A discrete uniformization theorem for decorated piecewise Euclidean metrics on surfaces, II

TL;DR

This paper introduces a discrete uniformization theorem for decorated Euclidean metrics on surfaces, establishing conditions under which a surface's discrete curvature can be uniformly prescribed, extending classical uniformization results to a discrete setting.

Contribution

It proves that surfaces with nonpositive Euler number can be discretely conformally transformed to metrics with constant discrete curvature, using a novel discrete conformal theory and variational methods.

Findings

Discrete curvature can be prescribed via conformal transformations.

Surfaces with nonpositive Euler number admit uniformization with constant discrete curvature.

The approach extends classical uniformization to discrete geometric settings.

Abstract

In this paper, we study a natural discretization of the smooth Gaussian curvature on surfaces, which is defined as the quotient of the angle defect and the area of a geodesic disk at a vertex of a polyhedral surface. It is proved that each decorated piecewise Euclidean metric on surfaces with nonpositive Euler number is discrete conformal to a decorated piecewise Euclidean metric with this discrete curvature constant. We further investigate the prescribing combinatorial curvature problem for a parametrization of this discrete curvature and prove some Kazdan-Warner type results. The main tools are Bobenko-Lutz's discrete conformal theory for decorated piecewise Euclidean metrics on surfaces and variational principles with constraints.