Surveys in Differential Geometry

Volume 25 (2020)

Volumes of quasifuchsian manifolds

Pages: 319 – 353



Jean-Marc Schlenker (Department of Mathematics, University of Luxembourg)


Quasifuchsian hyperbolic manifolds, or more generally convex co-compact hyperbolic manifolds, have infinite volume, but they have a well-defined “renormalized” volume. We outline some relations between this renormalized volume and the volume, or more precisely the “dual volume”, of the convex core. On one hand, there are striking similarities between them, for instance in their variational formulas. On the other, objects related to them tend to be within a bounded distance. These analogies and proximities lead to several questions. Both the renormalized volume and the dual volume can be used for instance to bound the volume of the convex core in terms of the Weil–Petersson distance between the conformal metrics at infinity.

The author was partially supported by UL IRP grant NeoGeo and FNR grants INTER/ANR/15/ 11211745 and OPEN/16/11405402. The author also acknowledge support from U.S. National Science Foundation grants DMS-1107452, 1107263, 1107367 “RNMS: GEometric structures And Representation varieties” (the GEAR Network).

Published 13 July 2022