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 language | English (US) |
|---|---|
| Pages (from-to) | 431-466 |
| Number of pages | 36 |
| Journal | Journal of Differential Geometry |
| Volume | 109 |
| Issue number | 3 |
| DOIs | |
| State | Published - 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
Cite this
Gu, X., Guo, R., Luo, F., Sun, J., & Wu, T. (2018). A discrete uniformization theorem for polyhedral surfaces II. Journal of Differential Geometry, 109(3), 431-466. https://doi.org/10.4310/jdg/1531188190