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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 ∈ 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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