The renormalized volume and the volume of the convex core of quasifuchsian manifolds

Abstract

We show that the renormalized volume of a quasifuchsian hyperbolic 3-manifold is equal, up to an additive constant, to the volume of its convex core. We also provide a precise upper bound on the renormalized volume in terms of the Weil-Petersson distance between the conformal structures at infinity. As a consequence we show that holomorphic disks in Teichmüller space which are large enough must have "enough" negative curvature.

arXiv comment9 pages, no figure. v2: 14pp, final version, to appear in MRL. v3: corrected an additive term in Thm 1.1, and other corrections without consequences on the main statements. v4: further corrections and better exposition, added a new/simpler formulation of the main variational formula (Cor 3.11). (Special thanks to C McMullen.)