Hypersurfaces of constant higher-order mean curvature in M x R

We consider hypersurfaces of products M x R with constant rth mean curvature H-r >= 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 H-F(m) (rank one symmetric spaces of noncompact type). We construct and classify complete rotational H-r(>= 0)-hypersurfaces in H-F(m) x R and in S-n x R as well. They include spheres, Delaunay-type annuli and, in the case of H-F(m) x R, entire graphs. We also construct and classify complete H-r(>= 0)-hypersurfaces of H-F(m) x 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 H-n x R or S-n x R (n >= 3) is a rotational embedded sphere. Other uniqueness results for complete H-r-hypersurfaces of these ambient spaces are obtained.