Differential forms on a Riemann surface

In mathematics, differential forms on a Riemann surface are an important special case of the general theory of differential forms on smooth manifolds, distinguished by the fact that the conformal structure on the Riemann surface intrinsically defines a Hodge star operator on 1-forms (or differentials) without specifying a Riemannian metric. This allows the use of Hilbert space techniques for studying function theory on the Riemann surface and in particular for the construction of harmonic and holomorphic differentials with prescribed singularities. These methods were first used by in his variational approach to the Dirichlet principle, making rigorous the arguments proposed by Riemann. Later found a direct approach using his method of orthogonal projection, a precursor of the modern theory of elliptic differential operators and Sobolev spaces. These techniques were originally applied to prove the uniformization theorem and its generalization to planar Riemann surfaces. Later they supplied the analytic foundations for the harmonic integrals of . This article covers general results on differential forms on a Riemann surface that do not rely on any choice of Riemannian structure.
Hodge star on 1-forms.
On a Riemann surface the Hodge star is defined on 1-forms by the local formula
It is well-defined because it is invariant under holomorphic changes of coordinate.
Indeed, if is holomorphic as a function of ,
then by the Cauchy–Riemann equations and . In the new coordinates
so that
proving the claimed invariance.
Note that for 1-forms and 
In particular if then
Note that in standard coordinates
Recall also that
so that
The decomposition formula_9 is independent of the choice of local coordinate. The 1-forms with only a formula_10 component are called (1,0) forms; those with only a formula_11 component are called (0,1) forms. The operators formula_12 and formula_13 are called the Dolbeault operators.
It follows that
The Dolbeault operators can similarly be defined on 1-forms and as zero on 2-forms. They have the properties
Poincaré lemma.
On a Riemann surface the Poincaré lemma states that every closed 1-form or 2-form is locally exact. Thus if "ω" is a smooth 1-form with then in some open neighbourhood of a given point there is a smooth function "f" such that in that neighbourhood; and for any smooth 2-form Ω there is a smooth 1-form "ω" defined in some open neighbourhood of a given point such that in that neighbourhood.
If is a closed 1-form on , then . If then and . Set
so that . Then must satisfy and . The right hand side here is independent of "x" since its partial derivative with respect to "x" is 0. So
and hence
Similarly, if then with . Thus a solution is given by and
Comment on differential forms with compact support. Note that if "ω" has compact support, so vanishes outside some smaller rectangle with and , then the same is true for the solution "f"("x","y"). So the Poincaré lemma for 1-forms holds with this additional conditions of compact support.
A similar statement is true for 2-forms; but, since there is some choices for the solution, a little more care has to be taken in making those choices.
In fact if Ω has compact support on and if furthermore , then with "ω" a 1-form of compact support on . Indeed, Ω must have support in some smaller rectangle with and . So vanishes for or and for or . Let "h"("y") be a smooth function supported in ("c"1,"d"1) with . Set : it is a smooth function supported in ("a"1,"b"1). Hence is smooth and supported in . It now satisfies . Finally set
Both "P" and "Q" are smooth and supported in with and . Hence is a smooth 1-form supported in with
Integration of 2-forms.
If Ω is a continuous 2-form of compact support on a Riemann surface "X", its support "K" can be covered by finitely many coordinate charts "U""i" and there is a partition of unity χ"i" of smooth non-negative functions with compact support such that ∑ χ"i" = 1 on a neighbourhood of "K". Then the integral of Ω is defined by
where the integral over "U""i" has its usual definition in local coordinates. The integral is independent of the choices here.
If Ω has the local representation "f"("x","y") "dx" ∧ "dy", then |Ω| is the density |"f"("x","y")| "dx" ∧ "dy", which is well defined and satisfies |∫"X" Ω| ≤ ∫"X" |Ω|. If Ω is a non-negative continuous density, not necessarily of compact support, its integral is defined by
If Ω is any continuous 2-form it is integrable if ∫"X" |Ω| < ∞. In this case, if ∫"X" |Ω| = lim ∫"X" ψ"n" |Ω|, then ∫"X" Ω can be defined as lim ∫"X" ψ"n" Ω. The integrable continuous 2-forms form a complex normed space with norm ||Ω||1 = ∫"X" |Ω|.
Integration of 1-forms along paths.
