Elliptic Weingarten hypersurfaces of Riemannian products

R. F. de Lima, A. K. Ramos and J. P. dos Santos

Mathematische Nachrichten, 2023, vol. 296, issue 10, 4712-4738

Abstract: Let Mn$M^n$ be either a simply connected space form or a rank‐one symmetric space of the noncompact type. We consider Weingarten hypersurfaces of M×R$M\times \mathbb {R}$, which are those whose principal curvatures k1,⋯,kn$k_1,\dots ,k_n$ and angle function 𝛩$\varTheta$ satisfy a relation W(k1,⋯,kn,𝛩2)=0$W(k_1,\dots ,k_n,\varTheta ^2)=0$, being W a differentiable function which is symmetric with respect to k1,⋯,kn$k_1,\dots , k_n$. When ∂W/∂ki>0$\partial W/\partial k_i>0$ on the positive cone of Rn$\mathbb {R} ^n$, 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×R$M\times \mathbb {R}$ 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 Sn×R$\mathbb {S}^n\times \mathbb {R}$ or Hn×R$\mathbb {H}^n\times \mathbb {R}$ 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 Sn×R$\mathbb {S}^n\times \mathbb {R}$ and Hn×R$\mathbb {H}^n\times \mathbb {R}$ 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 Sn×R$\mathbb {S}^n\times \mathbb {R}$ and Hn×R$\mathbb {H}^n\times \mathbb {R}$.

Date: 2023
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