On constant higher order mean curvature hypersurfaces in Hn×R ${\mathbb{H}}^{n}{\times}\mathbb{R}$

Abstract

We classify hypersurfaces with rotational symmetry and positive constant r-th mean curvature in Hn×R ${\mathbb{H}}^{n}{\times}\mathbb{R}$ . Specific constant higher order mean curvature hypersurfaces invariant under hyperbolic translation are also treated. Some of these invariant hypersurfaces are employed as barriers to prove a Ros–Rosenberg type theorem in Hn×R ${\mathbb{H}}^{n}{\times}\mathbb{R}$ : we show that compact connected hypersurfaces of constant r-th mean curvature embedded in Hn×[0,∞) ${\mathbb{H}}^{n}{\times}\left[0,\infty \right)$ with boundary in the slice Hn×{0} ${\mathbb{H}}^{n}{\times}\left\{0\right\}$ are topological disks under suitable assumptions.

Keywords

• higher order mean curvature
• alexandrov reflection technique
• hyperbolic space
• 53c42
• 53a10