the Cancellation Theorem, we create an elliptic and a hyperbolic point in $T$. Let us replace a neighbourhood of the elliptic point by a complex disk and observe that this disk is attached to a strictly pseudoconvex neighbourhood of the remaining part of the torus. The boundary of this disk bounds a real surface with one handle inside the domain. An important observation (due to Forstnerič \[F\]) is that, by Lai’s formulas, this construction does not work for a sphere. In fact, it is impossible to find an analytic disk and a disk with only hyperbolic complex points with common boundary and without common interior points.

We generalize these examples following \[N-3\] and prove that the boundary of a holomorphic disk attached to a strictly pseudoconvex domain in ${\Bbb C}^2$ is never sliced by a smooth disk inside the domain.

Theorem A5.5.1. [ *Suppose that an analytic disk $D$ is attached to a strictly pseudoconvex domain $U$ in a Stein surface $X$ with $\sfh_2(X,\rr)=0$. Then there is no smooth disk inside of $U$ that has the same boundary.* ]{}

Proof.

We assume that there is a smooth disk in $U$ with boundary $\partial D$. Smoothing the union of our two disks, we obtain an embedded sphere $S\subset X$. The naïve idea is to paste a tubular neighbourhood of $D$ to $U$ so that the result is a Stein domain. Then $S$ is not homologous to zero in this domain, which contradicts Theorem 9. The following basic example shows first that such a pasting is impossible and then provides a remedy for this problem.

Let us consider the standard disk $D=\{w=0,\,|z|\le 1\}$ in ${\Bbb C}^2$ with a strictly pseudoconvex “collar” $C=\{ |w|^2+(|z|-a)^2\le (a-1)^2,\,|z|\le 1+\eps\}$ for some $a>1$ and $\eps>0$. If we glue a small neighbourhood of $D$ to the collar, then the interior of this set contains a Hartogs figure with appropriate “walls,” and therefore this interior is not a Stein domain. However, we can construct a Stein neighbourhood “pinched” at the origin.

To see this, we note that, in logarithmic coordinates $\xi=\ln|z|$, $\eta=\ln|w|$, the domain on the $|z|>1$ side of $C$ corresponds to the subgraph of a smooth convex function $\eta=f(\xi)$, $\xi\in(0,\ln(1+\eps)]$. The $\eta$-axis is clearly a vertical asymptote of $f$. Let us choose a point $\xi_0\in(0,\ln(1+\eps)]$ such that the slope of $T_{\xi_0}\Gamma_f$ is a positive integer $n\ge 1$, and replace $f$ by this tangent for all $\xi\le\xi_0$. The subgraph of this new function defines a neighbourhood of $D\setminus\{0\}$ that is logarithmically convex, and hence a Stein domain. Near the origin, this neighbourhood is of the form $|w|\le K|z|^n$, $K>0$.

If we blow up the origin in ${\Bbb C}^2$, that is, make the change of variables $z=z'$, $w=w'z'$, then the proper preimage of our neighbourhood becomes $|w'|\le K|z'|^{n-1}$. Thus, after $n$ blow-ups, we obtain an ordinary neighbourhood of the proper preimage of $D\cup C$. By construction, this domain is locally pseudoconvex and intersects only the last exceptional curve by a disk. Hence, it is a Stein domain by the remark after Takeuchi’s.

The situation of the theorem is reduced to this model by a suitable choice of coordinates in a pinched neighbourhood of $D$ and by a small perturbation of $U$. Thus, we have the following construction of Stein neighbourhoods.

Lemma A5.5.2. [ *Suppose that an analytic disk $D$ is attached from the outside to a strictly pseudoconvex domain $U$ in a Stein surface $X$. Then the proper preimage of $D\cup U$, after several blow-ups at a point $p\in D$, is contained in a Stein domain.* ]{}

To complete the proof of the theorem, we recall that $\sfh_2(X,\rr)=0$, and thus $S^2=0$ and $c_1(X)\cdot S=0$. Consequently, for the proper preimage $\tilde S\subset\tilde X$ of the sphere $S$, after $n\ge 1$ blow-ups at $p\in S$ we have ${\tilde S}^2=-n$ and $|c_1({\tilde X})\cdot{\tilde S}|=n$, which contradicts the inequality of Theorem A5.3.1.

Corollary: Vitushkin’s conjecture. [ *It is impossible to attach an analytic disk from the outside to a strictly pseudoconvex domain in ${\Bbb C}^2$ diffeomorphic to the ball.* ]{}

Proof. The idea is to “reflect” the disk inside the domain by inversion.

Let $U\subset{\Bbb C}^2$ be a strictly pseudoconvex domain in ${\Bbb C}^2$ diffeomorphic to the ball. More precisely, we assume that there is a diffeomorphism $\psi\:\bar U\to\bar B$ of manifolds with boundary. Then $\psi$ extends to a diffeomorphism $\Psi$ of ${\Bbb C}^2$ that maps $U$ to the standard ball $B$.

If a holomorphic disk $D$ is attached to $U$ from the outside, then its image $\Psi(D)$ is a smooth disk attached to $B$. Applying the inversion with respect to $\partial B$, we obtain a smooth disk $D'$ in $B$ with the same boundary. The preimage $\Psi^{-1}(D')$ is a smooth disk in $U$ with boundary $\partial D$, which contradicts the previous theorem.

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2 Institute of Applied Problems &Institute of Applied problemsof Mechanics and Mathematics &of Mechanics and MathematicsUkrainian Acad. Sci., &Ukrainian Acad. Sci.,vul. Naukova 3b, 290053 L’viv &vul. Naukova 3b, 290053 L’viv Ukraine &Ukraine U.F.R. de Mahtématiques & Ruhr-Universität Bochum Université de Lille-I & Mathematisches Institut

Villeneuve d’Ascq Cedex & Universitätsstrasse 150 59655 France & NA 4/67 44780 Bochum Germany ivachkovgat.univ-lille1.fr & sewacplx.ruhr-uni-bochum.de
