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

2023/09/13 by Xu, Xu, Zheng, Chao
#52C26 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2309.06685

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.

Related

• A discrete uniformization theorem for decorated piecewise Euclidean metrics on surfaces
• A discrete uniformization theorem for polyhedral surfaces
• A discrete uniformization theorem for polyhedral surfaces II
• Parameterized discrete uniformization theorems and curvature flows for polyhedral surfaces, I
• Decorated discrete conformal equivalence in non-Euclidean geometries