Discrete Yamabe problem for polyhedral surfaces

TL;DR

This paper introduces a new way to discretize Gaussian curvature on polyhedral surfaces, proving the existence of constant curvature surfaces within each conformal class, and exploring their non-uniqueness.

Contribution

It proposes a novel discretization method for Gaussian curvature and extends discrete conformal theory to polyhedral surfaces, establishing existence results.

Findings

Existence of constant discrete Gaussian curvature surfaces in each conformal class

Explicit examples show non-uniqueness of these surfaces

New discretization approach based on angle defect and Voronoi cell area

Abstract

We study a new discretization of the Gaussian curvature for polyhedral surfaces. This discrete Gaussian curvature is defined on each conical singularity of a polyhedral surface as the quotient of the angle defect and the area of the Voronoi cell corresponding to the singularity. We divide polyhedral surfaces into discrete conformal classes using a generalization of discrete conformal equivalence pioneered by Feng Luo. We subsequently show that, in every discrete conformal class, there exists a polyhedral surface with constant discrete Gaussian curvature. We also provide explicit examples to demonstrate that this surface is in general not unique.