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Total mean curvature surfaces in the product space ${\mathbb{S}^{n}\times\mathbb{R}}$ and applications

Part of: Global differential geometry Classical differential geometry

Published online by Cambridge University Press:  26 April 2023

Alma L. Albujer Open the ORCID record for Alma L. Albujer [Opens in a new window] ,
Sylvia F. da Silva Open the ORCID record for Sylvia F. da Silva [Opens in a new window]  and
Fábio R. dos Santos Open the ORCID record for Fábio R. dos Santos [Opens in a new window]
Alma L. Albujer
Affiliation:
Departamento de Matemáticas, Universidad de Córdoba, Córdoba 14011, Spain (alma.albujer@uco.es)
Sylvia F. da Silva
Affiliation:
Departamento de Matemática, Universidade Federal de Pernambuco, Recife, Pernambuco 50.740-560, Brazil (sylvia.ferreira@ufrpe.br)
Fábio R. dos Santos
Affiliation:
Departamento de Matemática, Universidade Federal de Pernambuco, Recife, Pernambuco 50.740-560, Brazil (sylvia.ferreira@ufrpe.br)
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Abstract

The total mean curvature functional for submanifolds into the Riemannian product space $\mathbb{S}^n\times\mathbb{R}$ is considered and its first variational formula is presented. Later on, two second-order differential operators are defined and a nice integral inequality relating both of them is proved. Finally, we prove our main result: an integral inequality for closed stationary $\mathcal{H}$-surfaces in $\mathbb{S}^n\times\mathbb{R}$, characterizing the cases where the equality is attained.

Keywords

ℋ-surfaceproduct spaceminimal surfaceClifford torusVeronese surface

MSC classification

Primary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.)
Secondary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature 53C30: Homogeneous manifolds
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 66 , Issue 2 , May 2023 , pp. 346 - 365
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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