Uniformization theorem
In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of the three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. In particular it implies that every Riemann surface admits a Riemannian metric of constant curvature. For compact Riemann surfaces, those with universal cover the unit disk are precisely the hyperbolic surfaces of genus greater than 1, all with nonabelian fundamental group; those with universal cover the complex plane are the Riemann surfaces of genus 1, namely the complex tori or elliptic curves with fundamental group Z^{2}; and those with universal cover the Riemann sphere are those of genus zero, namely the Riemann sphere itself, with trivial fundamental group.
The uniformization theorem is a generalization of the Riemann mapping theorem from proper simply connected open subsets of the plane to arbitrary simply connected Riemann surfaces. The uniformization theorem also has an equivalent statement in terms of closed Riemannian 2manifolds: each such manifold has a conformally equivalent Riemannian metric with constant curvature.
Many classical proofs of the uniformization theorem rely on constructing a realvalued 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; the Dirichlet principle; and Weyl's method of orthogonal projection. In the context of closed Riemannian 2manifolds, 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 maps; Liouville's equation, already studied by Poincaré; and Ricci flow along with other nonlinear flows.
History
Felix Klein (1883) and Henri Poincaré (1882) conjectured the uniformization theorem for (the Riemann surfaces of) algebraic curves. Henri Poincaré (1883) extended this to arbitrary multivalued analytic functions and gave informal arguments in its favor. The first rigorous proofs of the general uniformization theorem were given by Poincaré (1907) and Paul Koebe (1907a, 1907b, 1907c). Paul Koebe later gave several more proofs and generalizations. The history is described in Gray (1994); a complete account of uniformization up to the 1907 papers of Koebe and Poincaré is given with detailed proofs in de SaintGervais (2016) (the Bourbakitype pseudonym of the group of fifteen mathematicians who jointly produced this publication).
Classification of compact Riemann surfaces
Every Riemann surface is the quotient of a free, proper and holomorphic action of a discrete group on its universal covering and this universal covering is holomorphically isomorphic (one also says: "conformally equivalent" or "biholomorphic") to one of the following:
 the Riemann sphere
 the complex plane
 the unit disk in the complex plane.
Rado's theorem shows that every Riemann surface is automatically 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.
Classification of closed oriented Riemannian 2manifolds
On an oriented 2manifold, a Riemannian metric induces a complex structure using the passage to isothermal coordinates. If the Riemannian metric is given locally as
then in the complex coordinate z = x + iy, it takes the form
where
so that λ and μ are smooth with λ > 0 and μ < 1. In isothermal coordinates (u, v) the metric should take the form
with ρ > 0 smooth. The complex coordinate w = u + i v satisfies
so that the coordinates (u, v) will be isothermal locally provided the Beltrami equation
has a locally diffeomorphic solution, i.e. a solution with nonvanishing Jacobian.
These conditions can be phrased equivalently in terms of the exterior derivative and the Hodge star operator ∗.^{[1]} u and v will be isothermal coordinates if ∗du = dv, where ∗ is defined on differentials by ∗(p dx + q dy) = −q dx + p dy. Let ∆ = ∗d∗d be the Laplace–Beltrami operator. By standard elliptic theory, u can be chosen to be harmonic near a given point, i.e. Δ u = 0, with du nonvanishing. By the Poincaré lemma dv = ∗du has a local solution v exactly when d(∗du) = 0. This condition is equivalent to Δ u = 0, so can always be solved locally. Since du is nonzero and the square of the Hodge star operator is −1 on 1forms, du and dv must be linearly independent, so that u and v give local isothermal coordinates.
The existence of isothermal coordinates can be proved by other methods, for example using the general theory of the Beltrami equation, as in Ahlfors (2006), or by direct elementary methods, as in Chern (1955) and Jost (2006).
From this correspondence with compact Riemann surfaces, a classification of closed orientable Riemannian 2manifolds follows. Each such is conformally equivalent to a unique closed 2manifold of constant curvature, so a quotient of one of the following by a free action of a discrete subgroup of an isometry group:
 the sphere (curvature +1)
 the Euclidean plane (curvature 0)
 the hyperbolic plane (curvature −1).

genus 0

genus 1

genus 2

genus 3
The first case gives the 2sphere, the unique 2manifold with constant positive curvature and hence positive Euler characteristic (equal to 2). The second gives all flat 2maniolds, i.e. the tori, which have Euler characteristic 0. The third case covers all 2manifolds of constant negative curvature, i.e. the hyperbolic 2manifolds all of which have negative Euler characteristic. The classification is consistent with the Gauss–Bonnet theorem, which implies that for a closed surface with constant curvature, the sign of that curvature must match the sign of the Euler characteristic. The Euler characteristic is equal to 2 – 2g, where g is the genus of the 2manifold, i.e. the number of "holes".
Methods of proof
Hilbert space methods
In 1913 Hermann Weyl published his classic textbook "Die Idee der Riemannschen Fläche" based on his Göttingen lectures from 19111912. 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 Hilbert's treatment of the Dirichlet problem using Hilbert space techniques; Brouwer's contributions to topology; and Koebe's proof of the uniformization theorem and its subsequent improvements. Much later 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 on elliptic regularity, was related to Hodge's theory of harmonic integrals; and both theories were subsumed into the modern theory of elliptic operators and L^{2} Sobolev spaces. In the third edition of his book from 1955, translated into English in Weyl (1964), Weyl adopted the modern definition of differential manifold, in preference to triangulations, but decided not to make use of his method of orthogonal projection. Springer (1957) followed Weyl's account of the uniformisation theorem, but used the method of orthogonal projection to treat the Dirichlet problem. This approach will be outlined below. 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 Donaldson (2011).
Nonlinear flows
In introducing the Ricci flow, Richard S. Hamilton showed that the 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 2sphere: a method for avoiding an appeal to the uniformization theorem (for genus 0) was provided by Chen, Lu & Tian (2006);^{[2]} a short selfcontained account of Ricci flow on the 2sphere was given in Andrews & Bryan (2009).
Generalizations
Koebe proved the general uniformization theorem that if a Riemann surface is homeomorphic to an open subset of the complex sphere (or equivalently if every Jordan curve separates it), then it is conformally equivalent to an open subset of the complex sphere.
In 3 dimensions, there are 8 geometries, called the eight Thurston geometries. Not every 3manifold admits a geometry, but Thurston's geometrization conjecture proved by Grigori Perelman states that every 3manifold can be cut into pieces that are geometrizable.
The simultaneous uniformization theorem of Lipman Bers shows that it is possible to simultaneously uniformize two compact Riemann surfaces of the same genus >1 with the same quasiFuchsian group.
The measurable Riemann mapping theorem shows more generally that the map to an open subset of the complex sphere in the uniformization theorem can be chosen to be a quasiconformal map with any given bounded measurable Beltrami coefficient.
Notes
 ↑ DeTurck & Kazdan 1981; Taylor 1996, pp. 377–378
 ↑ Brendle 2010
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