A discrete uniformization theorem for decorated piecewise Euclidean metrics on surfaces

TL;DR

This paper introduces a new way to discretize Gaussian curvature on surfaces and proves a uniformization theorem for these discrete metrics on surfaces with non-positive Euler number.

Contribution

It develops a novel discretization of Gaussian curvature and establishes a discrete uniformization theorem using discrete conformal theory and variational principles.

Findings

Discrete Gaussian curvature defined via angle defect and dual cell area

Proved a uniformization theorem for non-positive Euler number surfaces

Utilized Bobenko-Lutz's discrete conformal theory and variational methods

Abstract

In this paper, we introduce a new discretization of the Gaussian curvature on surfaces, which is defined as the quotient of the angle defect and the area of some dual cell of a weighted triangulation at the conic singularity. A discrete uniformization theorem for this discrete Gaussian curvature is established on surfaces with non-positive Euler number. The main tools are Bobenko-Lutz's discrete conformal theory for decorated piecewise Euclidean metrics on surfaces and variational principles with constraints.