In mathematics, the uniformization theorem says 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 and Henri conjectured the uniformization theorem for (the Riemann surfaces of) algebraic curves. 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 and . Paul Koebe later gave several more proofs and generalizations. The history is described in ; a complete account of uniformization up to the 1907 papers of Koebe and Poincaré is given with detailed proofs in (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 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 ; 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
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 non-vanishing Jacobian.
These conditions can be phrased equivalently in terms of the exterior derivative and the Hodge star operator . and will be isothermal coordinates if , where is defined on differentials by . Let be the Laplace–Beltrami operator. By standard elliptic theory, can be chosen to be harmonic near a given point, i.e. , with non-vanishing. By the Poincaré lemma has a local solution exactly when . This condition is equivalent to , so can always be solved locally. Since is non-zero and the square of the Hodge star operator is −1 on 1-forms, and must be linearly independent, so that and 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 , or by direct elementary methods, as in and .
From this correspondence with compact Riemann surfaces, a classification of closed orientable Riemannian 2-manifolds follows. Each such is conformally equivalent to a unique closed 2-manifold of constant curvature, so a quotient of one of the following by a free action of a discrete subgroup of an isometry group: