Complete hypersurfaces with two distinct principal curvatures in a locally symmetric Riemannian manifold. (March 2016)

• Record Type: Journal Article
• Authors: Gomes, José N.; de Lima, Henrique F.; dos Santos, Fábio R.; Velásquez, Marco Antonio L.
• Is Part Of: Nonlinear analysis. Volume 133(2016)
• Page Start: 15
• Page End: 27
• Publication Date: 2016-03
• Subjects: primary 53C42 -- secondary 53A10 53C20 53C50; Locally symmetric Riemannian manifolds -- Hypersurfaces with two distinct principal curvatures -- Complete linear Weingarten hypersurfaces -- Isoparametric hypersurfaces; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2015.11.026
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We deal with complete hypersurfaces with two distinct principal curvatures in a locally symmetric Riemannian manifold, which is supposed to obey some appropriated curvature constraints. Initially, considering the case that such a hypersurface has constant mean curvature, we apply a Simons type formula jointly with the well known generalized maximum principle of Omori–Yau to show that it must be isometric to an isoparametric hypersurface of the ambient space. Afterwards, we use a Cheng–Yau modified operator in order to obtain a sort of extension of this previously mentioned result for the context of linear Weingarten hypersurfaces, that is, hypersurfaces whose mean and scalar curvatures are linearly related.