The shape of hyperbolic Dehn surgery space - Mathematics > Geometric Topology

Abstract: In this paper we develop a new theory of infinitesimal harmonic deformationsfor compact hyperbolic 3-manifolds with tubular boundary-. In particular,this applies to complements of tubes of radius at least $R 0 =\arctanh1-\sqrt{3} \approx 0.65848$ around the singular set of hyperboliccone manifolds, removing the previous restrictions on cone angles.We then apply this to obtain a new quantitative version of Thurston-shyperbolic Dehn surgery theorem, showing that all generalized Dehn surgerycoefficients outside a disc of uniform- size yield hyperbolic structures.Here the size of a surgery coefficient is measured using the Euclidean metricon 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 ingeometry e.g. volumes and core geodesic lengths during hyperbolic Dehnfilling.This new harmonic deformation theory has also been used by Bromberg and hiscoworkers in their proofs of the Bers Density Conjecture for Kleinian groups.

Author: Craig D. Hodgson, Steven P. Kerckhoff

Source: https://arxiv.org/