Hypersurfaces of constant higher-order mean curvature in $$M\times {\mathbb{R}}$$

Abstract

We consider hypersurfaces of products $$M\times {\mathbb{R}}$$ with constant rth 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 one 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.