UC Santa Cruz

The relationship between the geometry of a conformally compact manifold and the conformal geometry of its conformal infinity is of particular interest due to its association with the AdS/CFT correspondence of physics, a conjectured correlation between a string theory on a negatively curved Einstein manifold and a conformal field theory on its boundary at infinity. In the case of hyperbolic space $\mathbb{H}^{n+1}$ with conformal infinity the round sphere $\mathbb{S}^n$, a very precise relationship has been established between conformal invariants (the eigenvalues of the Schouten tensor) on $\mathbb{S}^n$ and Weingarten curvatures of immersed hypersurfaces \cite{MR2538508}. This same relationship has been extended to hyperbolic Poincar {e} manifolds \cite{MR2679632}. We establish a correspondence between constant scalar curvature metrics on conformal infinity and families of hypersurfaces of constant Weingarten curvature in a neighborhood of infinity. This generalizes results of Mazzeo and Pacard \cite{MR2815739} on existence of constant curvature foliations of asymptotically hyperbolic spaces to more arbitrary Weingarten curvatures.