A discrete uniformization theorem for polyhedral surfaces II

Xianfeng Gu, Ren Guo, Feng Luo, Jian Sun, Tianqi Wu

Research output: Contribution to journalArticle

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Abstract

A notion of 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.

Original languageEnglish (US)
Pages (from-to)431-466
Number of pages36
JournalJournal of Differential Geometry
Volume109
Issue number3
DOIs
StatePublished - Jul 2018

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All Science Journal Classification (ASJC) codes

  • Analysis
  • Algebra and Number Theory
  • Geometry and Topology

Keywords

  • And Delaunay triangulation.
  • Discrete Yamabe flow
  • Discrete conformality
  • Discrete uniformization
  • Hyperbolic metrics
  • Variational principle

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