Beltrami equation

In mathematics, the Beltrami equation, named after Eugenio Beltrami, is the partial differential equation
for "w" a complex distribution of the complex variable "z" in some open set "U", with derivatives that are locally "L"2, and where "μ" is a given complex function in "L"∞("U") of norm less than 1, called the Beltrami coefficient. Classically this differential equation was used by Gauss to prove the existence locally of isothermal coordinates on a surface with analytic Riemannian metric. Various techniques have been developed for solving the equation. The most powerful, developed in the 1950s, provides global solutions of the equation on C and relies on the L"p" theory of the Beurling transform, a singular integral operator defined on L"P"(C) for all 1 < "p" < ∞. The same method applies equally well on the unit disk and upper half plane and plays a fundamental role in Teichmüller theory and the theory of quasiconformal mappings. Various uniformization theorems can be proved using the equation, including the measurable Riemann mapping theorem and the simultaneous uniformization theorem. The existence of conformal weldings can also be derived using the Beltrami equation. One of the simplest applications is to the Riemann mapping theorem for simply connected bounded open domains in the complex plane. When the domain has smooth boundary, elliptic regularity for the equation can be used to show that the uniformizing map from the unit disk to the domain extends to a C∞ function from the closed disk to the closure of the domain.
Metrics on planar domains.
Consider a 2-dimensional Riemannian manifold, say with an ("x", "y") coordinate system on it. The curves of constant "x" on that surface typically don't intersect the curves of constant "y" orthogonally. A new coordinate system ("u", "v") is called isothermal when the curves of constant "u" do intersect the curves of constant "v" orthogonally and, in addition, the parameter spacing is the same — that is, for small enough "h", the little region with formula_2 and formula_3 is nearly square, not just nearly rectangular. The Beltrami equation is the equation that has to be solved in order to construct isothermal coordinate systems.
To see how this works, let "S" be an open set in C and let
be a smooth metric "g" on "S". The first fundamental form of "g"
is a positive real matrix ("E" > 0, "G" > 0, "EG" − "F"2 > 0) that varies smoothly with "x" and "y".
The Beltrami coefficient of the metric "g" is defined to be
This coefficient has modulus strictly less than one since the identity
implies that
Let "f"("x","y") =("u"("x","y"),"v"("x","y")) be a smooth diffeomorphism of "S" onto another open set "T" in C. The map "f" preserves orientation just when its Jacobian is positive:
And using "f" to pull back to "S" the standard Euclidean metric "ds"2 = "du"2 + "dv"2 on "T" induces a metric on "S" given by
a metric whose first fundamental form is
When "f" both preserves orientation and induces a metric that differs from the original metric "g" only by a positive, smoothly varying scale factor "r"("x", "y"), the new coordinates "u" and "v" defined on "S" by "f" are called isothermal coordinates.
To determine when this happens, we reinterpret "f" as a complex-valued function of a complex variable "f"("x"+i"y") = "u"("x"+i"y") + i"v"("x"+i"y") so that we can apply the Wirtinger derivatives:
Since
the metric induced by "f" is given by
The Beltrami quotient of this induced metric is defined to be formula_16.
The Beltrami quotient formula_16 of formula_18 equals the Beltrami coefficient formula_19 of the original metric "g" just when
The real and imaginary parts of this identity linearly relate formula_22 formula_23 formula_24 and formula_25 and solving for formula_26 and formula_27 gives
It follows that the metric induced by "f" is then "r"("x", "y") "g"("x","y"), where formula_29 which is positive, while the Jacobian of "f" is then formula_30 which is also positive. So, when formula_31 the new coordinate system given by "f" is isothermal.
Conversely, consider a diffeomorphism "f" that does give us isothermal coordinates. We then have
where the scale factor "r"("x", "y") has dropped out and the expression inside the square root is the perfect square formula_33 Since "f" must preserve orientation to give isothermal coordinates, the Jacobian formula_34 is the positive square root; so we have
The right-hand factors in the numerator and denominator are equal and, since the Jacobian is positive, their common value can't be zero; so formula_36
Thus, the local coordinate system given by a diffeomorphism "f" is isothermal just when "f" solves the Beltrami equation for formula_37
Isothermal coordinates for analytic metrics.