If "ω" is a 1-form on a Riemann surface "X" and "γ"("t") for is a smooth path in "X", then the mapping "γ" induces a 1-form "γ"∗"ω" on ["a","b"]. The integral of "ω" along "γ" is defined by
This definition extends to piecewise smooth paths "γ" by dividing the path up into the finitely many segments on which it is smooth. In local coordinates if and then
so that
Note that if the 1-form "ω" is exact on some connected open set "U", so that for some smooth function "f" on "U" (unique up to a constant), and "γ"("t"), , is a smooth path in "U", then
This depends only on the difference of the values of "f" at the endpoints of the curve, so is independent of the choice of "f". By the Poincaré lemma, every closed 1-form is locally exact, so this allows ∫γ ω to be computed as a sum of differences of this kind and for the integral of closed 1-forms to be extended to continuous paths:
Monodromy theorem. If "ω" is a closed 1-form, the integral can be extended to any continuous path "γ"("t"), so that it is invariant under any homotopy of paths keeping the end points fixed. 
The same argument shows that a homotopy between closed continuous loops does not change their integrals over closed 1-forms. Since , the integral of an exact form over a closed loop vanishes. 
Conversely if the integral of a closed 1-form "ω" over any closed loop vanishes, then the 1-form must be exact.
A closed 1-form is exact if and only if its integral around any piecewise smooth or continuous Jordan curve vanishes.
The above argument also shows that given a continuous Jordan curve "γ"("t"), there is a finite set of simple smooth Jordan curves "γ""i"("t") with nowhere zero derivatives such that
for any closed 1-form "ω". Thus to check exactness of a closed form it suffices to show that the vanishing of the integral around any regular closed curve, i.e. a simple smooth Jordan curve with nowhere vanishing derivative.
The same methods show that any continuous loop on a Riemann surface is homotopic to a smooth loop with nowhere zero derivative.
Green–Stokes formula.
If "U" is a bounded region in the complex plane with boundary consisting of piecewise smooth curves and "ω" is a 1-form defined on a neighbourhood of the closure of "U", then the Green–Stokes formula states that
In particular if "ω" is a 1-form of compact support on C then
since the formula may be applied to a large disk containing the support of ω.
Similar formulas hold on a Riemann surface "X" and can be deduced from the classical formulas using partitions of unity. Thus if is a connected region with compact closure and piecewise smooth boundary ∂"U" and "ω" is a 1-form defined on a neighbourhood of the closure of "U", then the Green–Stokes formula states that
Moreover, if "ω" is a 1-form of compact support on "X" then
To prove the second formula take a partition of unity "ψ""i" supported in coordinate charts covering the support of "ω". Then , by the planar result. Similarly to prove the first formula it suffices to show that
when "ψ" is a smooth function compactly supported in some coordinate patch. If the coordinate patch avoids the boundary curves, both sides vanish by the second formula above. Otherwise it can be assumed that the coordinate patch is a disk, the boundary of which cuts the curve transversely at two points. The same will be true for a slightly smaller disk containing the support of "ψ". Completing the curve to a Jordan curve by adding part of the boundary of the smaller disk, the formula reduces to the planar Green-Stokes formula.
The Green–Stokes formula implies an adjoint relation for the Laplacian on functions defined as Δ"f" = −"d"∗"df". This gives a 2-form, given in local coordinates by the formula
Then if "f" and "g" are smooth and the closure of "U" is compact
Moreover, if "f" or "g" has compact support then
Duality between 1-forms and closed curves.
Theorem. If "γ" is a continuous Jordan curve on a Riemann surface "X", there is a smooth closed 1-form "α" of compact support such that for any closed smooth 1-form "ω" on "X".
Corollary 1. A closed smooth 1-form "ω" is exact if and only if for all smooth 1-forms "α" of compact support.
Corollary 2. If "γ" is a continuous closed curve on a Riemann surface "X", there is a smooth closed 1-form "α" of compact support such that for any closed smooth 1-form "ω" on "X". The form "α" is unique up to adding an exact form and can be taken to have support in any open neighbourhood of the image of "γ".
Intersection number of closed curves.
The intersection number of two closed curves "γ"1, "γ"2 in a Riemann surface "X" can be defined analytically by the formula
where "α"1 and "α"2 are smooth 1-forms of compact support corresponding to "γ"1 and "γ"2. From the definition it follows that . Since "αi" can be taken to have its support in a neighbourhood of the image of γ"i", it follows that if "γ"1 and "γ"2 are disjoint. By definition it depends only on the homotopy classes of "γ"1 and "γ"2.
More generally the intersection number is always an integer and counts the number of times "with signs" that the two curves intersect. A crossing at a point is a positive or negative crossing according to whether "d"γ1 ∧ "d"γ2 has the same or opposite sign to , for a local holomorphic parameter "z" = "x" + "iy".
Holomorphic and harmonic 1-forms.
