Liouville action and Weil-Petersson metric on deformation spaces, global Kleinian reciprocity and holography

Details Liouville action and Weil-Petersson metric on deformation spaces, global Kleinian reciprocity and holography Liouville action and Weil-Petersson metric on deformation spaces, global Kleinian reciprocity and holography

2002-04-25

arxiv.org/abs/math/0204318v2

Commun.Math.Phys. 239 (2003) 183-240

Mathematics

Complex Variables

60 pages, proof of the Lemma 5.1 corrected, references and section 5.3 added

Scientific paper

10.1007/s00220-003-0878-5

We rigorously define the Liouville action functional for finitely generated, purely loxodromic quasi-Fuchsian group using homology and cohomology double complexes naturally associated with the group action. We prove that the classical action - the critical point of the Liouville action functional, considered as a function on the quasi-Fuchsian deformation space, is an antiderivative of a 1-form given by the difference of Fuchsian and quasi-Fuchsian projective connections. This result can be considered as global quasi-Fuchsian reciprocity which implies McMullen's quasi-Fuchsian reciprocity. We prove that the classical action is a Kahler potential of the Weil-Petersson metric. We also prove that Liouville action functional satisfies holography principle, i.e., it is a regularized limit of the hyperbolic volume of a 3-manifold associated with a quasi-Fuchsian group. We generalize these results to a large class of Kleinian groups including finitely generated, purely loxodromic Schottky and quasi-Fuchsian groups and their free combinations.

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