We prove that given two metrics with curvature less than −1 on a closed, oriented surface of genus greater than 2, there exists an Anti-de-Sitter manifold with smooth, space-like, strictly convex boundary such that the induced metrics on the two connected components are equal to the two metrics we started with. Using the duality between convex space-like surfaces in Anti-de-Sitter space, we obtain an equivalent result about the prescription of the third fundamental form.

We prove that, given an acausal curve in the boundary at infinity of Anti-de Sitter space which is the graph of a quasi-symmetric homeomorphism, there exists a foliation of its domain of dependence by constant mean curvature surfaces with bounded second fundamental form. Moreover, these surfaces provide a family of quasi-conformal extensions of the quasi-symmetric homeomorphism we started with.