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arXiv · 1703.09338

Rigidity of Circle Polyhedra in the 2-Sphere and of Hyperideal Polyhedra in Hyperbolic 3-Space

John C. Bowers·Philip L. Bowers·Kevin Pratt
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Abstract

We generalize Cauchy's celebrated theorem on the global rigidity of convex polyhedra in Euclidean $3$-space $\mathbb{E}^{3}$ to the context of circle polyhedra in the $2$-sphere $\mathbb{S}^{2}$. We prove that any two convex and proper non-unitary c-polyhedra with M\"obius-congruent faces that are consistently oriented are M\"obius-congruent. Our result implies the global rigidity of convex inversive distance circle packings in the Riemann sphere as well as that of certain hyperideal hyperbolic polyhedra in $\mathbb{H}^{3}$.

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John C. Bowers, Philip L. Bowers, Kevin Pratt. 2017-03-27. Rigidity of Circle Polyhedra in the 2-Sphere and of Hyperideal Polyhedra in Hyperbolic 3-Space. https://arxiv.org/abs/1703.09338

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