  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 $\del M$ 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 \cite{Ri} and Neves-Tian \cite{NT1},
\cite{NT2}, 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.
