Hypersurfaces in Hyperbolic Poincaré Manifolds and Conformally Invariant PDEs

Details Hypersurfaces in Hyperbolic Poincaré Manifolds and Conformally Invariant PDEs Hypersurfaces in Hyperbolic Poincaré Manifolds and Conformally Invariant PDEs

2009-10-15

arxiv.org/abs/0910.2769v2

Mathematics

Differential Geometry

16 pages

Scientific paper

We derive a relationship between the eigenvalues of the Weyl-Schouten tensor of a conformal representative of the conformal infinity of a hyperbolic Poincar\'e 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 ${\H}^{n+1}$ and their connection to the conformal geometry of ${\SS}^n$ as exhibited in [7] and gives a correspondence between Weingarten hypersurfaces in hyperbolic Poincar\'e 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 ${\H}^{n+1}$ and scalar curvature problems on the conformal infinity ${\SS}^n$ to hyperbolic Poincar\'e manifolds.

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