Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc.

Published online May 12, 2022

Copyright © The Korean Mathematical Society.

A sharp integral inequality for compact linear Weingarten hypersurfaces

Henrique de Lima, Fábio dos Santos, and Lucas Rocha

Federal University of Campina Grande, Federal University of Pernambuco

Abstract

We establish a sharp integral inequality related to compact (without boundary) linear Weingarten hypersurfaces immersed in a locally symmetric Einstein manifold and we apply it to characterize totally umbilical hypersurfaces and isoparametric hypersurfaces with two distinct principal curvature, one which is simple, in such an ambient space. Our approach is based on the ideas and techniques introduced by Alías and Meléndez in reference [3] for the case of hypersurfaces with constant scalar curvature in the Euclidean round sphere.

Keywords: Locally symmetric Einstein manifold; compact linear Weingarten hypersurfaces; totally umbilical hypersurfaces; isoparametric hypersurfaces

MSC numbers: Primary: 53C42; Secondary: 53C24 and 53C40.

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