  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.
