Deformations of hyperbolic convex polyhedra and 3-cone-manifolds

Abstract

The Stoker problem, first formulated in 1968, consists in understanding to what extent a convex polyhedron is determined by its dihedral angles. By means of the double construction, this problem is intimately related to rigidity issues for 3-dimensional cone-manifolds. In a former paper, two such rigidity results were proven, implying that the infinitesimal version of the Stoker conjecture is true in the hyperbolic and Euclidean cases. In this second article, we prove that local rigidity holds and obtain that the space of convex hyperbolic polyhedra with given combinatorial type is locally parameterized by the set of dihedral angles, together with a similar statement for hyperbolic 3-cone-manifolds.