Revista Matemática Iberoamericana


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Volume 27, Issue 1, 2011, pp. 303–333
DOI: 10.4171/RMI/637

Constant curvature foliations in asymptotically hyperbolic spaces

Rafe Mazzeo[1] and Frank Pacard[2]

(1) Department of Mathematics, Stanford University, CA 94305-2125, STANFORD, UNITED STATES
(2) Laboratoire d'Analyse et de Mathématiques Appliqué, Université Paris 12 – Val de Marne, 61, avenue du Général de Gaulle, F-94010, CRÉTEIL CEDEX, FRANCE

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.

Keywords: Constant mean curvature, foliations, constant scalar curvature, Schouten tensor.

Mazzeo R, Pacard F. Constant curvature foliations in asymptotically hyperbolic spaces. Rev. Mat. Iberoamericana 27 (2011), 303-333. doi: 10.4171/RMI/637