Elliptic Weingarten Hypersurfaces of Riemannian Products

TL;DR

This paper studies elliptic Weingarten hypersurfaces in Riemannian product spaces, establishing existence, uniqueness, and classification results for convex, rotational, and invariant hypersurfaces with prescribed curvature conditions.

Contribution

It introduces new existence and classification theorems for elliptic Weingarten hypersurfaces in product spaces, extending prior results on constant curvature hypersurfaces.

Findings

Existence of rotational convex Weingarten hypersurfaces as spheres or entire graphs.

A Jellett-Liebmann-type theorem classifying compact elliptic hypersurfaces as rotational spheres.

New proofs for classification of constant sectional curvature hypersurfaces in product spaces.

Abstract

Let $M^n$ be either a simply connected space form or a rank-one symmetric space of noncompact type. We consider Weingarten hypersurfaces of $M\times\mathbb R$, which are those whose principal curvatures $k_1,\dots ,k_n$ and angle function $\theta$ satisfy a relation $W(k_1,\dots,k_n,\theta^2)=0,$ being $W$ a differentiable function which is symmetric with respect to $k_1,\dots, k_n.$ When $\partial W/\partial k_i>0$ on the positive cone of $\mathbb R^n,$ a strictly convex Weingarten hypersurface determined by $W$ is said to be elliptic. We show that, for a certain class of Weingarten functions $W,$ 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 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 $\mathbb H^n\times\mathbb R.$