Communications in Mathematical Physics

, Volume 279, Issue 3, pp 637-668

On the Renormalized Volume of Hyperbolic 3-Manifolds

  • Kirill KrasnovAffiliated withSchool of Mathematical Sciences, University of NottinghamPerimeter Institute for Theoretical Physics Email author 
  • , Jean-Marc SchlenkerAffiliated withInstitut de Mathématiques, UMR CNRS 5219, Université Toulouse III

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The 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 its geometrical meaning. We use another regularization procedure based on surfaces equidistant to a given convex surface ∂N. The renormalized volume computed via this procedure is 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 ∂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 Schläfli identities. These variational formulas are invariant under a certain transformation that replaces the data at ∂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 Kähler potential on various moduli spaces.