Harmonic deformations of hyperbolic 3-manifolds

Abstract

This paper gives an exposition of the authors' harmonic deformation theory for 3-dimensional hyperbolic cone-manifolds. We discuss topological applications to hyperbolic Dehn surgery as well as recent applications to Kleinian group theory. A central idea is that local rigidity results (for deformations fixing cone angles) can be turned into effective control on the deformations that do exist. This leads to precise analytic and geometric versions of the idea that hyperbolic structures with short geodesics are close to hyperbolic structures with cusps. The paper also outlines a new harmonic deformation theory which applies whenever there is a sufficiently large embedded tube around the singular locus, removing the previous restriction to cone angles at most 2pi.