In this paper we develop a new theory of infinitesimal harmonic deformations for compact hyperbolic
–manifolds
with “tubular boundary”. In particular, this applies to complements of tubes of radius at
least
around the singular set of hyperbolic cone manifolds, removing the previous
restrictions on cone angles.
We then apply this to obtain a new quantitative version of Thurston’s
hyperbolic Dehn surgery theorem, showing that all generalized Dehn surgery
coefficients outside a disc of “uniform” size yield hyperbolic structures. Here the
size of a surgery coefficient is measured using the Euclidean metric on a
horospherical cross section to a cusp in the complete hyperbolic metric,
rescaled to have area 1. We also obtain good estimates on the change in
geometry (eg volumes and core geodesic lengths) during hyperbolic Dehn
filling.
This new harmonic deformation theory has also been used by Bromberg and
his coworkers in their proofs of the Bers Density Conjecture for Kleinian
groups.