Rigidity of Koebe Polyhedra and Inversive Distance Circle Packings

TL;DR

This paper proves the global rigidity of hyperbolic inversive distance circle packings and Koebe polyhedra on the sphere, removing previous restrictions on edge tangency, thus generalizing key rigidity and uniqueness results in hyperbolic geometry.

Contribution

It establishes the global rigidity of inversive distance circle packings and Koebe polyhedra without the previous tangency restrictions, extending fundamental geometric theorems.

Findings

Proves global rigidity under mild vertex link assumptions

Removes restrictions on edge tangency in hyperbolic polyhedra

Generalizes classical rigidity and uniqueness theorems

Abstract

Hyperbolic inversive distance circle packings on the $2$-sphere correspond to Koebe polyhedra in the Beltrami-Klein model $\mathbb{B}^{3}$ of hyperbolic $3$-space. Koebe polyhedra are triangulated convex hyperbolic polyhedra with hyperideal vertices whose faces meet $\mathbb{B}^{3}$. We prove the global rigidity of these circle packings or, equivalently, of these Koebe polyhedra under mild assumptions on the links of their vertices. Previous rigidity results apply only when all edges of the Koebe polyhedron are tangent or, alternatively, when no edge is tangent to the ideal boundary of hyperbolic space. We remove these restrictions. This generalizes the global rigidity results of both Bao-Bonahon and Bowers-Bowers-Pratt (arXiv:1703.09338), as well as the uniqueness part of the celebrated Koebe-Andre'ev-Thurston Theorem to the case where adjacent circles need not touch.