Characterizations of infinite circle patterns and convex polyhedra in hyperbolic 3-space

TL;DR

This work advances the theory of infinite circle patterns and their 3D hyperbolic counterparts by proving rigidity and uniformization for regular circle patterns with $0\leq Θ<π$ and by characterizing infinite trivalent hyperbolic polyhedra (THP) through the CP–THP correspondence. The authors develop existence results via a Ring Lemma for obtuse angles, Thurston’s CP construction, and $(Z_1)$–$(Z_4)$ angle conditions, then establish a uniformization framework using vertex-extremal-length theory to relate CPs to recurrent/escaping networks. They prove rigidity for hyperbolic RCPs and, separately, for parabolic RCPs, showing that the geometry is determined up to isometries or Euclidean similarities under natural hypotheses. Finally, they connect infinite THP to RCPs, delineating parabolic vs hyperbolic accumulation, and obtain existence, uniformization, and rigidity statements that extend classical finite results of Andreev and Rivin–Hodgson to the infinite setting, providing a unified view of infinite circle packings and infinite hyperbolic polyhedra.

Abstract

Since Thurston pioneered the connection between circle packing (abbr. CP) and three-dimensional geometric topology, the characterization of CPs and hyperbolic polyhedra has become increasingly profound. Some milestones have been achieved, for example, Rodin-Sullivan \cite{Rodin-Sullivan} and Schramm \cite{schramm91} proved the rigidity of infinite CPs with the intersection angle $Θ=0$. Rivin-Hodgson \cite{RH93} fully characterized the existence and rigidity of compact convex polyhedra in $\mathbb{H}^3$. He \cite{He} proved the rigidity and uniformization theorem for infinite CPs with $0\leqΘ\leq π/2$. \cite{He} also envisioned that "in a future paper, the techniques of this paper will be extended to the case when $0\leqΘ<π$. In particular, we will show a rigidity property for a class of infinite convex polyhedra in the 3-dimentional hyperbolic space". The article aims to accomplish the work claimed in \cite{He} by proving the rigidity and uniformization theorem for infinite CPs with $0\leqΘ<π$, as well as infinite trivalent hyperbolic polyhedra. We will pay special attention to CPs whose contact graphs are disk triangulation graphs. Such CPs are called regular because they exclude some singular configurations and correspond well to hyperbolic polyhedra. We will establish the existence and the rigidity of infinite regular CPs. Moreover, we will prove a uniformization theorem for regular CPs, which solves the classification problem for regular CPs. Thereby, the existence and rigidity of infinite convex trivalent polyhedra are obtained.

Figures (16)

• Figure 1: two circle configuration
• Figure 2: two reducible edges
• Figure 3: He's reducible configuration (Bowers-Stephenson's extraneous tangencies)
• Figure 4: two adjacent triangle configuration
• Figure 5: three circle configuration

Theorems & Definitions (113)

• Theorem 1.1: Existence of RCPs
• Theorem 1.2: Existence of THP
• Theorem 1.3: Rigidity of CPs
• Theorem 1.4: Uniformization of RCP/THP
• Theorem 1.5: Rigidity of THP
• Remark 1.6
• Lemma 2.2
• Example 2.3
• Definition 2.4: RCP
• Lemma 2.5