Bulletin of the
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BKMS

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

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Bull. Korean Math. Soc. 2022; 59(3): 789-799

Online first article May 31, 2022      Printed May 31, 2022

https://doi.org/10.4134/BKMS.b210479

Copyright © The Korean Mathematical Society.

A sharp integral inequality for compact linear Weingarten hypersurfaces

Henrique F. de~Lima , F\'{a}bio R. dos~Santos, Lucas S. Rocha

Universidade Federal de Campina Grande; Universidade Federal de Pernambuco; Universidade Federal de Campina Grande

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 curvatures, one which is simple, in such an ambient space. Our approach is based on the ideas and techniques introduced by Al\'{\i}as and Mel\'{e}ndez in [4] for the case of hypersurfaces with constant scalar curvature in an Euclidean round sphere.

Keywords: Locally symmetric Einstein manifolds, compact linear Weingarten hypersurfaces, totally umbilical hypersurfaces, isoparametric hypersurfaces

MSC numbers: Primary 53C42; Secondary 53C25, 53C35

Supported by: The first author is partially supported by CNPq, Brazil, grant 301970/2019-0. The second author is partially supported by CNPq, Brazil, grants 431976/2018-0 and 311124/2021-6, Fundaci\'on S\'eneca project reference 19901/GERM/15, Spain, and Propesqi/UFPE, Brazil. The third author is partially supported by CAPES, Brazil.

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