Séance Séminaire

Séminaire Gaston Darboux

Volume of Anti-de Sitter 3-manifolds and L^1 energy.

Let S be a closed , connected, oriented surface of genus g>1. A celebrated result by J. Brock states that the volume of the convex core of a quasi-Fuchsian manifold is coarsely equivalent to the Weil-Petersson distance between the two conformal structures at infinity. In this talk, we will study a similar problem in a Lorentzian setting. The Lorentzian analogue of quasi-Fuchsian manifolds are GHMC Anti-de Sitter manifolds, whose deformation space is parametrised by two copies of the Teichmuller space of S. We will show that the volume of the convex core of these manifolds behaves coarsely like the L^1 energy of a map between the two corresponding hyperbolic surfaces. We will deduce that the volume is bounded from above by Thurston asymmetric distance and from below by the Weil-Petersson distance. We will derive also important consequences on the behaviour of the length function of a measured geodesic lamination.