This work is devoted to the study of deformations of hyperbolic cone structures under the assumption that the lengths of the singularity remain uniformly bounded over the deformation. Given a sequence (M_{ i},p_{ i}) of pointed hyperbolic cone-manifolds with topological type (M,{ \Sigma}), where M is a closed, orientable and irreducible 3-manifold and { \Sigma} an embedded link in M. Assuming that the lengths of the singularity remain uniformly bounded, we prove that either the sequence M_{ i} collapses and M is Seifert fibered or a Sol manifold, or the sequence M_{ i} does not collapse and in this case a subsequence of (M_{ i},p_{ i}) converges to a complete Alexandrov space of dimension 3 endowed with a hyperbolic metric of finite volume on the complement of a finite union of quasi-geodesics. We apply this result to a conjecture of Thurston and to the case where { \Sigma} is a small link in M.
Categories: math.GT