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

TL;DR

This paper develops a method to construct and classify hypersurfaces with constant higher order mean curvature in product spaces, providing new examples, classifications, and uniqueness results for such hypersurfaces in various Riemannian manifolds.

Contribution

It introduces a general construction method for $H_r$-hypersurfaces and classifies complete rotational examples in key product spaces, extending understanding of their geometry.

Findings

Constructed many explicit $H_r$-hypersurfaces in various manifolds.

Classified complete rotational $H_r$-hypersurfaces in hyperbolic and spherical products.

Proved uniqueness of compact, convex $H_r$-hypersurfaces as rotational spheres.

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.