Constant mean curvature hypersurfaces in Anti-de Sitter space

TL;DR

This paper classifies entire constant mean curvature hypersurfaces in Anti-de Sitter space based on their asymptotic boundary, shows they foliate the domain as mean curvature varies, and extends Cheng-Yau's theorem to this setting.

Contribution

It provides a classification of hypersurfaces by boundary, demonstrates their foliation property, and extends a key completeness theorem to Anti-de Sitter space.

Findings

Unique hypersurfaces for each admissible boundary sphere and mean curvature.

Hypersurfaces form an analytic foliation of the domain as H varies.

Extension of Cheng-Yau Theorem confirming completeness of these hypersurfaces.

Abstract

We study spacelike entire constant mean curvature hypersurfaces in Anti-de Sitter space of any dimension. First, we give a classification result with respect to their asymptotic boundary, namely we show that every admissible sphere $\Lambda$ is the boundary of a unique such hypersurface, for any given value $H$ of the mean curvature. We also demonstrate that, as $H$ varies in $\mathbb{R}$, these hypersurfaces analytically foliate the invisible domain of $\Lambda$. Finally, we extend Cheng-Yau Theorem to the Anti-de Sitter space, which establishes the completeness of any entire constant mean curvature hypersurface.