CORRESPONDENCES OF HYPERSURFACES IN HYPERBOLIC POINCARE MANIFOLDS AND CONFORMALLY INVARIANT PDES

On a hyperbolic Poincare manifold, we derive an explicit relationship between the eigenvalues of Weyl-Schouten tensor of a conformal representative of the conformal infinity and the principal curvatures of the level sets of the associated geodesic defining function. This considerably simplifies the arguments and generalizes the results of Galvez, Mira and the second author. In particular, we obtain the equivalence between Christoffel-type problems for hypersurfaces in a hyperbolic Poincare manifold and scalar curvature problems on the conformal infinity.