Gauss proved the existence of isothermal coordinates locally in the analytic case by reducing the Beltrami to an ordinary differential equation in the complex domain. Here is a cookbook presentation of Gauss's technique.
An isothermal coordinate system, say in a neighborhood of the origin ("x", "y") = (0, 0), is given by the real and imaginary parts of a complex-valued function "f"("x", "y") that satisfies
Let formula_18 be such a function, and let formula_40 be a complex-valued function of a complex variable that is holomorphic and whose derivative is nowhere zero. Since any holomorphic function formula_40 has formula_42 identically zero, we have
Thus, the coordinate system given by the real and imaginary parts of formula_44 is also isothermal. Indeed, if we fix formula_18 to give one isothermal coordinate system, then all of the possible isothermal coordinate systems are given by formula_44 for the various holomorphic formula_40 with nonzero derivative.
When "E", "F", and "G" are real analytic, Gauss constructed a particular isothermal coordinate system formula_48 the one that he chose being the one with formula_49 for all "x". So the "u" axis of his isothermal coordinate system coincides with the "x" axis of the original coordinates and is parameterized in the same way. All other isothermal coordinate systems are then of the form formula_50 for a holomorphic formula_40 with nonzero derivative.
Gauss lets "q"("t") be some complex-valued function of a real variable "t" that satisfies the following ordinary differential equation:
where "E", "F", and "G" are here evaluated at "y" = "t" and "x" = "q"("t"). If we specify the value of "q"("s") for some start value "s", this differential equation determines the values of "q"("t") for "t" either less than or greater than "s". Gauss then defines his isothermal coordinate system "h" by setting "h"("x", "y") to be formula_53 along the solution path of that differential equation that passes through the point ("x", "y"), and thus has "q"("y") = "x".
This rule sets "h"("x", 0) to be formula_54, since the starting condition is then "q"(0)="x". More generally, suppose that we move by an infinitesimal vector ("dx", "dy") away from some point ("x", "y"), where "dx" and "dy" satisfy
Since formula_56, the vector ("dx", "dy") is then tangent to the solution curve of the differential equation that passes through the point ("x", "y"). Because we are assuming the metric to be analytic, it follows that
for some smooth, complex-valued function formula_58 We thus have
We form the quotient formula_61 and then multiply numerator and denominator by formula_62, which is the complex conjugate of the denominator. Simplifying the result, we find that
Gauss's function "h" thus gives the desired isothermal coordinates.
Solution in "L"2 for smooth Beltrami coefficients.
In the simplest cases the Beltrami equation can be solved using only Hilbert space techniques and the Fourier transform. The method of proof is the prototype for the general solution using L"p" spaces, although Adrien Douady has indicated a method for handling the general case using only Hilbert spaces: the method relies on the classical theory of quasiconformal mappings to establish Hölder estimates that are automatic in the L"p" theory for "p" > 2.
Let "T" be the Beurling transform on L2(C) defined on the Fourier transform of an L2 function "f" as a multiplication operator:
It is a unitary operator and if "f" is a tempered distribution on C with partial derivatives in
L2 then
where the subscripts denote complex partial derivatives.
The fundamental solution of the operator
is given by the distribution
a locally integrable function on C. Thus on Schwartz functions "f"
The same holds for distributions of compact support on C. In particular if "f" is an L2 function with compact support, then its Cauchy transform, defined as
is locally square integrable. The above equation can be written
Moreover, still regarding "f" and "Cf" as distributions,
Indeed, the operator "D" is given on Fourier transforms as multiplication by "iz"/2 and "C" as multiplication by its inverse.
