proof of the uniformization theorem

Our proof relies on the well-known Newlander-Niremberg theorem which implies, in particular, that any Riemmanian metric on an oriented 2-dimensional real manifold defines a unique analytic structure.

We will merely use the fact that H1(X,)=0. If X is compact, then X is a complex curve of genus 0, so X1. On the other hand, the elementary Riemann mapping theoremMathworldPlanetmath says that an open set Ω with H1(Ω,)=0 is either equal to or biholomorphic to the unit disk. Thus, all we have to show is that a non compact Riemann surface X with H1(X,)=0 can be embedded in the complex plane .

Let Ων be an exhausting sequence of relatively compact connected open sets with smooth boundary in X. We may assume that XΩν has no relatively compact connected components, otherwise we “fill the holes” of Ων by taking the union with all such components. We let Yν be the double of the manifold with boundary (Ω¯ν,Ων), i.e. the union of two copies of Ω¯ν with opposite orientationsPlanetmathPlanetmath and the boundaries identified. Then Yν is a compact oriented surface without boundary.

Fact: we have H1(Yν,)=0. We postpone the proof of this fact to the end of the present paragraph and we continue with the proof of the uniformization theoremMathworldPlanetmath.

Extend the almost complex structure of Ω¯ν in an arbitrary way to Yν, e.g. by an extension of a Riemmanian metric. Then Yν becomes a compact Riemann surface of genus 0, thus Yν1 and we obtain in particular a holomorphic embedding Φν:Ων. Fix a point aΩ0 and a non zero linear form ξ*TaX. We can take the compositionMathworldPlanetmathPlanetmath of Φν with an affine linear map so that Φν(a)=0 and dΦν(a)=ξ*. By the well-known properties of injective holomorphic maps, (Φν) is then uniformly bounded on every small disk centered at a, thus also on every compact subset of X by a connectedness argument. Hence (Φν) has a subsequence converging towards an injective holomorphic map Φ:X.

Proof of the ”fact”: Let us first compute the cohomology with compact support Hc1(Ων,). Let u be a closed 1-form with compact support in Ων. By Poincaré duality Hc1(X,)=0, so u=df for some ”test” function f𝒟(X). As df=0 on a neighborhoodMathworldPlanetmath of XΩν and as all connected components of this set are non compact, f must be equal to the constant zero near XΩν. Hence u=df is the zero class in Hc1(Ων,) and we get Hc1(Ων,)=H1(Ων,)=0. The exact sequence of the pair (Ω¯ν,Ωnu) yelds


thus H0(Ων,)=. Finally, the Mayer-Vietoris sequence applied to small neighborhoods of the two copies of Ω¯ν in Yν gives an exact sequence


where the first map is onto. Hence H1(Yν,)=0.


J.-P. Demailly, Complex AnalyticPlanetmathPlanetmath and Algebraic GeometryMathworldPlanetmathPlanetmath.

Title proof of the uniformization theorem
Canonical name ProofOfTheUniformizationTheorem
Date of creation 2013-03-22 15:37:50
Last modified on 2013-03-22 15:37:50
Owner Simone (5904)
Last modified by Simone (5904)
Numerical id 14
Author Simone (5904)
Entry type Proof
Classification msc 30F20
Classification msc 30F10