Abstract
We use minimal surface techniques to show that the set of quasi-fuchsian hyperbolic manifolds containing a closed surface with principal curvatures less than 1 is parametrized by a subset of the cotangent of Teichmüller space. This also yields a parametrization of the space of all GHMC AdS manifolds by the whole cotangent of Teichmüller space. The same techniques work for hyperbolic or AdS manifolds with singular curvatures (physically, particles) and provides a description in terms of the Teichmüller space with marked points which should be well adapted to quantization.