Now in the Beltrami equation
with "μ" a smooth function of compact support, set
and assume that the first derivatives of "g" are L2. Let "h" = "g""z" = "f""z" – 1. Then
If "A" and "B" are the operators defined by
then their operator norms are strictly less that 1 and
Hence
where the right hand sides can be expanded as Neumann series. It follows that
has the same support as "μ" and "g". Hence "f" is given by
Elliptic regularity can now be used to deduce that "f" is smooth.
In fact, off the support of "μ",
so by Weyl's lemma "f" is even holomorphic for |"z"| > "R". Since "f" = "CT*h" + "z", it follows that
"f" tends to 0 uniformly as |"z"| tends to ∞.
The elliptic regularity argument to prove smoothness, however, is the same everywhere and uses the theory of L2 Sobolev spaces on the torus. Let ψ be a smooth function of compact support on C, identically equal to 1 on a neighbourhood of the support of "μ" and set "F" = "ψ" "f". The support of "F" lies in a large square |"x"|, |"y"| ≤ "R", so, identifying opposite sides of the square, "F" and "μ" can be regarded as a distribution and smooth function on a torus T2. By construction "F" is in "L"2(T2). As a distribution on T2 it satisfies
where "G" is smooth. On the canonical basis "e""m" of L2(T2) with "m" in Z + "i" Z, define
Thus "U" is a unitary and on trigonometric polynomials or smooth functions "P"
Similarly it extends to a unitary on each Sobolev space H"k"(T2) with the same property. It is the counterpart on the torus of the Beurling transform. The standard theory of Fredholm operators shows that the operators corresponding to "I" – "μ" "U" and "I" – "U" "μ" are invertible on each Sobolev space. On the other hand,
Since "UG" is smooth, so too is ("I" – "μU")"F" and hence also "F".
Thus the original function "f" is smooth. Regarded as a map of C = R2 into itself, the Jacobian is given by
This Jacobian is nowhere vanishing by a classical argument of . In fact formally writing
"f""z" = "e""k", it follows that
This equation for "k" can be solved by the same methods as above giving a solution tending to 0 at ∞.
By uniqueness "h" + 1 = "e""k" so that
is nowhere vanishing. Since "f" induces a smooth map of the Riemann sphere C ∪ ∞ into itself which is locally a diffeomorphism, "f" must be a diffeomorphism. In fact "f" must be onto by connectedness of the sphere, since its image is an open and closed subset; but then, as a covering map, "f" must cover each point of the sphere the same number of times. Since only ∞ is sent to ∞, it follows that "f" is one-to-one.
The solution "f" is a quasiconformal conformal diffeomorphism. These form a group and their Beltrami coefficients can be computed according to the following rule:
Moreover, if "f"(0) = 0 and
then
This formula reflects the fact that on a Riemann surface, a Beltrami coefficient is not a function.
Under a holomorphic change of coordinate "w" = "w"("z"), the coefficient is transformed to
Defining a smooth Beltrami coefficient on the sphere in this way, if "μ" is such a coefficient then, taking a smooth bump function ψ equal to 0 near 0, equal 1 for |"z"| > 1 and satisfying 0 ≤ "ψ" ≤ 1, "μ" can be written as a sum of two Beltrami coefficients:
Let "g" be the quasiconformal diffeomorphism of the sphere fixing 0 and ∞ with coefficient 
"μ"∞. Let λ be the Beltrami coefficient of compact support on C defined by
If "f" is the quasiconformal diffeomorphism of the sphere fixing 0 and ∞ with coefficient λ, then
the transformation formulas above show that "f" ∘ "g"−1 is a quasiconformal diffeomorphism of the sphere fixing 0 and ∞ with coefficient "μ".
The solutions of Beltrami's equation restrict to diffeomorphisms of the upper halfplane or unit disk if the coefficient "μ" has extra symmetry properties; since the two regions are related by a Möbius transformation (the Cayley transform), the two cases are essentially the same.
For the upper halfplane Im "z" > 0, if "μ" satisfies
then by uniqueness the solution "f" of the Beltrami equation satisfies
so leaves the real axis and hence the upper halfplane invariant.
