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 Teichmuller space which are large must have enough negative curvature. 1. Results 1.1. Notations. In all the paper we consider a closed surface S of genus g ≥ 2, and we call TS the Teichmuller space of S. Given a complex structure c ∈ TS, we denote by Qc the vector space of holomorphic quadratic differentials on (S,c). We also call GS the space of quasifuchsian metrics on S × R, considered up to isotopy. The Bers Simultaneous Uniformization Theorem provides a homeomorphism between T S∪S and GS. So GS is parameterized by T+ × T−, where T+ and T− are two copies of TS corresponding respectively to the upper and lower boundaries at infinity of S × R. For q ∈ G, we denote by VR(q) the renormalized volume of (S × R,g) (as defined in Section 3 following e.g. (12)), while C(q) is the convex core of (S × R,g) and VC(q) is its volume. 1.2. Comparing the renormalized volume to the volume of the convex core. The first result pre- sented here is a precise comparison between the volume of the convex core and the renormalized volume of a quasifuchsian hyperbolic manifold.