Rigidity of circle polyhedra and hyperideal polyhedra: the tangency case
John C. Bowers, Philip L. Bowers, Carl O. R. Lutz
TL;DR
This work extends global rigidity for proper triangulated convex hyperbolic circle polyhedra on the sphere to include tangency between neighboring vertex-disks, thereby generalizing the Koebe–Andre'ev–Thurston framework. It develops an extended notion of properness, adapts Minkowski/de Sitter geometry to the circle-polyhedron setting, and uses a deformation-and-limit strategy that bypasses the non-unitary obstruction previously required. Central to the approach is the inversive-measure map on the configuration space, which, under strict convexity, has full rank, enabling local-to-global rigidity via Möbius congruences and limiting arguments with non-unitary perturbations. The paper also clarifies conditions guaranteeing properness (e.g., global shallow-ness) and connects these results to hyperideal and Koebe-type polyhedra, while proposing conjectures about shallowness and duality.
Abstract
We prove the global rigidity of proper triangulated convex hyperbolic circle polyhedra on the sphere $\mathbb{S}^2$. These circle polyhedra correspond to proper triangulated convex hyperbolic polyhedra in the Beltrami-Klein model $\mathbb{B}^{3}$ of hyperbolic space with hyperideal vertices whose faces meet $\mathbb{B}^{3}$. Although the vertices of these polyhedra lie outside $\mathbb{B}^{3} \cup \mathbb{S}^{2}$ and the faces meet $\mathbb{B}^{3}$, the edges may miss $\mathbb{B}^{3}$ entirely, meet $\mathbb{B}^{3}$, or, more importantly, lie tangent to $\mathbb{B}^{3}$ at ideal points on the boundary $\partial \mathbb{B}^{3} = \mathbb{S}^{2}$. The latter case is new and generalizes the global rigidity results of both Bao-Bonahon and arXiv:1703.09338. This result also generalizes the uniqueness part of the celebrated Koebe-Andre'ev-Thurston theorem to the case where adjacent circles need not touch.