Similarly for the unit disc |"z"| < 1, if "μ" satisfies
then by uniqueness the solution "f" of the Beltrami equation satisfies
so leaves the unit circle and hence the unit disk invariant.
Conversely Beltrami coefficients defined on the closures of the upper halfplane or unit disk which satisfy these conditions on the boundary can be "reflected" using the formulas above. If the extended functions are smooth the preceding theory can be applied. Otherwise the extensions will be continuous but with a jump in the derivatives at the boundary. In that case the more general theory for measurable coefficients "μ" is required and is most directly handled within the L"p" theory.
Smooth Riemann mapping theorem.
Let "U" be an open simply connected domain in the complex plane with smooth boundary containing 0 in its interior and let "F" be a diffeomorphism of the unit disk "D" onto "U" extending smoothly to the boundary and the identity on a neighbourhood of 0. Suppose that in addition the induced metric on the closure of the unit disk can be reflected in the unit circle to define a smooth metric on C. The corresponding Beltrami coefficient is then a smooth function on C vanishing near 0 and ∞ and satisfying
The quasiconformal diffeomorphism "h" of C satisfying
preserves the unit circle together with its interior and exterior. From the composition formulas for Beltrami coefficients
so that "f" = "F"∘ "h"−1 is a smooth diffeomorphism between the closures of "D" and "U" which is holomorphic on the interior. Thus, if a suitable diffeomorphism "F" can be constructed, the mapping "f" proves the smooth Riemann mapping theorem for the domain "U".
To produce a diffeomorphism "F" with the properties above, it can be assumed after an affine transformation that the boundary of "U" has length 2π and that 0 lies in "U". The smooth version of the Schoenflies theorem produces a smooth diffeomorphism "G" from the closure of "D" onto the closure of "u" equal to the identity on a neighbourhood of 0 and with an explicit form on a tubular neighbourhood of the unit circle. In fact taking polar coordinates ("r","θ") in "R"2 and letting ("x"("θ"),"y"("θ")) ("θ" in [0,2]) be a parametrization of ∂"U" by arclength, "G" has the form
Taking "t" = 1 − "r" as parameter, the induced metric near the unit circle is given by
where
is the curvature of the plane curve ("x"("θ"),"y"("θ")).
Let
After a change of variable in the "t" coordinate and a conformal change in the metric, the metric takes the form
where ψ is an analytic real-valued function of "t":
A formal diffeomorphism sending ("θ","t") to ("f"("θ","t"),"t") can be defined as a formal power series in "t":
where the coefficients "f""n" are smooth functions on the circle. These coefficients can be defined by recurrence so that the transformed metric only has even powers of "t" in the coefficients. This condition is imposed by demanding that no odd powers of "t" appear in the formal power series expansion:
By Borel's lemma, there is a diffeomorphism defined in a neighbourhood of the unit circle, "t" = 0, for which the formal expression "f"("θ","t") is the Taylor series expansion in the "t" variable. It follows that, after composing with this diffeomorphism, the extension of the metric obtained by reflecting in the line "t" = 0 is smooth.
Hölder continuity of solutions.
Douady and others have indicated ways to extend the "L"2 theory to prove the existence and uniqueness of solutions when the Beltrami coefficient "μ" is bounded and measurable with "L"∞ norm "k" strictly less than one. Their approach involved the theory of quasiconformal mappings to establish directly the solutions of Beltrami's equation when "μ" is smooth with fixed compact support are uniformly Hölder continuous. In the L"p" approach Hölder continuity follows automatically from operator theory.
The "L""p" theory when "μ" is smooth of compact support proceeds as in the L2 case. By the Calderón–Zygmund theory the Beurling transform and its inverse are known to be continuous for the L"p" norm. The Riesz–Thorin convexity theorem implies that the norms "C""p" are continuous functions of "p". In particular "C""p" tends to 1 when "p" tends to 2.
