Correspondences of hypersurfaces in hyperbolic Poincaré manifolds and conformally invariant PDEs

Bonini, Vincent; Espinar, José; Qing, Jie

doi:10.1090/S0002-9939-2010-10512-9

Correspondences of hypersurfaces in hyperbolic Poincaré manifolds and conformally invariant PDEs
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by Vincent Bonini, José M. Espinar and Jie Qing PDF

Proc. Amer. Math. Soc. 138 (2010), 4109-4117 Request permission

Abstract:

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 Poincaré manifold and scalar curvature problems on the conformal infinity.

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