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25 September 2022
 
  » arxiv » math/0607081

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On the renormalized volume of hyperbolic 3-manifolds
Kirill Krasnov ; Jean-Marc Schlenker ;
Date 4 Jul 2006
Subject Differential Geometry
AbstractThe renormalized volume of hyperbolic manifolds is a quantity motivated by the AdS/CFT correspondence of string theory and computed via a certain regularization procedure. The main aim of the present paper is to elucidate the geometrical meaning of this quantity. We propose a new regularization procedure based on surfaces equidistant to a given convex surface $partial N$. The renormalized volume computed via this procedure is shown to be equal to what we call the $W$-volume of the convex region $N$ given by the usual volume of $N$ minus the quarter of the integral of the mean curvature over $partial N$. The $W$-volume satisfies some remarkable properties. First, this quantity is self-dual in the sense explained in the paper. Second, it verifies some simple variational formulas analogous to the classical geometrical Schl"afli identities. These variational formulas are invariant under a certain transformation that replaces the data at $partial N$ by those at infinity of $M$. We use the variational formulas in terms of the data at infinity to give a simple geometrical proof of results of Takhtajan et al on the K"ahler potential on various moduli spaces.
Source arXiv, math/0607081
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