Hostname: page-component-5cf477f64f-2wr7h Total loading time: 0 Render date: 2025-04-03T06:44:57.191Z Has data issue: false hasContentIssue false
  • English
  • Français

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)
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Albujer, A. L. and dos Santos, F. R., Willmore surfaces and Hopf tori in homogeneous 3-manifolds, Ann. Global Anal. Geom. 62 (2022), 181–200.10.1007/s10455-022-09844-2CrossRefGoogle Scholar
Barbosa, J. L. M. and Colares, A. G., Stability of hypersurfaces with constant r-mean curvature, Ann. Global Anal. Geom. 15 (1997), 277–297.10.1023/A:1006514303828CrossRefGoogle Scholar
Chen, B.-Y., On the total curvature of immersed manifolds. I. An inequality of Fenchel–Borsuk–Willmore, Amer. J. Math. 93 (1971), 148–162.10.2307/2373454CrossRefGoogle Scholar
Chen, B.-Y., Some conformal invariants of submanifolds and their applications, Boll. Unione Mat. Ital. 10 (1974), 380–385.Google Scholar
Chern, S. S., do Carmo, M. P. and Kobayashi, S.. Minimal submanifolds of a sphere with second fundamental form of constant length, Functional Analysis and Related Fields (Proc. Conf. for M. Stone, Univ. Chicago, Chicago, Ill. 1968), Springer, New York, 1970, .Google Scholar
Daniel, B., Isometric immersions into 3-dimensional homogeneous manifolds, Comment. Math. Helv. 82 (2007), 87–131.10.4171/CMH/86CrossRefGoogle Scholar
dos Santos, F. R. and da Silva, S. F., A Simons type integral inequality for closed submanifolds in the product space $\mathbb{S}^n\times\mathbb{R}$, Nonlinear Anal. 209 (2021), .10.1016/j.na.2021.112366CrossRefGoogle Scholar
dos Santos, F. R. and da Silva, S. F., On complete submanifolds with parallel normalized mean curvature in product spaces, Proc. Roy. Soc. Edinburgh Sect. A 152 (2022), 331–355.10.1017/prm.2021.6CrossRefGoogle Scholar
Guo, Z. and Yin, B., Variational problems of total mean curvature of submanifolds in a sphere, Proc. Amer. Math. Soc. 144 (2016), 3563–3568.10.1090/proc/13009CrossRefGoogle Scholar
Lawson, H. B.Jr., Local rigidity theorems for minimal hypersurfaces, Ann. Math. 89 (1969), 187–197.10.2307/1970816CrossRefGoogle Scholar
Li, A. M. and Li, J. M., An intrinsic rigidity theorem for minimal submanifolds in a sphere, Arch. Math. 58 (1992), 582–594.Google Scholar
Marques, F. C. and Neves, A., Min-max theory and the Willmore conjecture, Ann. Math. 179 (2014), 683–782.10.4007/annals.2014.179.2.6CrossRefGoogle Scholar
O’Neill, B., Semi-Riemannian Geometry, With Applications to Relativity (Academic Press, New York, 1983).Google Scholar
Simons, J., Minimal varieties in Riemannian manifolds, Ann. Math. 88 (1968), 62–105.10.2307/1970556CrossRefGoogle Scholar
Weiner, J. L., On a problem of Chen, Willmore, et al., Indiana Univ. Math. J. 27 (1978), 19–35.10.1512/iumj.1978.27.27003CrossRefGoogle Scholar
Willmore, T. J., Note on embedded surfaces, An. Stiint. Univ. Al. I. Cuza Iasi Mat. 11B (1965), 493–496.Google Scholar