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 [J. Differential Geom. 44 (1996), 807–817].