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

Abstract

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

arXiv comment30 pages, 3 tables, 14 figures. Figures 9-14 added, minor changes in the exposition. Accepted for publication in Advanced Nonlinear Studies