A holomorphic 1-form ω is one that in local coordinates is given by an expression "f"("z") "dz" with "f" holomorphic. Since formula_43 it follows that "dω" = 0, so any holomorphic 1-form is closed. Moreover, since ∗"dz" = −"i" "dz", ω must satisfy ∗"ω" = −"iω". These two conditions characterize holomorphic 1-forms. For if ω is closed, locally it can be written as "dg" for some "g", The condition ∗"dg" = "i" "dg" forces formula_44, so that "g" is holomorphic and "dg" = "g" '("z") "dz", so that ω is holomorphic.
Let "ω" = "f" "dz" be a holomorphic 1-form. Write "ω" = "ω"1 + "iω"2 with "ω"1 and "ω"2 real. Then "dω"1 = 0 and "dω"2 = 0; and since ∗"ω" = −"iω", ∗"ω"1 = "ω"2. Hence "d"∗"ω"1 = 0. This process can clearly be reversed, so that there is a one-one correspondence between holomorphic 1-forms and real 1-forms ω1 satisfying "d"ω1 = 0 and "d"∗ω1 = 0. Under this correspondence, "ω"1 is the real part of "ω" while "ω" is given by "ω" = "ω"1 + "i"∗"ω"1. Such forms ω1 are called harmonic 1-forms. By definition ω1 is harmonic if and only if ∗"ω"1 is harmonic.
Since holomorphic 1-forms locally have the form "df" with "f" a holomorphic function and since the real part of a holomorphic function is harmonic, harmonic 1-forms locally have the form "dh" with "h" a harmonic function. Conversely if "ω"1 can be written in this way locally, "d"∗ω1 = "d"∗"dh" = ("h""xx" + "h""yy") "dx"∧"dy" so that "h" is harmonic.
Remark. The definition of harmonic functions and 1-forms is intrinsic and only relies on the underlying Riemann surface structure. If, however, a conformal metric is chosen on the Riemann surface (see below), the adjoint "d"* of "d" can be defined and the Hodge star operation extended to functions and 2-forms. The Hodge Laplacian can be defined on "k"-forms as ∆"k" = "dd"* +"d"*"d" and then a function "f" or a 1-form "ω" is harmonic if and only if it is annihilated by the Hodge Laplacian, i.e. ∆0"f" = 0 or ∆1"ω" = 0. The metric structure, however, is not required for the application to the uniformization of simply connected or planar Riemann surfaces.
Sobolev spaces on T2.
The theory of Sobolev spaces on can be found in , an account which is followed in several later textbooks such as and . It provides an analytic framework for studying function theory on the torus C/Z+"i" Z = R2 / Z2 using Fourier series, which are just eigenfunction expansions for the Laplacian . The theory developed here essentially covers tori C / Λ where Λ is a lattice in C. Although there is a corresponding theory of Sobolev spaces on any compact Riemann surface, it is elementary in this case, because it reduces to harmonic analysis on the compact Abelian group . Classical approaches to Weyl's lemma use harmonic analysis on the non-compact Abelian group C = R2, i.e. the methods of Fourier analysis, in particular convolution operators and the fundamental solution of the Laplacian.
Let T2 = {("e""ix","e""iy": "x", "y" ∊ [0,2"π")} = R2/Z2 = C/Λ where Λ = Z + "i" Z.
For λ = "m" + "i" "n" ≅ ("m","n") in Λ, set . Furthermore, set "D""x" = −"i"∂/∂"x" and "D""y" = −"i"∂/∂"y". For α = ("p","q") set "D"α =("D""x")"p" ("D""y")"q", a differential operator of total degree |"α"| = "p" + "q". Thus , where . The ("e"λ) form an orthonormal basis in C(T2) for the inner product , so that .
For "f" in C∞("T"'2) and "k" an integer, define the "k"th Sobolev norm by
The associated inner product
makes C∞(T2) into an inner product space. Let "H""k"(T2) be its Hilbert space completion. It can be described equivalently as the Hilbert space completion of the space of trigonometric polynomials—that is finite sums —with respect to the "k"th Sobolev norm, so that "H""k"(T2) = {∑ "a"λ "e"λ : ∑ |"a"λ|2(1 + |λ|2)"k" < ∞} with inner product
As explained below, the elements in the intersection "H"∞(T2) = formula_47 "H""k"(T2) are exactly the smooth functions on T2; elements in the union "H"−∞(T2) = formula_48 "H""k"(T2) are just distributions on T2 (sometimes referred to as "periodic distributions" on R2).
The following is a (non-exhaustive) list of properties of the Sobolev spaces.