Convergence of discrete conformal geometry and computation of uniformization maps

Abstract

The classical uniformization theorem of Poincaré and Koebe states that any simply connected surface with a Riemannian metric is conformally diffeomorphic to the Riemann sphere, or the complex plane or the unit disk. Using the work by Gu-Luo-Sun-Wu [9] on discrete conformal geometry for polyhedral surfaces, we show that the uniformization maps for simply connected Riemann surfaces are computable.