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

Abstract

We classify hypersurfaces with rotational symmetry and positive constant r-th mean curvature in H n × 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 H n × R ${\mathbb{H}}^{n}{\times}\mathbb{R}$ : we show that compact connected hypersurfaces of constant r-th mean curvature embedded in H n × [ 0 , ∞ ) ${\mathbb{H}}^{n}{\times}\left[0,\infty \right)$ with boundary in the slice H n × { 0 } ${\mathbb{H}}^{n}{\times}\left\{0\right\}$ are topological disks under suitable assumptions.