Elliptic Weingarten Hypersurfaces of Riemannian Products

Let $M^n$ be either the unit sphere $\mathbb S^n$ or a rank-one symmetric space of noncompact type. We consider elliptic Weingarten hypersurfaces of $M\times\mathbb R$, which are defined as those whose principal curvatures $k_1,\dots ,k_n$ and angle function $\theta$ satisfy a relation $W(k_1,\dots,k_n,\theta^2)=c,$ being $c$ a constant and $W$ an elliptic Weingarten function. We show that, for a certain class of Weingarten functions $W$ (which includes the higher order mean curvatures, the norm of the second fundamental form, and the scalar curvature), and for certain values of $c>0,$ there exist rotational strictly convex Weingarten hypersurfaces of $M\times\mathbb R$ which are either topological spheres or entire graphs over $M.$ We establish a Jellett-Liebmann-type theorem by showing that a compact, connected and strictly convex elliptic Weingarten hypersurface of either $\mathbb S^n\times\mathbb R$ or $\mathbb H^n\times\mathbb R$ is a rotational embedded sphere. Other uniqueness results for complete elliptic Weingarten hypersurfaces of these ambient spaces are obtained. We also obtain existence results for constant scalar curvature hypersurfaces of $\mathbb S^n\times\mathbb R$ and $\mathbb H^n\times\mathbb R$ which are either rotational or invariant by translations (parabolic or hyperbolic). We apply our methods to give new proofs of the main results by Manfio and Tojeiro on the classification of constant sectional curvature hypersurfaces of $\mathbb S^n\times\mathbb R$ and $\mathbb H^n\times\mathbb R.$