A discrete uniformization theorem for polyhedral surfaces II

TL;DR

This paper introduces a computable notion of discrete conformality for hyperbolic polyhedral surfaces, proving a unique correspondence between metrics with prescribed curvature via a discrete Yamabe flow.

Contribution

It establishes a discrete uniformization theorem for hyperbolic polyhedral surfaces, including existence, uniqueness, and a flow-based method for metric deformation.

Findings

Discrete conformality is computable.

Unique hyperbolic metric with prescribed curvature exists.

Discrete Yamabe flow with surgery converges to the uniformized metric.

Abstract

A discrete conformality for hyperbolic polyhedral surfaces is introduced in this paper. This discrete conformality is shown to be computable. It is proved that each hyperbolic polyhedral metric on a closed surface is discrete conformal to a unique hyperbolic polyhedral metric with a given discrete curvature satisfying Gauss-Bonnet formula. Furthermore, the hyperbolic polyhedral metric with given curvature can be obtained using a discrete Yamabe flow with surgery. In particular, each hyperbolic polyhedral metric on a closed surface with negative Euler characteristic is discrete conformal to a unique hyperbolic metric.