Deformation of three-dimensional hyperbolic cone structures: the noncollapsing case

Abstract

This work is devoted to the study of deformations of hyperbolic cone structures under the assumption that the length of the singularity remains uniformly bounded over the deformation. Let.Mi; pi/ be a sequence of pointed hyperbolic cone manifolds with cone angles of at most 2 and topological type .M;6/, where M is a closed, orientable and irreducible 3-manifold and6 an embedded link in M. Assuming that the length of the singularity remains uniformly bounded, we prove that either the sequence Mi collapses and M is Seifert fibered or a Sol manifold, or the sequence Mi does not collapse and, in this case, a subsequence of.Mi; pi/ converges to a complete three dimensional Alexandrov space endowed with a hyperbolic metric of finite volume on the complement of a finite union of quasigeodesics. We apply this result to a question proposed by Thurston and to provide universal constants for hyperbolic cone structures when6 is a small link in M.