On the Renormalized Volume of Hyperbolic 3-Manifolds

• Published in: Communications in mathematical physics Vol. 279; no. 3; pp. 637 - 668
• Main Authors: Krasnov, Kirill, Schlenker, Jean-Marc
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.05.2008; Springer; Springer Verlag
• Subjects: Classical and Quantum Gravitation; Complex Systems; Exact sciences and technology; Manifolds and cell complexes; Mathematical analysis; Mathematical and Computational Physics; Mathematical methods in physics; Mathematical Physics; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Other topics in mathematical methods in physics; Physics; Physics and Astronomy; Quantum Physics; Relativity Theory; Sciences and techniques of general use; Several complex variables and analytic spaces; Theoretical; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Modulus Space; Manifold; Principal Curvature; Fundamental Form; Boundary Component; Regularization method; Surfaces; Three manifold; Regularization; Mathematical physics
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-008-0423-7

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.