2009/10/15 by Vincent Bonini, Bonini, Vincent, José M. Espinar +3
Mathematics · #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #math.DG #msc:53C30 #msc:53C40 #msc:58J05
paper · pdf · doi:10.48550/arxiv.0910.2769
16 pages
arxiv created 2009/10/16 · arxiv updated 2009/12/01
We derive a relationship between the eigenvalues of the Weyl-Schouten tensor of a conformal representative of the conformal infinity of a hyperbolic Poincaré manifold and the principal curvatures on the level sets of its uniquely associated defining function with calculations based on [9] [10]. This relationship generalizes the result for hypersurfaces in \Hn+1 and their connection to the conformal geometry of \SSn as exhibited in [7] and gives a correspondence between Weingarten hypersurfaces in hyperbolic Poincaré manifolds and conformally invariant equations on the conformal infinity. In particular, we generalize an equivalence exhibited in [7] between Christoffel-type problems for hypersurfaces in \Hn+1 and scalar curvature problems on the conformal infinity \SSn to hyperbolic Poincaré manifolds.