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The Weil–Petersson gradient flow of renormalized volume and $3$–dimensional convex cores

Martin Bridgeman, Jeffrey Brock and Kenneth Bromberg

Geometry & Topology 27 (2023) 3183–3228
Abstract

We use the Weil–Petersson gradient flow for renormalized volume to study the space CC(N;S,X) of convex cocompact hyperbolic structures on the relatively acylindrical 3–manifold (N;S). Among the cases of interest are the deformation space of an acylindrical manifold and the Bers slice of quasifuchsian space associated to a fixed surface. To treat the possibility of degeneration along flow-lines to peripherally cusped structures, we introduce a surgery procedure to yield a surgered gradient flow that limits to the unique structure Mgeod CC(N;S,X) with totally geodesic convex core boundary facing S. Analyzing the geometry of structures along a flow line, we show that if V R(M) is the renormalized volume of M, then V R(M) V R(Mgeod ) is bounded below by a linear function of the Weil–Petersson distance dWP(cM,cMgeod ), with constants depending only on the topology of S. The surgered flow gives a unified approach to a number of problems in the study of hyperbolic 3–manifolds, providing new proofs and generalizations of well-known theorems such as Storm’s result that Mgeod has minimal volume for N acylindrical and the second author’s result comparing convex core volume and Weil–Petersson distance for quasifuchsian manifolds.

Keywords
hyperbolic 3–manifold, renormalized volume, Weil–Petersson metric
Mathematical Subject Classification
Primary: 32G15, 30F40, 30F60
Secondary: 32Q45, 51P05
References
Publication
Received: 1 February 2021
Revised: 21 January 2022
Accepted: 1 April 2022
Published: 9 November 2023
Proposed: Benson Farb
Seconded: Tobias H Colding, David Gabai
Authors
Martin Bridgeman
Department of Mathematics
Boston College
Chestnut Hill, MA
United States
https://sites.google.com/bc.edu/martin-bridgeman
Jeffrey Brock
Department of Mathematics
Yale University
New Haven, CT
United States
http://www.jeffbrock.net
Kenneth Bromberg
Department of Mathematics
University of Utah
Salt Lake City, UT
United States
http://math.utah.edu/~bromberg

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