V.34 The Uniformization Theorem
The uniformization theorem is a remarkable classification of RIEMANN SURFACES [III.79]. Two surfaces are biholomorphically equivalent if there is a HOLOMORPHIC FUNCTION [I.3 §5.6] from one to the other that has a holomorphic inverse. If a Riemann surface is SIMPLY CONNECTED [III.93], then the uniformization theorem states that it is biholomorphically equivalent to the sphere, the Euclidean plane, or the HYPERBOLIC PLANE [I.3 §6.6]. These three spaces can all be viewed as Riemann surfaces, and they are all particularly symmetric: they have constant CURVATURE [III.78] (positive, zero, and negative, respectively); more generally, given any two points x and y in such a space, one can find a symmetry of the space that takes x to y, and one can ensure that a little arrow at x ends up pointing in any desired direction at y. Loosely speaking, these spaces “look the same from every point.”
It can be shown that an open subset of 
that is not the whole of cannot be biholomorphically equivalent to the sphere or to . Therefore, by the uniformization theorem, a simply connected open subset of that is not the whole of must be biholomorphically equivalent to the hyperbolic plane. This proves that any such set, no matter how irregular its boundary might be, can be mapped biholomorphically to any other. This result is called the Riemann mapping theorem. Biholomorphic maps are conformal: that is, if two curves in one set meet at an angle θ, then the angle between their images in the other set is also θ. So the Riemann mapping theorem implies that the interior of any simple closed curve can be mapped in an angle-preserving way to the open unit disk. Recall that one of the main models of the hyperbolic plane is Poincaré’s disk model. Thus, the hyperbolic metric on the disk together with the biholomorphic map that is given by the uniformization theorem can be used to define a hyperbolic metric on any simply connected proper open subset of .
If a Riemann surface is not simply connected, it is at least a QUOTIENT [I.3 §3.3] of a simply connected surface, namely its UNIVERSAL COVER [III.93]. For example, a torus is a quotient of the complex plane (in many possible ways that are topologically but not biholomorphically equivalent). Thus, the uniformization theorem tells us that a general Riemann surface is a quotient of the sphere, the Euclidean plane, or the hyperbolic plane. For a more detailed discussion of what such a quotient might be like, see FUCHSIAN GROUPS [III.28].