Volume 12, issue 2 (2008)

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The shape of hyperbolic Dehn surgery space

Craig D Hodgson and Steven P Kerckhoff

Geometry & Topology 12 (2008) 1033–1090

In this paper we develop a new theory of infinitesimal harmonic deformations for compact hyperbolic 3–manifolds with “tubular boundary”. In particular, this applies to complements of tubes of radius at least R0 = arctanh(13) 0.65848 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.

Dedicated to Bill Thurston on his 60th birthday

hyperbolic Dehn surgery, harmonic deformation
Mathematical Subject Classification 2000
Primary: 57M50
Secondary: 57N10
Forward citations
Received: 22 September 2007
Accepted: 20 February 2008
Published: 12 May 2008
Proposed: Dave Gabai
Seconded: Cameron Gordon, Walter Neumann
Craig D Hodgson
Department of Mathematics and Statistics
University of Melbourne
Victoria 3010
Steven P Kerckhoff
Department of Mathematics
Stanford University
Stanford, CA 94305