Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery

Abstract

The local rigidity theorem of WeiI [28] and Garland [12] for complete, finite volume hyperbolic manifolds states that there is no non-trivial deformation of such a structure through complete hyperbolic structures if the manifold has dimension at least 3. If the manifold is closed, the condition that the structures be complete is automatically satisfied. However, if the manifold is non-compact, there may be deformations through incomplete structures. This cannot happen in dimensions greater than 3 (Garland-Raghunathan [13]); but there are always non-trivial deformations in dimension 3 (Thurston [24]) in the non-compact case.