Discrete Uniformization of Polyhedral Surfaces with Non-positive Curvature and Branched Covers over the Sphere via Hyper-ideal Circle Patterns

Abstract

With the help of hyper-ideal circle pattern theory, we develop a discrete version of the classical uniformization theorems for closed polyhedral surfaces with non-positive curvature and for surfaces represented as finite branched covers over the Riemann sphere. We show that in these cases discrete uniformization via hyper-ideal circle patterns always exists and is unique. We also propose a numerical algorithm, utilizing convex optimization, that constructs the desired discrete uniformization.