Infinitesimal rigidity of cone-manifolds and the Stoker problem for hyperbolic and Euclidean polyhedra

TL;DR

This paper proves infinitesimal rigidity for hyperbolic and Euclidean cone-manifolds with cone angles less than 2π, advancing understanding of their deformation theory and implications for polyhedral moduli and 3-manifold topology.

Contribution

It establishes a new infinitesimal rigidity result for cone angles less than 2π, extending previous work to a broader range of cone angles and using analytic methods on stratified spaces.

Findings

Infinitesimal deformations fixing dihedral angles are trivial for hyperbolic cone-manifolds.

In Euclidean cases, deformations are limited to simple, well-understood types.

The methods involve elliptic operators on stratified spaces, linking deformation theory to Einstein metrics.

Abstract

The deformation theory of hyperbolic and Euclidean cone-manifolds with all cone angles less then 2{\pi} plays an important role in many problems in low dimensional topology and in the geometrization of 3-manifolds. Furthermore, various old conjectures dating back to Stoker about the moduli of convex hyperbolic and Euclidean polyhedra can be reduced to the study of deformations of cone-manifolds by doubling a polyhedron across its faces. This deformation theory has been understood by Hodgson and Kerckhoff when the singular set has no vertices, and by Wei{\ss} when the cone angles are less than {\pi}. We prove here an infinitesimal rigidity result valid for cone angles less than 2{\pi}, stating that infinitesimal deformations which leave the dihedral angles fixed are trivial in the hyperbolic case, and reduce to some simple deformations in the Euclidean case. The method is to treat this as a problem concerning the deformation theory of singular Einstein metrics, and to apply analytic methods about elliptic operators on stratified spaces. This work is an important ingredient in the local deformation theory of cone-manifolds by the second author, see also the concurrent work by Wei{\ss}.