Foliations of Hyperbolic Space by Constant Mean Curvature Hypersurfaces

TL;DR

This paper proves that certain constant mean curvature hypersurfaces in hyperbolic space form a foliation, providing a new geometric understanding of hyperbolic space boundaries and their associated hypersurfaces.

Contribution

It establishes conditions under which constant mean curvature hypersurfaces foliate hyperbolic space, extending previous results to more general boundary conditions.

Findings

Hypersurfaces span boundary of star-shaped domains in asymptotic sphere.

Unique CMC hypersurfaces exist for given boundary conditions and mean curvature.

These hypersurfaces form a foliation of hyperbolic space.

Abstract

We show that the constant mean curvature hypersurfaces in the hyperbolic n-space spanning the boundary of a star shaped C^{1,1} domain in the asymptotic sphere give a foliation of the hyperbolic n-space. We also show that if C is a closed codimension-1 C^{2,a} submanifold in the asymptotic sphere bounding a unique constant mean curvature hypersurface S_H in the hyperbolic n-space with asymptotic boundary C for any -1<H<1, then the constant mean curvature hypersurfaces {S_H} foliates the hyperbolic n-space.