Hypersurfaces in Hyperbolic Poincar\'e Manifolds and Conformally Invariant PDEs

TL;DR

This paper establishes a link between the eigenvalues of the Weyl-Schouten tensor in hyperbolic Poincaré manifolds and principal curvatures, generalizing known results to relate hypersurfaces and conformally invariant PDEs.

Contribution

It generalizes the relationship between hypersurface geometry and conformal invariants from hyperbolic space to hyperbolic Poincaré manifolds, connecting Weingarten hypersurfaces with conformally invariant equations.

Findings

Derived a new relationship between eigenvalues and principal curvatures.

Extended classical results to the setting of hyperbolic Poincaré manifolds.

Established a correspondence between geometric hypersurfaces and conformal PDEs.

Abstract

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.