In the Beltrami equation
with "μ" a smooth function of compact support, set
and assume that the first derivatives of "g" are L"p". Let "h" = "g""z" = "f""z" – 1. Then
If "A" and "B" are the operators defined by "AF" = "TμF" and "BF" = "μTF", then their operator norms are strictly less that 1 and ("I" − "A")"h" = "T"μ. Hence
where the right hand sides can be expanded as Neumann series. It follows that
has the same support as "μ" and "g". Hence, up to the addition of a constant, "f" is given by
Convergence of functions with fixed compact support in the L"p" norm for "p" > 2 implies convergence in
L2, so these formulas are compatible with the L2 theory if "p" > 2.
The Cauchy transform "C" is not continuous on L2 except as a map into functions of vanishing mean oscillation.
 On L"p" its image is contained in Hölder continuous functions with Hölder exponent 1 − 2"p"−1 once a suitable constant is added. In fact for a function "f" of compact support define
Note that the constant is added so that "Pf"(0) = 0. Since "Pf" only differs from "Cf" by a constant, it follows exactly as in the "L"2 theory that
Moreover, "P" can be used instead of "C" to produce a solution:
On the other hand, the integrand defining "Pf" is in L"q" if "q"−1 = 1 − "p"−1. The Hölder inequality implies that "Pf" is Hölder continuous with an explicit estimate:
where
For any "p" > 2 sufficiently close to 2, "C""p""k" <1. Hence the Neumann series for ("I" − "A")−1 and ("I" − "B")−1 converge. The Hölder estimates for "P" yield the following uniform estimates for the normalized solution of the Beltrami equation:
If "μ" is supported in |"z"| ≤ "R", then
Setting "w"1 = "z" and "w"2 = 0, it follows that for |"z"| ≤ "R"
where the constant "C" > 0 depends only on the L∞ norm of "μ". So the Beltrami coefficient of "f"−1 is smooth and supported in
"z"| ≤ "CR". It has the same L∞ norm as that of "f". So the inverse diffeomorphisms also satisfy uniform Hölder estimates.
Solution for measurable Beltrami coefficients.
Existence.
The theory of the Beltrami equation can be extended to measurable Beltrami coefficients "μ". For simplicity only a special class of "μ" will be considered—adequate for most applications—namely those functions which are smooth an open set Ω (the regular set) with complement Λ a closed set of measure zero (the singular set). Thus Λ is a closed set that is contained in open sets of arbitrarily small area. For measurable Beltrami coefficients "μ" with compact support in |"z"| < "R", the solution of the Beltrami equation can be obtained as a limit of solutions for smooth Beltrami coefficients.
In fact in this case the singular set Λ is compact. Take smooth functions φ"n" of compact support with 0 ≤ φ"n" ≤ 1, equal to 1 on a neighborhood of Λ and 0 off a slightly larger neighbourhood, shrinking to Λ as "n" increases. Set
The "μ""n" are smooth with compact support in |"z"| < "R" and
The "μ""n" tend to "μ" in any "L""p" norm with "p" < ∞.
The corresponding normalised solutions "f""n" of the Beltrami equations and their inverses "g""n" satisfy uniform Hölder estimates. They are therefore equicontinuous on any compact subset of C; they are even holomorphic for |"z"| > "R". So by the Arzelà–Ascoli theorem, passing to a subsequence if necessary, it can be assumed that both "f""n" and "g""n" converge uniformly on compacta to "f" and "g". The limits will satisfy the same Hölder estimates and be holomorphic for |"z"| > "R". The relations "f""n"∘"g""n" = id = "g""n"∘"f""n" imply that in the limit "f"∘"g" = id = "g"∘"f", so that "f" and "g" are homeomorphisms.
This establishes the existence of homeomorphic solutions of Beltrami's equation in the case of Beltrami coefficients of compact support. It also shows that the inverse homeomorphisms and composed homeomorphisms satisfy Beltrami equations and that all computations can be performed by restricting to regular sets.
