Constant mean curvature hypersurfaces in ${\mathbb{H}}^n\times \mathbb{R}$ with small planar boundary

Abstract

We show that constant mean curvature hypersurfaces in ${\mathbb{H}}^n\times \mathbb{R}$, with small and pinched boundary contained in a horizontal slice $P$ are topological disks, provided they are contained in one of the two halfspaces determined by $P$. This is the analogous in ${\mathbb{H}}^n\times \mathbb{R}$ of a result in ${\mathbb{R}}^3$ by A. Ros and H. Rosenberg.

18 pages, 3 figures, comments are welcome