Discrete uniformization of finite branched covers over the Riemann sphere via hyper-ideal circle patterns

TL;DR

This paper develops a discrete uniformization method for finite branched covers over the Riemann sphere using hyper-ideal circle patterns, proving existence, uniqueness, and providing a convex optimization algorithm for construction.

Contribution

It introduces a novel discrete uniformization approach for branched covers and polyhedral surfaces using hyper-ideal circle patterns, with proven existence and uniqueness.

Findings

Discrete uniformization always exists for the considered surfaces.

The uniformization is unique for these surfaces.

A convex optimization algorithm effectively constructs the uniformization.

Abstract

With the help of hyper-ideal circle pattern theory, we have developed a discrete version of the classical uniformization theorems for surfaces represented as finite branched covers over the Riemann sphere as well as compact polyhedral surfaces with non-positive curvature. We show that in the case of such surfaces 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.