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We show that the renormalized volume of a quasifuchsian hyperbolic 3-manifoldis equal, up to an additive constant, to the volume of its convex core. We alsoprovide a precise upper bound on the renormalized volume in terms of theWeil-Petersson distance between the conformal structures at infinity. As aconsequence we show that holomorphic disks in Teichm\"uller space which arelarge enough must have "enough" negative curvature.

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