Prescribing discrete Gaussian curvature on polyhedral surfaces

Xu Xu; Chao Zheng

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arXiv · 2110.12326

Prescribing discrete Gaussian curvature on polyhedral surfaces

Xu Xu·Chao Zheng

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Abstract

Vertex scaling of piecewise linear metrics on surfaces introduced by Luo is a straightforward discretization of smooth conformal structures on surfaces. Combinatorial $\alpha$-curvature for vertex scaling of piecewise linear metrics on surfaces is a discretization of Gaussian curvature on surfaces. In this paper, we investigate the prescribing combinatorial $\alpha$-curvature problem on polyhedral surfaces. Using Gu-Luo-Sun-Wu's discrete conformal theory for piecewise linear metrics on surfaces and variational principles with constraints, we prove some Kazdan-Warner type theorems for prescribing combinatorial $\alpha$-curvature problem, which generalize the results obtained by Gu-Luo-Sun-Wu on prescribing combinatorial curvatures on surfaces. Gu-Luo-Sun-Wu conjectured that one can prove Kazdan-Warner's theorems via approximating smooth surfaces by polyhedral surfaces. This paper takes the first step in this direction.

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Xu Xu, Chao Zheng. 2021-10-24. Prescribing discrete Gaussian curvature on polyhedral surfaces. https://arxiv.org/abs/2110.12326

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