If the support is not compact the same trick used in the smooth case can be used to construct a solution in terms of two homeomorphisms associated to compactly supported Beltrami coefficients. Note that, because of the assumptions on the Beltrami coefficient, a Möbius transformation of the extended complex plane can be applied to make the singular set of the Beltrami coefficient compact. In that case one of the homeomorphisms can be chosen to be a diffeomorphism.
Uniqueness.
There are several proofs of the uniqueness of solutions of the Beltrami equation with a given Beltrami coefficient. Since applying a Möbius transformation of the complex plane to any solution gives another solution, solutions can be normalised so that they fix 0, 1 and ∞. The method of solution of the Beltrami equation using the Beurling transform also provides a proof of uniqueness for coefficients of compact support "μ" and for which the distributional derivatives are in 1 + L"p" and L"p". The relations
for smooth functions ψ of compact support are also valid in the distributional sense for L"p" functions "h" since they can be written as L"p" of ψ"n"'s. If "f" is a solution of the Beltrami equation with "f"(0) = 0 and "f""z" - 1 in L"p" then
satisfies
So "F" is weakly holomorphic. Applying Weyl's lemma it is possible to conclude that there exists a holomorphic function "G" that is equal to "F" almost everywhere. Abusing notation redefine "F:=G". The conditions "F" '(z) − 1 lies in L"p" and "F"(0) = 0 force "F"("z") = "z". Hence
and so differentiating
If "g" is another solution then
Since "T"μ has operator norm on L"p" less than 1, this forces
But then from the Beltrami equation
Hence "f" − "g" is both holomorphic and antiholomorphic, so a constant. Since "f"(0) = 0 = "g"(0), it follows that "f" = "g". Note that since "f" is holomorphic off the support of "μ" and "f"(∞) = ∞, the conditions that the derivatives are locally in L"p" force
For a general "f" satisfying Beltrami's equation and with distributional derivatives locally in L"p", it can be assumed after applying a Möbius transformation that 0 is not in the singular set of the Beltrami coefficient "μ". If "g" is a smooth diffeomorphism "g" with Beltrami coefficient λ supported near 0, the Beltrami coefficient "ν" for "f" ∘ "g"−1 can be calculated directly using the change of variables formula for distributional derivatives:
"λ" can be chosen so that ν vanishes near zero. Applying the map "z"−1 results in a solution of Beltrami's equation with a Beltrami coefficient of compact support. The directional derivatives are still locally in L"p". The coefficient ν depends only on "μ", "λ" and "g", so any two solutions of the original equation will produce solutions near 0 with distributional derivatives locally in "L""p" and the same Beltrami coefficient. They are therefore equal. Hence the solutions of the original equation are equal.
Uniformization of multiply connected planar domains.
The method used to prove the smooth Riemann mapping theorem can be generalized to multiply connected planar regions with smooth boundary. The Beltrami coefficient in these cases is smooth on an open set, the complement of which has measure zero. The theory of the Beltrami equation with measurable coefficients is therefore required.
Doubly connected domains. If Ω is a doubly connected planar region, then there is a diffeomorphism "F" of an annulus "r" ≤ |z| ≤ 1 onto the closure of Ω, such that after a conformal change the induced metric on the annulus can be continued smoothly by reflection in both boundaries. The annulus is a fundamental domain for the group generated by the two reflections, which reverse orientation. The images of the fundamental domain under the group fill out C with 0 removed and the Beltrami coefficient is smooth there. The canonical solution "h" of the Beltrami equation on C, by the L"p" theory is a homeomorphism. It is smooth on away from 0 by elliptic regularity. By uniqueness it preserves the unit circle, together with its interior and exterior. Uniqueness of the solution also implies that reflection there is a conjugate Möbius transformation "g" such that "h" ∘ "R" = "g" ∘ "h" where "R" denotes reflection in |"z"| = "r". Composing with a Möbius transformation that fixes the unit circle it can be assumed that "g" is a reflection in a circle |"z"| = "s" with "s" < 1. It follows that "F" ∘ "h"−1 is a smooth diffeomorphism of the annulus "s" ≤ |"z"| ≤ 1 onto the closure of Ω, holomorphic in the interior.
