A Simons type integral inequality for closed submanifolds in the product space Sn×R. (August 2021)

• Record Type: Journal Article
• Authors: dos Santos, Fábio R.; da Silva, Sylvia F.
• Is Part Of: Nonlinear analysis. Volume 209(2021)
• Publication Date: 2021-08
• Subjects: primary 53C42 -- secondary 53A10 53C20; Closed submanifolds with parallel normalized mean curvature vector field -- Riemannian product -- Second mean curvature -- Totally umbilical hypersurfaces -- Clifford torus; 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.2021.112366
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: A Simons type formula for submanifolds with parallel normalized mean curvature vector field (pnmc submanifolds) in the product spaces S n × R, where S n is the unit Euclidean sphere, is proved. As an application, an integral inequality for pnmc closed submanifolds in S n × R is obtained. By this, it is shown that the sharpness in this inequality is attained, in the totally umbilical hypersurfaces, and in a certain family of standard tori of the form S 1 ( 1 − r 2 ) × S n − 1 ( r ) with 0 < r < 1 .