Published in Proceedings of the American Mathematical Society, Volume 138, Issue 11, November 1, 2010, pages 4109-4117.
This article was first published in Proceedings of the American Mathematical Society, published by the American Mathematical Society. Copyright © 2010 American Mathematical Society. The definitive version is available at http://dx.doi.org/10.1090/S0002-9939-2010-10512-9 .
On a hyperbolic Poincaré 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 Gálvez, Mira and the second author. In particular, we obtain the equivalence between Christoffel-type problems for hypersurfaces in a hyperbolic Poincar´e manifold and scalar curvature problems on the conformal infinity.