In mathematics, the uniformization theorem states that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. The theorem is a generalization of the Riemann mapping theorem from simply connected open subsets of the plane to arbitrary simply connected Riemann surfaces. Since every Riemann surface has a universal cover which is a simply connected Riemann surface, the uniformization theorem leads to a classification of Riemann surfaces into three types: those that have the Riemann sphere as universal cover ("elliptic"), those with the plane as universal cover ("parabolic") and those with the unit disk as universal cover ("hyperbolic"). It further follows that every Riemann surface admits a Riemannian metric of constant curvature, where the curvature can be taken to be 1 in the elliptic, 0 in the parabolic and -1 in the hyperbolic case. The uniformization theorem also yields a similar classification of closed orientable Riemannian 2-manifolds into elliptic/parabolic/hyperbolic cases. Each such manifold has a conformally equivalent Riemannian metric with constant curvature, where the curvature can be taken to be 1 in the elliptic, 0 in the parabolic and -1 in the hyperbolic case.
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 Saint-Gervais (2016) (the Bourbaki-type pseudonym of the group of fifteen mathematicians who jointly produced this publication).
Classification of connected Riemann surfaces Every Riemann surface is the quotient of the free, proper and holomorphic action of a discrete group on its universal covering and this universal covering, being a simply connected Riemann surface, 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. For compact Riemann surfaces, those with universal cover the unit disk are precisely the hyperbolic surfaces of genus greater than 1, all with non-abelian 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 Z2; and those with universal cover the Riemann sphere are those of genus zero, namely the Riemann sphere itself, with trivial fundamental group.
Classification of closed oriented Riemannian 2-manifolds On an oriented 2-manifold, a Riemannian metric induces a complex structure using the passage to isothermal coordinates. If the Riemannian metric is given locally as
d s 2 = E d x 2 + 2 F d x d y + G d y 2 , {\displaystyle ds^{2}=E\,dx^{2}+2F\,dx\,dy+G\,dy^{2},}
then in the complex coordinate z = x + iy, it takes the form
d s 2 = λ | d z + μ d z ¯ | 2 , {\displaystyle ds^{2}=\lambda |dz+\mu \,d{\overline {z}}|^{2},}
where
λ = 1 4 ( E + G + 2 E G − F 2 ) , μ = 1 4 λ ( E − G + 2 i F ) , {\displaystyle \lambda ={\frac {1}{4}}\left(E+G+2{\sqrt {EG-F^{2}}}\right),\ \ \mu ={\frac {1}{4\lambda }}(E-G+2iF),}
so that λ and μ are smooth with λ > 0 and |μ| < 1. In isothermal coordinates (u, v) the metric should take the form
d s 2 = ρ ( d u 2 + d v 2 ) {\displaystyle ds^{2}=\rho (du^{2}+dv^{2})}
with ρ > 0 smooth. The complex coordinate w = u + i v satisfies
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