Editing Uniformization theorem (section)

==Methods of proof==
Many classical proofs of the uniformization theorem rely on constructing a real-valued [[harmonic function]] on the simply connected Riemann surface, possibly with a singularity at one or two points and often corresponding to a form of [[Green's function]]. Four methods of constructing the harmonic function are widely employed: the [[Perron method]]; the [[Schwarz alternating method]]; [[Dirichlet's principle]]; and [[Hermann Weyl|Weyl]]'s method of orthogonal projection. In the context of closed Riemannian 2-manifolds, several modern proofs invoke nonlinear differential equations on the space of conformally equivalent metrics. These include the [[Beltrami equation]] from [[Teichmüller theory]] and an equivalent formulation in terms of [[harmonic map]]s; [[Liouville's equation]], already studied by Poincaré; and [[Ricci flow]] along with other nonlinear flows.

[[Radó's theorem (Riemann surfaces)|Rado's theorem]] shows that every Riemann surface is automatically [[second-countable space|second-countable]]. Although Rado's theorem is often used in proofs of the uniformization theorem, some proofs have been formulated so that Rado's theorem becomes a consequence. Second countability is automatic for compact Riemann surfaces.

===Hilbert space methods===
{{See also|Planar Riemann surface#Uniformization theorem}}
In 1913 Hermann Weyl published his classic textbook "Die Idee der Riemannschen Fläche" based on his Göttingen lectures from 1911 to 1912. It was the first book to present the theory of Riemann surfaces in a modern setting and through its three editions has remained influential. Dedicated to [[Felix Klein]], the first edition incorporated [[David Hilbert|Hilbert's]] treatment of the [[Dirichlet problem]] using [[Hilbert space]] techniques; [[L. E. J. Brouwer|Brouwer's]] contributions to topology; and [[Paul Koebe|Koebe's]] proof of the uniformization theorem and its subsequent improvements. Much later {{harvtxt|Weyl|1940}} developed his method of orthogonal projection which gave a streamlined approach to the Dirichlet problem, also based on Hilbert space; that theory, which included [[Weyl's lemma (Laplace equation)|Weyl's lemma]] on [[elliptic regularity]], was related to [[W. V. D. Hodge|Hodge's]] [[Hodge theory|theory of harmonic integrals]]; and both theories were subsumed into the modern theory of [[elliptic operator]]s and {{math|''L''<sup>2</sup>}} [[Sobolev space]]s. In the third edition of his book from 1955, translated into English in {{harvtxt|Weyl|1964}}, Weyl adopted the modern definition of differential manifold, in preference to [[triangulation (topology)|triangulations]], but decided not to make use of his method of orthogonal projection. {{harvtxt|Springer|1957}} followed Weyl's account of the uniformisation theorem, but used the method of orthogonal projection to treat the Dirichlet problem. {{harvtxt|Kodaira|2007}} describes the approach in Weyl's book and also how to shorten it using the method of orthogonal projection. A related account can be found in {{harvtxt|Donaldson|2011}}.

===Nonlinear flows===
{{see also|Ricci flow#Relationship to uniformization and geometrization}}

[[Richard S. Hamilton]] showed that the [[Ricci flow|normalized Ricci flow]] on a closed surface uniformizes the metric (i.e., the flow converges to a constant curvature metric).  However, his proof relied on the uniformization theorem. The missing step involved Ricci flow on the 2-sphere: a method for avoiding an appeal to the uniformization theorem (for genus 0) was provided by {{harvtxt|Chen|Lu|Tian|2006}};<ref>{{harvnb|Brendle|2010}}</ref> a short self-contained account of Ricci flow on the 2-sphere was given in {{harvtxt|Andrews|Bryan|2010}}.