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

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.

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Published 13 March 2014