Constant curvature foliations in asymptotically hyperbolic spaces

Abstract

Let $(M,g)$ be an asymptotically hyperbolic manifold with a smooth conformal compactification. We establish a general correspondence between semilinear elliptic equations of scalar curvature type on $\partial M$ and Weingarten foliations in some neighbourhood of infinity in $M$. We focus mostly on foliations where each leaf has constant mean curvature, though our results apply equally well to foliations where the leaves have constant ${\sigma }_k$-curvature. In particular, we prove the existence of a unique foliation near infinity in any quasi-Fuchsian 3-manifold by surfaces with constant Gauss curvature. There is a subtle interplay between the precise terms in the expansion for $g$ and various properties of the foliation. Unlike other recent works in this area, by Rigger ([The foliation of asymptotically hyperbolic manifolds by surfaces of constant mean curvature (including the evolution equations and estimates). Manuscripta Math. 113 (2004), 403-421]) and Neves-Tian ([Existence and uniqueness of constant mean curvature foliation of asymptotically hyperbolic 3-manifolds. Geom. Funct. Anal. 19 (2009), no.3, 910-942], [Existence and uniqueness of constant mean curvature foliation of asymptotically hyperbolic 3-manifolds. II. J. Reine Angew. Math. 641 (2010), 69-93]), we work in the context of conformally compact spaces, which are more general than perturbations of the AdS-Schwarzschild space, but we do assume a nondegeneracy condition.