Correspondences of hypersurfaces in hyperbolic poincaré manifolds and conformally invariant pdes

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Proceedings of the American Mathematical Society, ISSN: 0002-9939, Vol: 138, Issue: 11, Page: 4109-4117

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Bonini, Vincent; Espinar, José M.; Qing, Jie
American Mathematical Society (AMS)
article description
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 Ǵalvez, Mira and the second author. In particular, we obtain the equivalence between Christoffel-type problems for hypersurfaces in a hyperbolic Poincaré manifold and scalar curvature problems on the conformal infinity. © 2010 American Mathematical Society.