Hypersurfaces of Constant Higher Order Mean Curvature in $M\times \mathbb{R}$

Abstract

We consider hypersurfaces of products $M\times \mathbb{R}$ with constant $r$-th mean curvature $H_r\ge 0$ (to be called $H_r$-hypersurfaces), where $M$ is an arbitrary Riemannian $n$-manifold. We develop a general method for constructing them, and employ it to produce many examples for a variety of manifolds $M,$ including all simply connected space forms and the hyperbolic spaces ${\mathbb{H}}_{\mathbb{F}}^m$ (rank $1$ symmetric spaces of noncompact type). We construct and classify complete rotational $H_r(\ge 0)$-hypersurfaces in ${\mathbb{H}}_{\mathbb{F}}^m\times \mathbb{R}$ and in ${\mathbb{S}}^n\times \mathbb{R}$ as well. They include spheres, Delaunay-type annuli and, in the case of ${\mathbb{H}}_{\mathbb{F}}^m\times \mathbb{R},$ entire graphs. We also construct and classify complete $H_r(\ge 0)$-hypersurfaces of ${\mathbb{H}}_{\mathbb{F}}^m\times \mathbb{R}$ which are invariant by either parabolic isometries or hyperbolic translations. We establish a Jellett-Liebmann-type theorem by showing that a compact, connected and strictly convex $H_r$-hypersurface of ${\mathbb{H}}^n\times \mathbb{R}$ or ${\mathbb{S}}^n\times \mathbb{R}$ $(n\ge 3)$ is a rotational embedded sphere. Other uniqueness results for complete $H_r$-hypersurfaces of these ambient spaces are obtained.