Journal of Differential Geometry

A discrete uniformization theorem for polyhedral surfaces II

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

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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.

Article information

Source
J. Differential Geom., Volume 109, Number 3 (2018), 431-466.

Dates
Received: 16 April 2014
First available in Project Euclid: 10 July 2018

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1531188190

Digital Object Identifier
doi:10.4310/jdg/1531188190

Mathematical Reviews number (MathSciNet)
MR3825607

Zentralblatt MATH identifier
06877019

Subjects
Primary: 52C26: Circle packings and discrete conformal geometry 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.) 58E30: Variational principles

Keywords
hyperbolic metrics discrete uniformization discrete conformality discrete Yamabe flow variational principle Delaunay triangulation

Citation

Gu, Xianfeng; Guo, Ren; Luo, Feng; Sun, Jian; Wu, Tianqi. A discrete uniformization theorem for polyhedral surfaces II. J. Differential Geom. 109 (2018), no. 3, 431--466. doi:10.4310/jdg/1531188190. https://projecteuclid.org/euclid.jdg/1531188190


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