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Revisiting linear Weingarten hypersurfaces immersed into a locally symmetric Riemannian manifold

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Abstract

We deal with complete linear Weingarten hypersurfaces immersed into locally symmetric Riemannian manifolds whose sectional curvature obeys certain standard constraints. Under an assumption that such a hypersurface satisfies a suitable Okumura type inequality, we apply a version of the Omori–Yau maximum principle to prove that it must be either totally umbilical or isometric to an isoparametric hypersurface having two distinct principal curvatures. When the ambient space is Einstein, we also use a technique recently developed by Alías and Meléndez (Mediterr J Math 17(2): 61, 2020 [6]) to establish a sharp integral inequality for compact linear Weingarten hypersurfaces.

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Acknowledgements

The authors would like to thank the referee for his/her valuable suggestions which improved the paper.

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Authors and Affiliations

  1. Unidade Acadêmica de Ciências Exatas e da Natureza, Universidade Federal de Campina Grande, 58.900-000, Cajazeiras, Paraíba, Brazil

    Eudes L. de Lima

  2. Departamento de Matemática, Universidade Federal de Campina Grande, 58.429-970, Campina Grande, Paraíba, Brazil

    Henrique F. de Lima & Lucas S. Rocha

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  1. Eudes L. de Lima
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  2. Henrique F. de Lima
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  3. Lucas S. Rocha
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Correspondence to Henrique F. de Lima.

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The second author is partially supported by CNPq, Brazil, Grant 301970/2019-0. The third author is partially supported by CAPES, Brazil.

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de Lima, E.L., de Lima, H.F. & Rocha, L.S. Revisiting linear Weingarten hypersurfaces immersed into a locally symmetric Riemannian manifold. European Journal of Mathematics 8, 388–402 (2022). https://doi.org/10.1007/s40879-021-00467-8

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  • DOI: https://doi.org/10.1007/s40879-021-00467-8

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