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arXiv · 2309.06685

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

Xu Xu·Chao Zheng
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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.

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Xu Xu, Chao Zheng. 2023-09-13. A discrete uniformization theorem for decorated piecewise Euclidean metrics on surfaces, II. https://arxiv.org/abs/2309.06685

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