Multiply connected domains. For regions with a higher degree of connectivity "k" + 1, the result is essentially Bers' generalization of the retrosection theorem. There is a smooth diffeomorphism "F" of the region Ω1, given by the unit disk with "k" open disks removed, onto the closure of Ω. It can be assumed that 0 lies in the interior of the domain. Again after a modification of the diffeomorphism and conformal change near the boundary, the metric can be assumed to be compatible with reflection. Let "G" be the group generated by reflections in the boundary circles of Ω1. The interior of Ω1 iz a fundamental domain for "G". Moreover, the index two normal subgroup "G"0 consisting of orientation-preserving mappings is a classical Schottky group. Its fundamental domain consists of the original fundamental domain with its reflection in the unit circle added. If the reflection is "R"0, it is a free group with generators "R""i"∘"R"0 where "R""i" are the reflections in the interior circles in the original domain. The images of the original domain by the "G", or equivalently the reflected domain by the Schottky group, fill out the regular set for the Schottky group. It acts properly discontinuously there. The complement is the limit set of "G"0. It has measure zero. The induced metric on Ω1 extends by reflection to the regular set. The corresponding Beltrami coefficient is invariant for the reflection group generated by the reflections "R""i" for "i" ≥ 0. Since the limit set has measure zero, the Beltrami coefficient extends uniquely to a bounded measurable function on C. smooth on the regular set. The normalised solution of the Beltrami equation "h" is a smooth diffeomorphism of the closure of Ω1 onto itself preserving the unit circle, its exterior and interior. Necessarily "h" ∘ "R""i" = "S""i" ∘ "h". where "S""i" is the reflection in another circle in the unit disk. Looking at fixed points, the circles arising this way for different "i" must be disjoint. It follows that "F" ∘ "h"−1 defines a smooth diffeomorphism of the unit disc with the interior of these circles removed onto the closure of Ω, which is holomorphic in the interior.
Simultaneous uniformization.
 showed that two compact Riemannian 2-manifolds "M"1, "M"2 of genus "g" > 1 can be simultaneously uniformized.
As topological spaces "M"1 and "M"2 are homeomorphic to a fixed quotient of the upper half plane H by a discrete cocompact subgroup Γ of PSL(2,R). Γ can be identified with the fundamental group of the manifolds and H is a universal covering space. The homeomorphisms can be chosen to be piecewise linear on corresponding triangulations. A result of implies that the homeomorphisms can be adjusted near the edges and the vertices of the triangulation to produce diffeomorphisms. The metric on "M"1 induces a metric on H which is Γ-invariant. Let "μ" be the corresponding Beltrami coefficient on H. It can be extended to C by reflection
It satisfies the invariance property
for "g" in Γ. The solution "f" of the corresponding Beltrami equation defines a homeomorphism of C, preserving the real axis and the upper and lower half planes. Conjugation of the group elements by "f"−1 gives a new cocompact subgroup Γ1 of PSL(2,R). Composing the original diffeomorphism with the inverse of "f" then yield zero as the Beltrami coefficient. Thus the metric induced on H is invariant under Γ1 and conformal to the Poincaré metric on H. It must therefore be given by multiplying by a positive smooth function that is Γ1-invariant. Any such function corresponds to a smooth function on "M"1. Dividing the metric on "M"1 by this function results in a conformally equivalent metric on "M"1 which agrees with the Poincaré metric on H / Γ1. In this way "M"1 becomes a compact Riemann surface, i.e. is uniformized and inherits a natural complex structure.
