Anti-de Sitter geometry: convex domains, foliations and volume

We study various aspects of the geometry of globally hyperbolic anti-de Sitter 3-manifolds.
For manifolds with convex space-like boundaries, homeomorphic to the product of a closed, connected and oriented surface of genus at least two with an interval, we prove that every pair of metrics with curvature less than -1 on the surface can be realised on the two boundary components.
For globally hyperbolic maximal compact (GHMC) anti-de Sitter manifolds, we study various geometric quantities, such as the volume, the Hausdorff dimension of the limit set, the width of the convex core and the Holder exponent of the manifold, in terms of the parameters that describe the deformation space of GHMC anti-de Sitter structures.
Moreover, we prove existence and uniqueness of a foliation by constant mean curvature surfaces of the domain of dependence of any quasi-circle in the boundary at infinity of anti-de Sitter space.