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A discrete uniformization theorem for decorated piecewise hyperbolic metrics on surfaces

Xu Xu, Chao Zheng

Abstract

In this paper, we study a natural discretization of the smooth Gaussian curvature on surfaces. A discrete uniformization theorem is established for this discrete Gaussian curvature. We further investigate the prescribing combinatorial curvature problem for a parametrization of this discrete Gaussian curvature, which is called the combinatorial $α$-curvature. To find decorated piecewise hyperbolic metrics with prescribed combinatorial $α$-curvatures, we introduce the combinatorial $α$-Ricci flow for decorated piecewise hyperbolic metrics. To handle the potential singularities along the combinatorial $α$-Ricci flow, we do surgery along the flow by edge flipping under the weighted Delaunay condition. Then we prove the longtime existence and convergence of the combinatorial $α$-Ricci flow with surgery. As an application of the combinatorial $α$-Ricci flow with surgery, we give the existence of decorated piecewise hyperbolic metrics with prescribed combinatorial $α$-curvatures. We further introduce the combinatorial $α$-Calabi flow with surgery and study its longtime behavior.

A discrete uniformization theorem for decorated piecewise hyperbolic metrics on surfaces

Abstract

In this paper, we study a natural discretization of the smooth Gaussian curvature on surfaces. A discrete uniformization theorem is established for this discrete Gaussian curvature. We further investigate the prescribing combinatorial curvature problem for a parametrization of this discrete Gaussian curvature, which is called the combinatorial -curvature. To find decorated piecewise hyperbolic metrics with prescribed combinatorial -curvatures, we introduce the combinatorial -Ricci flow for decorated piecewise hyperbolic metrics. To handle the potential singularities along the combinatorial -Ricci flow, we do surgery along the flow by edge flipping under the weighted Delaunay condition. Then we prove the longtime existence and convergence of the combinatorial -Ricci flow with surgery. As an application of the combinatorial -Ricci flow with surgery, we give the existence of decorated piecewise hyperbolic metrics with prescribed combinatorial -curvatures. We further introduce the combinatorial -Calabi flow with surgery and study its longtime behavior.
Paper Structure (14 sections, 27 theorems, 86 equations)

This paper contains 14 sections, 27 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Combinatorial curvature and discrete uniformization theorem
  3. Combinatorial $\alpha$-curvature flows
  4. Organization of the paper
  5. Combinatorial curvature flows for fixed triangulations
  6. Discrete conformal equivalence
  7. Local convergence of combinatorial $\alpha$-curvature flows
  8. Rigidity of the combinatorial $\alpha$-curvatures
  9. Bobenko-Lutz's discrete conformal theory
  10. Rigidity of the combinatorial $\alpha$-curvatures
  11. Combinatorial $\alpha$-Ricci flow with surgery
  12. An equivalent theorem
  13. Existence of decorated PH metrics with the prescribed combinatorial $\alpha$-curvatures
  14. Combinatorial $\alpha$-Calabi flow with surgery

Key Result

Theorem 1.1

Let $(d,r)$ be a decorated PH metric on a marked surface $(S,V)$ with Euler characteristic $\chi(S)<0$ and let $\overline{R}: V\rightarrow(-\infty,0]$ be a given function defined on the vertices. There exists a unique decorated PH metric $(\widetilde{d},\widetilde{r})$ discrete conformal equivalent

Theorems & Definitions (42)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Remark 1.5
  • Definition 1.6
  • Theorem 1.7
  • Definition 2.1: BL, Proposition 2.7
  • Remark 2.2
  • Lemma 2.3
  • ...and 32 more