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Combinatorial Ricci flows on infinite disk triangulations

Huabin Ge, Bobo Hua, Puchun Zhou

TL;DR

The paper develops combinatorial Ricci flows on infinite disk triangulations in both Euclidean and hyperbolic geometries, establishing global existence and uniqueness under curvature bounds and proving convergence results that realize a discrete uniformization: hyperbolic CRFs converge to zero curvature when starting from non-positive curvature, and Euclidean CRFs on the hexagonal lattice converge to the regular packing for small initial perturbations. The analysis combines exhaustion by finite subcomplexes, maximum principles, and energy methods, with a semilinear parabolic reformulation on the hexagonal lattice to obtain explicit convergence criteria. A key contribution is linking CRF convergence to vertex extremal length (VEL) properties and circle-packing geometries on infinite graphs, including corollaries about VEL-hyperbolicity and random walks. The work also lays out conjectures positioning CRFs as tools for uniformization and ideal circle patterns in noncompact settings, supported by technical appendices on angle derivatives and related estimates.

Abstract

In this paper, we introduce combinatorial Ricci flows (CRFs in short) in Euclidean and hyperbolic background geometries on infinite triangulations of the open disk, which are discrete analogs of Ricci flows on simply connected open surfaces. We establish well-posedness results, the existence and the uniqueness, of CRFs in both Euclidean and hyperbolic background geometries. Moreover, we prove convergence results of CRFs, which indicate a uniformization theorem for CRFs on infinite disk triangulations. As an application, we prove an existence result of circle-packing metrics with infinite prescribed cone angles in hyperbolic background geometry. To our knowledge, these are the first results of CRFs on infinite triangulations.

Combinatorial Ricci flows on infinite disk triangulations

TL;DR

The paper develops combinatorial Ricci flows on infinite disk triangulations in both Euclidean and hyperbolic geometries, establishing global existence and uniqueness under curvature bounds and proving convergence results that realize a discrete uniformization: hyperbolic CRFs converge to zero curvature when starting from non-positive curvature, and Euclidean CRFs on the hexagonal lattice converge to the regular packing for small initial perturbations. The analysis combines exhaustion by finite subcomplexes, maximum principles, and energy methods, with a semilinear parabolic reformulation on the hexagonal lattice to obtain explicit convergence criteria. A key contribution is linking CRF convergence to vertex extremal length (VEL) properties and circle-packing geometries on infinite graphs, including corollaries about VEL-hyperbolicity and random walks. The work also lays out conjectures positioning CRFs as tools for uniformization and ideal circle patterns in noncompact settings, supported by technical appendices on angle derivatives and related estimates.

Abstract

In this paper, we introduce combinatorial Ricci flows (CRFs in short) in Euclidean and hyperbolic background geometries on infinite triangulations of the open disk, which are discrete analogs of Ricci flows on simply connected open surfaces. We establish well-posedness results, the existence and the uniqueness, of CRFs in both Euclidean and hyperbolic background geometries. Moreover, we prove convergence results of CRFs, which indicate a uniformization theorem for CRFs on infinite disk triangulations. As an application, we prove an existence result of circle-packing metrics with infinite prescribed cone angles in hyperbolic background geometry. To our knowledge, these are the first results of CRFs on infinite triangulations.

Paper Structure

This paper contains 15 sections, 30 theorems, 118 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Organization of the paper and proof strategies
  3. Preliminaries
  4. Circle-packing metrics and discrete conformal factors
  5. Variational principles
  6. Vertex extremal length
  7. Well-posedness of CRF on infinite disk triangulations
  8. The long-time existence of the flow \ref{['infinite_flow']}
  9. The uniqueness of the flow
  10. Convergence of the hyperbolic CRF
  11. Convergence of the Euclidean CRF on $\mathcal{T}_H$
  12. Discrete Laplacian and the analysis on the hexagonal lattice
  13. The Euclidean CRF as a semilinear parabolic equation
  14. The conjectures
  15. Appendix

Key Result

Theorem 1.1

Let $\mathcal{T}=(V,E,F)$ be an infinite disk triangulation with intersection angle $\Phi\in [0,\frac{\pi}{2}]^E.$ For any initial data $r(0),$ there exists a solution $r(t), t\in[0,\infty),$ to the flow unnormalized or hyperbolic_flow . Moreover, the solution to the flow with uniformly bounded disc

Figures (3)

  • Figure 1: A part of a regular hexagonal packing.
  • Figure 2: The circle-packing metric in Euclidean background geometry on a triangle.
  • Figure 3: A circle packing of a triangle with power center locating at the origin in the Poincaré disk model.

Theorems & Definitions (58)

  • Theorem 1.1: Well-posedness of the CRF
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4: Convergence of the hyperbolic CRF
  • Remark 1.5
  • Corollary 1.6
  • Theorem 1.7: Convergence of Euclidean CRF on $\mathcal{T}_H$
  • Conjecture 1.1: Uniformization of the CRF
  • Lemma 2.1
  • Proposition 2.2
  • ...and 48 more