How large is a Riemann surface: the type problem

Abstract

The uniformization theorem asserts that a simply-connected non-compact Riemann surface S is conformally equivalent to precisely one of the unit disk D or the finite complex plane C. While this result (nearly a century old) closes one chapter in the theory of analytic functions of one complex variable, it opens another: given a surface S described in some explicit manner, determine from intrinsic considerations which of the conformal types S is. While this subject reached a zenith of activity in the 1930s, recent developments and the availability of new tools suggest a resurgence of interest.