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

Abstract

We survey the renormalized volume of hyperbolic 3-manifolds, as a tool for Teichmüller 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 Kähler potential for the Weil–Petersson metric on Teichmüller 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.