JournalsrmiVol. 27, No. 1pp. 303–333

Constant curvature foliations in asymptotically hyperbolic spaces

  • Rafe Mazzeo

    Stanford University, USA
  • Frank Pacard

    École Polytechnique, Palaiseau, France
Constant curvature foliations in asymptotically hyperbolic spaces cover
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Abstract

Let (M,g)(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 M\partial M and Weingarten foliations in some neighbourhood of infinity in MM. 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 σk\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 gg 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.

Cite this article

Rafe Mazzeo, Frank Pacard, Constant curvature foliations in asymptotically hyperbolic spaces. Rev. Mat. Iberoam. 27 (2011), no. 1, pp. 303–333

DOI 10.4171/RMI/637