On the renormalized volume of hyperbolic 3-manifolds (Q926254)

The renormalized volume is usually introduced by a limiting procedure involving the volume of a convex domain \(N\) and the area of its parallel surfaces in increasing distance. It is motivated by string theory and the renormalized gravity action. For a hyperbolic 3-manifold the authors replace this definition by the so-called \(W\)-volume defined by the volume of \(N\) minus one quarter of the total mean curvature of the boundary. In the paper geometric properties of the \(W\)-volume are explained, including a certain self-duality. In particular a variational formula is derived which is similar to the Schläfli formula. Furthermore, the authors show that the boundary term can be interpreted by a certain quantity ``at infinity''. Finally a conformal deformation of the metric at infinity is studied, and the renormalized volume is interpreted as a function on the Teichmüller space.