With this conformal change in metric "M"1 can be identified with H / Γ1. The diffeomorphism between onto "M"2 induces another metric on H which is invariant under Γ1. It defines a Beltrami coefficient λ
omn H which this time is extended to C by defining λ to be 0 off H. The solution "h" of the Beltrami equation is a homeomorphism of C which is holomorphic on the lower half plane and smooth on the upper half plane. The image of the real axis is a Jordan curve dividing C into two components. Conjugation of Γ1 by "h"−1 gives a quasi-Fuchsian subgroup Γ2 of PSL(2,C). It leaves invariant the Jordan curve and acts properly discontinuously on each of the two components. The quotients of the two components by Γ2 are naturally identified with "M"1 and "M"2. This identification is compatible with the natural complex structures on both "M"1 and "M"2.
Conformal welding.
An orientation-preserving homeomorphism "f" of the circle is said to be quasisymmetric if there are positive constants "a" and "b" such that
If
Conversely if this condition is satisfied for all such triples of points, then "f" is quasisymmetric.
An apparently weaker condition on a homeomorphism "f" of the circle is that it be "quasi-Möbius", that is there are constants "c", "d" > 0 such that
where
denotes the cross-ratio. In fact if "f" is quasisymmetric then it is also quasi-Möbius, with "c" = "a"2 and "d" = "b": this follows by multiplying the first inequality above for ("z"1,"z"3,"z"4) and ("z"2,"z"4,"z"3).
Conversely if "f" is a quasi-Möbius homeomorphism then it is also quasisymmetric. Indeed, it is immediate that if "f" is quasi-Möbius so is its inverse. It then follows that "f" (and hence "f"−1) is Hölder continuous. To see this let "S" be the set of cube roots of unity, so that if "a" ≠ "b" in "S", then |"a" − "b"| = 2 sin /3 = . To prove a Hölder estimate, it can be assumed that "x" – "y" is uniformly small. Then both "x" and "y" are greater than a fixed distance away from "a", "b" in "S" with "a" ≠ "b", so the estimate follows by applying the quasi-Möbius inequality to "x", "a", "y", "b". To check that "f" is quasisymmetric, it suffices to find a uniform upper bound for |"f"("x") − "f"("y")| / |"f"("x") − "f"("z")| in the case of a triple with |"x" − "z"| = |"x" − "y"|, uniformly small. In this case there is a point "w" at a distance greater than 1 from "x", "y" and "z". Applying the quasi-Möbius inequality to "x", "w", "y" and "z" yields the required upper bound.
A homeomorphism "f" of the unit circle can be extended to a homeomorphism "F" of the closed unit disk which is diffeomorphism on its interior. , generalizing earlier results of Ahlfors and Beurling, produced such an extension with the additional properties that it commutes with the action of SU(1,1) by Möbius transformations and is quasiconformal if "f" is quasisymmetric. (A less elementary method was also found independently by : Tukia's approach has the advantage of also applying in higher dimensions.) When "f" is a diffeomorphism of the circle, the Alexander extension provides another way of extending "f":
where ψ is a smooth function with values in [0,1], equal to 0 near 0 and 1 near 1, and
with "g"("θ" + 2) = "g"("θ") + 2. give a survey of various methods of extension, including variants of the Ahlfors-Beurling extension which are smooth or analytic in the open unit disk.
In the case of a diffeomorphism, the Alexander extension "F" can be continued to any larger disk |"z"| < "R" with "R" > 1. Accordingly, in the unit disc
This is also true for the other extensions when "f" is only quasisymmetric.
Now extend "μ" to a Beltrami coefficient on the whole of C by setting it equal to 0 for |"z"| ≥ 1. Let "G" be the corresponding solution of the Beltrami equation. Let "F"1("z") = "G" ∘ "F"−1("z") for |"z"| ≤ 1 and
"F"2("z") = "G" ("z") for |"z"| ≥ 1. Thus "F"1 and "F"2 are univalent holomorphic maps of |"z"| < 1 and |"z"| > 1 onto the inside and outside of a Jordan curve. They extend continuously to homeomorphisms "f""i" of the unit circle onto the Jordan curve on the boundary. By construction they satisfy the
conformal welding condition: