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Generalized circle patterns on surfaces with cusps

Zhiwen Xiong, Xu Xu

TL;DR

This work extends Guo-Luo’s generalized circle patterns to surfaces with cusps by establishing existence results for $(1,1,0)$ and $(0,0,\delta)$ types via a variational framework with strictly convex potentials and convex admissible spaces. It proves that prescribed generalized curvatures $\hat{K}$ can be realized by radii when natural edge-face substructure inequalities hold, and develops two curvature flows—the combinatorial Ricci flow and the combinatorial Calabi flow—that are negative-gradient flows of convex energies and converge exponentially to the desired patterns under the stated conditions. The results generalize Bobenko-Springborn’s hyperbolic circle patterns to cusped, generalized hyperbolic settings, and provide long-time existence and convergence guarantees for the associated flows, offering algorithmic means to construct generalized circle patterns on cusped surfaces. Collectively, this work broadens the applicability of discrete conformal geometry in noncompact settings and advances understanding of geometric structures via variational and flow methods.

Abstract

Guo and Luo introduced generalized circle patterns on surfaces and proved their rigidity. In this paper, we prove the existence of Guo-Luo's generalized circle patterns with prescribed generalized intersection angles on surfaces with cusps, which partially answers a question raised by Guo-Luo and generalizes Bobenko-Springborn's hyperbolic circle patterns on closed surfaces to generalized hyperbolic circle patterns on surfaces with cusps. We further introduce the combinatorial Ricci flow and combinatorial Calabi flow for generalized circle patterns on surfaces with cusps, and prove the longtime existence and convergence of the solutions for these combinatorial curvature flows.

Generalized circle patterns on surfaces with cusps

TL;DR

This work extends Guo-Luo’s generalized circle patterns to surfaces with cusps by establishing existence results for and types via a variational framework with strictly convex potentials and convex admissible spaces. It proves that prescribed generalized curvatures can be realized by radii when natural edge-face substructure inequalities hold, and develops two curvature flows—the combinatorial Ricci flow and the combinatorial Calabi flow—that are negative-gradient flows of convex energies and converge exponentially to the desired patterns under the stated conditions. The results generalize Bobenko-Springborn’s hyperbolic circle patterns to cusped, generalized hyperbolic settings, and provide long-time existence and convergence guarantees for the associated flows, offering algorithmic means to construct generalized circle patterns on cusped surfaces. Collectively, this work broadens the applicability of discrete conformal geometry in noncompact settings and advances understanding of geometric structures via variational and flow methods.

Abstract

Guo and Luo introduced generalized circle patterns on surfaces and proved their rigidity. In this paper, we prove the existence of Guo-Luo's generalized circle patterns with prescribed generalized intersection angles on surfaces with cusps, which partially answers a question raised by Guo-Luo and generalizes Bobenko-Springborn's hyperbolic circle patterns on closed surfaces to generalized hyperbolic circle patterns on surfaces with cusps. We further introduce the combinatorial Ricci flow and combinatorial Calabi flow for generalized circle patterns on surfaces with cusps, and prove the longtime existence and convergence of the solutions for these combinatorial curvature flows.

Paper Structure

This paper contains 18 sections, 16 theorems, 73 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Generalized circle patterns on surfaces
  3. Combinatorial curvature flows for the generalized circle patterns
  4. Organization of the paper
  5. Generalized hyperbolic circle patterns in the case of (1, 1, 0)
  6. The admissible space of generalized circle patterns in the case of (1, 1, 0)
  7. Limit behaviors of the generalized angles
  8. Existence theorem
  9. Prescribed curvature flows in the case of (1, 1, 0)
  10. The longtime behavior of the combinatorial Ricci flow
  11. The longtime behavior of the combinatorial Calabi flow
  12. Generalized hyperbolic circle patterns in the case of (0, 0, $\delta$)
  13. The admissible space of generalized circle patterns in the case of (0, 0, $\delta$)
  14. Limit behaviors of the generalized angles
  15. Existence theorem
  16. ...and 3 more sections

Key Result

Theorem 1.1

Suppose $\Sigma$ is a cellular decomposed closed surface. For any two functions $\theta: E\to (0,\pi)$ and $K: F\to (0,+\infty)$, there exists a unique $(1,1,1)$ type hyperbolic circle pattern with the generalized angle $\theta$ and the generalized combinatorial curvature $K$ if and only if Here $F^\prime$ is a nonempty subset of $F$ and $E^\prime$ is the set of all edges incident to any face in

Figures (2)

  • Figure 1: Generalized hyperbolic triangles for $\epsilon=1$, $\delta\in \{0,-1\}$
  • Figure 2: Generalized hyperbolic triangles for $\epsilon=0$, $\delta\in \{1,-1,0\}$

Theorems & Definitions (19)

  • Theorem 1.1: B-S, Theorem 3
  • Theorem 1.2: G-L, Theorem 1.6
  • Theorem 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • ...and 9 more