Deformations of hyperbolic convex polyhedra and 3-cone-manifolds

TL;DR

This paper proves local rigidity of convex hyperbolic polyhedra and hyperbolic 3-cone-manifolds, showing they are locally determined by dihedral angles, advancing understanding of the Stoker problem in hyperbolic geometry.

Contribution

It establishes local rigidity results for convex hyperbolic polyhedra and 3-cone-manifolds, demonstrating their local parameterization by dihedral angles, extending previous infinitesimal results.

Findings

Local rigidity of convex hyperbolic polyhedra proven

Space of polyhedra with fixed combinatorics is locally parameterized by dihedral angles

Rigidity results for hyperbolic 3-cone-manifolds established

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.