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.
Vincent Bonini, José M. Espinar and Jie Qing. "Correspondences Of Hypersurfaces in Hyperbolic Poincaré Manifolds and Conformally Invariant PDEs" Proceedings of the American Mathematical Society
Vol. 138 Iss. 11 p. 4109 - 4117
Available at: http://works.bepress.com/vbonini/3/