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

Bridgeman, Martin; Brock, Jeffrey; Bromberg, Kenneth

doi:10.2140/gt.2023.27.3183

Download this article
	For screen
	For printing

Recent Issues

The Journal

Submission Guidelines

Submission Page

Policies for Authors

Ethics Statement

ISSN 1364-0380 (online)

ISSN 1465-3060 (print)

Author Index

To Appear

Other MSP Journals

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

DOI: 10.2140/gt.2023.27.3183

Abstract

We use the Weil–Petersson gradient flow for renormalized volume to study the space $C C (N; S, X)$ 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 $M_{geod} \in C C (N; S, X)$ 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 $V_{R} (M) - V_{R} (M_{geod})$ is bounded below by a linear function of the Weil–Petersson distance $d_{W P} (\partial_{c} M, \partial_{c} M_{geod})$ , 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 $M_{geod}$ 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

Open Access made possible by participating institutions via Subscribe to Open.

Volume 27, issue 8 (2023)