Abstract
Two problems concerning asymptotically hyperbolic manifolds with an inner boundary are studied. First, we study scalar curvature presciption with either Dirichlet or mean curvature prescription interior boundary condition. Then we apply those results to the Lichnerowicz equation with (future or past) apparent horizon interior boundary condition. In the last part we show how to construct TT-tensors. Thus we obtain Cauchy data with constant mean curvature for Einstein vacuum equations.