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Infinite rigidity of inversive distance circle packings in the Poincaré disk

Yanwen Luo, Xu Xu, Chao Zheng

TL;DR

This work develops a unified maximum principle framework for hyperbolic inversive distance circle packings on triangulated surfaces and leverages it to derive a discrete Schwarz–Ahlfors lemma. By bridging Euclidean and hyperbolic inversive distance theories and introducing weighted Delaunay conditions, the authors extend the maximum principle to non-compact settings and prove an infinite rigidity theorem in the Poincaré disk. The results generalize prior rigidity theorems (notably He's) to a broader hyperbolic inversive distance context and provide a robust toolkit for analyzing discrete conformal structures on non-compact surfaces within $\mathbb{D}$. The methods rely on transferring Euclidean maximal principles to hyperbolic geometry via the mutual induction of PE and PH metrics and exploiting invariance of inversive distance under these transitions, yielding both local and global rigidity conclusions with potential applications in discrete differential geometry and geometric topology.

Abstract

The maximum principle for hyperbolic inversive distance circle packings on polyhedral surfaces is established,which unifies and generalizes existing maximum principles for various types of circle packings in the literature.As an application of this principle, a discrete Schwarz-Ahlfors lemma is established.Furthermore, an infinite rigidity theorem for weighted Delaunay triangulations of the Poincaré disk is proved,which generalizes He's hyperbolic rigidity result \cite{He2}.

Infinite rigidity of inversive distance circle packings in the Poincaré disk

TL;DR

This work develops a unified maximum principle framework for hyperbolic inversive distance circle packings on triangulated surfaces and leverages it to derive a discrete Schwarz–Ahlfors lemma. By bridging Euclidean and hyperbolic inversive distance theories and introducing weighted Delaunay conditions, the authors extend the maximum principle to non-compact settings and prove an infinite rigidity theorem in the Poincaré disk. The results generalize prior rigidity theorems (notably He's) to a broader hyperbolic inversive distance context and provide a robust toolkit for analyzing discrete conformal structures on non-compact surfaces within . The methods rely on transferring Euclidean maximal principles to hyperbolic geometry via the mutual induction of PE and PH metrics and exploiting invariance of inversive distance under these transitions, yielding both local and global rigidity conclusions with potential applications in discrete differential geometry and geometric topology.

Abstract

The maximum principle for hyperbolic inversive distance circle packings on polyhedral surfaces is established,which unifies and generalizes existing maximum principles for various types of circle packings in the literature.As an application of this principle, a discrete Schwarz-Ahlfors lemma is established.Furthermore, an infinite rigidity theorem for weighted Delaunay triangulations of the Poincaré disk is proved,which generalizes He's hyperbolic rigidity result \cite{He2}.

Paper Structure

This paper contains 11 sections, 11 theorems, 36 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. Organization of the paper
  5. Hyperbolic maximal principle and discrete Schwarz-Ahlfors lemma
  6. Euclidean inversive distance circle packings
  7. Inversive distance
  8. Weighted Delaunay condition
  9. Euclidean maximum principle
  10. Hyperbolic maximum principle and discrete Schwarz-Ahlfors lemma
  11. Infinite rigidity of hyperbolic inversive distance circle packings

Key Result

Theorem 1.2

Let $\eta$ be a regular weight on $(P_n,\mathcal{T})$ satisfying the structure condition (Eq: SC) with $\eta: E\rightarrow (-1,1]$ or $\eta: E\rightarrow [0,+\infty)$. Let $r$ and $\bar{r}$ be two weighted Delaunay hyperbolic inversive distance circle packings on $(P_n,\mathcal{T},\eta)$, with all c

Figures (3)

  • Figure 1: The configuration of the four circles.
  • Figure 2: A star triangulation of a polygon.
  • Figure 3: The circles $C_0$ and $C_1$ with $\eta_{01}\in (-1,0)$.

Theorems & Definitions (15)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4: Discrete Schwarz-Ahlfors lemma
  • Theorem 1.5: Rigidity theorem
  • Theorem 2.1: L-X-Z, Theorem 3.1
  • Lemma 2.2
  • Theorem 2.3
  • Theorem 2.4: Discrete Schwarz-Ahlfors lemma
  • Corollary 2.5
  • ...and 5 more