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Correspondences of hypersurfaces in hyperbolic Poincaré manifolds and conformally invariant PDEs

Authors: Vincent Bonini, José M. Espinar and Jie Qing
Journal: Proc. Amer. Math. Soc. 138 (2010), 4109-4117
MSC (2010): Primary 53C30, 53C40; Secondary 58J05
Published electronically: June 11, 2010
MathSciNet review: 2679632
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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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Additional Information

Vincent Bonini
Affiliation: Department of Mathematics, California Polytechnic State University, San Luis Obispo, California 93407

José M. Espinar
Affiliation: Departmento de Geometría y Topología, Universidad de Granada, E-18071 Granda, Spain

Jie Qing
Affiliation: Department of Mathematics, University of California, Santa Cruz, California 95064
MR Author ID: 268101

Keywords: Hyperbolic Poincaré manifold, hypersurfaces, second fundamental form, Schouten tensor, conformally invariant PDE
Received by editor(s): October 16, 2009
Received by editor(s) in revised form: February 12, 2010
Published electronically: June 11, 2010
Communicated by: Matthew J. Gursky
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.