Rigidity of circle polyhedra in the $2$-sphere and of hyperideal polyhedra in hyperbolic $3$-space

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 nonunitary c-polyhedra with Möbius-congruent faces that are consistently oriented are Möbius 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 $\mathbb {H}^{3}$.