The Weil-Petersson metric and the renormalized volume of hyperbolic 3-manifolds

Details The Weil-Petersson metric and the renormalized volume of hyperbolic 3-manifolds The Weil-Petersson metric and the renormalized volume of hyperbolic 3-manifolds

2009-07-15

arxiv.org/abs/0907.2590v2

Mathematics

Differential Geometry

44 pages, no figures, survey paper

Scientific paper

We survey the renormalized volume of hyperbolic 3-manifolds, as a tool for Teichmuller theory, using simple differential geometry arguments to recover results sometimes first achieved by other means. One such application is McMullen's quasifuchsian (or more generally Kleinian) reciprocity, for which different arguments are proposed. Another is the fact that the renormalized volume of quasifuchsian (or more generally geometrically finite) hyperbolic 3-manifolds provides a Kahler potential for the Weil-Petersson metric on Teichmuller space. Yet another is the fact that the grafting map is symplectic, which is proved using a variant of the renormalized volume defined for hyperbolic ends.

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