A Discrete Uniformization Theorem for Polyhedral Surfaces II

Xianfeng Gu Stony Brook University Feng Luo Rutgers University Jian Sun Tsinghua University Tianqi Wu Tsinghua University

Differential Geometry Convex and Discrete Geometry mathscidoc:1705.10003

Accepted by Journal of Differential Geometry (JDG), 2016
A discrete conformality for hyperbolic polyhedral surfaces is introduced in this paper. This discrete conformality is shown to be computable. It is proved that each hyperbolic polyhedral metric on a closed surface is discrete conformal to a unique hyperbolic polyhedral metric with a given discrete curvature satisfying Gauss-Bonnet formula. Furthermore, the hyperbolic polyhedral metric with given curvature can be obtained using a discrete Yamabe flow with surgery. In particular, each hyperbolic polyhedral metric on a closed surface with negative Euler characteristic is discrete conformal to a unique hyperbolic metric.
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@inproceedings{xianfeng2016a,
  title={A Discrete Uniformization Theorem for Polyhedral Surfaces II},
  author={Xianfeng Gu, Feng Luo, Jian Sun, and Tianqi Wu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170530162956364308774},
  booktitle={Accepted by Journal of Differential Geometry (JDG)},
  year={2016},
}
Xianfeng Gu, Feng Luo, Jian Sun, and Tianqi Wu. A Discrete Uniformization Theorem for Polyhedral Surfaces II. 2016. In Accepted by Journal of Differential Geometry (JDG). http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170530162956364308774.
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