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New characterizations of linear Weingarten hypersurfaces immersed in the hyperbolic space

Cícero P. Aquino; Henrique F. de Lima — 2015

Archivum Mathematicum

In this paper, we deal with complete linear Weingarten hypersurfaces immersed in the hyperbolic space $ℍ^{n + 1}$ , that is, complete hypersurfaces of $ℍ^{n + 1}$ whose mean curvature $H$ and normalized scalar curvature $R$ satisfy $R = a H + b$ for some $a$ , $b \in ℝ$ . In this setting, under appropriate restrictions on the mean curvature and on the norm of the traceless part of the second fundamental form, we prove that such a hypersurface must be either totally umbilical or isometric to a hyperbolic cylinder of $ℍ^{n + 1}$ . Furthermore, a rigidity result...

On the quadric CMC spacelike hypersurfaces in Lorentzian space forms

Cícero P. Aquino; Henrique F. de Lima; Fábio R. dos Santos — 2016

Colloquium Mathematicae

We deal with complete spacelike hypersurfaces immersed with constant mean curvature in a Lorentzian space form. Under the assumption that the support functions with respect to a fixed nonzero vector are linearly related, we prove that such a hypersurface must be either totally umbilical or isometric to a hyperbolic cylinder of the ambient space.

On complete linear Weingarten hypersurfaces in locally symmetric Riemannian manifolds

Cícero P. Aquino; Henrique F. de Lima; Fábio R. dos Santos; Marco Antonio L. Velásquez — 2015

Commentationes Mathematicae Universitatis Carolinae

Our aim is to apply suitable generalized maximum principles in order to obtain characterization results concerning complete linear Weingarten hypersurfaces immersed in a locally symmetric Riemannian manifold, whose sectional curvature is supposed to obey standard constraints. In this setting, we establish sufficient conditions to guarantee that such a hypersurface must be either totally umbilical or an isoparametric hypersurface with two distinct principal curvatures one of which is simple.

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Currently displaying 1 – 3 of 3

New characterizations of linear Weingarten hypersurfaces immersed in the hyperbolic space

On the quadric CMC spacelike hypersurfaces in Lorentzian space forms

On complete linear Weingarten hypersurfaces in locally symmetric Riemannian manifolds