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COMPLETE WEINGARTEN HYPERSURFACES SATISFYING AN OKUMURA TYPE INEQUALITY

Part of: Global differential geometry

Published online by Cambridge University Press:  11 March 2019

EUDES L. DE LIMA Open the ORCID record for EUDES L. DE LIMA [Opens in a new window]  and
HENRIQUE F. DE LIMA
EUDES L. DE LIMA*
Affiliation:
Unidade Acadêmica de Ciências Exatas e da Natureza, Universidade Federal de Campina Grande, Cajazeiras 58.900-000, Paraíba, Brazil email eudes.lima@ufcg.edu.br
HENRIQUE F. DE LIMA
Affiliation:
Departamento de Matemática,Universidade Federal de Campina Grande, Campina Grande 58.429-970, Paraíba, Brazil email henrique@mat.ufcg.edu.br
*
✉eudes.lima@ufcg.edu.br
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Abstract

In this paper we deal with complete linear Weingarten hypersurfaces immersed into Riemannian space forms. Assuming an Okumura type inequality on the total umbilicity tensor of such hypersurfaces, we prove that either the hypersurface is totally umbilical or it holds an estimate for the norm of the total umbilicity tensor, which is also shown be sharp in the sense that the product of space forms realize them. Our approach is based on a version of the Omori–Yau maximum principle for a suitable Cheng–Yau type operator.

Keywords

Riemannian space formsliner Weingarten hypersurfacesOkumura type inequalitytotally umbilical hypersurfacescircular and hyperbolic cylinders

MSC classification

Primary: 53C24: Rigidity results
Secondary: 53C40: Global submanifolds 53C42: Immersions (minimal, prescribed curvature, tight, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 109 , Issue 1 , August 2020 , pp. 81 - 92
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Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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Footnotes

The second author is partially supported by CNPq, Brazil, grant 303977/2015-9.

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