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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

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.

hyperbolic 3–manifold, renormalized volume, Weil–Petersson metric
Mathematical Subject Classification
Primary: 32G15, 30F40, 30F60
Secondary: 32Q45, 51P05
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
Martin Bridgeman
Department of Mathematics
Boston College
Chestnut Hill, MA
United States
Jeffrey Brock
Department of Mathematics
Yale University
New Haven, CT
United States
Kenneth Bromberg
Department of Mathematics
University of Utah
Salt Lake City, UT
United States

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