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On complete submanifolds with parallel normalized mean curvature in product spaces

Part of: Global differential geometry Classical differential geometry

Published online by Cambridge University Press:  27 January 2022

Fábio R. dos Santos Open the ORCID record for Fábio R. dos Santos [Opens in a new window]  and
Sylvia F. da Silva
Fábio R. dos Santos
Affiliation:
Departamento de Matemática, Universidade Federal de Pernambuco, 50.740-540 Recife, Pernambuco, [email protected]@gmail.com
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Abstract

A Simons type formula for submanifolds with parallel normalized mean curvature vector field (pnmc submanifolds) in the product spaces $M^{n}(c)\times \mathbb {R}$, where $M^{n}(c)$ is a space form with constant sectional curvature $c\in \{-1,1\}$, it is shown. As an application is obtained rigidity results for submanifolds with constant second mean curvature.

Keywords

Submanifolds with parallel normalized mean curvature vector fieldSimons type formulaconstant second mean curvaturetotally umbilical submanifolds

MSC classification

Primary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.)
Secondary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature 53C20: Global Riemannian geometry, including pinching
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 2 , April 2022 , pp. 331 - 355
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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References

Abresch, U. and Rosenberg, H.. A Hopf differential for constant mean curvature surfaces in $\mathbb {S}^2\times \mathbb {R}$ and $\mathbb {H}^2\times \mathbb {R}$. Acta Math. 193 (2004), 141–174.CrossRefGoogle Scholar
Alencar, H. and do Carmo, M.. Hypersurfaces with constant mean curvature in spheres. Proc. Am. Math. Soc. 120 (1994), 1223–1229.CrossRefGoogle Scholar
Alías, L. J., Impera, D. and Rigoli, M.. Hypersurfaces of constant higher order mean curvature in warped products. Trans. Am. Math. Soc. 365 (2013), 591–621.CrossRefGoogle Scholar
Alías, L. J., Mastrolia, P. and Rigoli, M.. Maximum principles and geometric applications (Springer, Cham: Springer Monographs in Mathematics, 2016).CrossRefGoogle Scholar
Araújo, K. O. and Tenenblat, K.. On submanifolds with parallel mean curvature vector. Kodai Math. J. 32 (2009), 59–76.CrossRefGoogle Scholar
Batista, M.. Simons type equation in $\mathbb {S}^2\times \mathbb {R}$ and $\mathbb {H}^2\times \mathbb {R}$ and applications. Ann. Inst. Fourier (Grenoble) 61 (2011), 1299–1322.CrossRefGoogle Scholar
Cao, L. and Li, H.. $r$-Minimal submanifolds in space forms. Ann. Glob. Anal. Geom. 32 (2007), 311–341.CrossRefGoogle Scholar
Cheng, S. Y. and Yau, S-T.. Hypersurfaces with constant scalar curvature. Math. Ann. 225 (1977), 195–204.CrossRefGoogle Scholar
Chern, S. S., do Carmo, M. P. and Kobayashi, S.. Minimal submanifolds of a sphere with second fundamental form of constant length. Funct. Anal. Relat. Fields (1970), 59–75. https://link.springer.com/chapter/10.1007/978-3-642-25588-5_5.Google Scholar
Dajczer, M.. Submanifolds and isometric immersions, Mathematics Lecture Series, 13 (Houston, TX: Inc. Publish or Perish, 1990).Google Scholar
Daniel, B.. Isometric immersions into $3$-dimensional homogeneous manifolds. Comment. Math. Helv. 82 (2007), 87–131.CrossRefGoogle Scholar
Dillen, F., Fastenakels, J., Van der Veken, J. and Vrancken, L.. Constant angle surfaces in $\mathbb {S}^2\times \mathbb {R}$. Monatsh. Math. 152 (2007), 89–96.CrossRefGoogle Scholar
Dillen, F. and Munteanu, M. I.. Constant angle surfaces in $\mathbb {S}^2\times \mathbb {R}$. Bull. Braz. Math. Soc. 40 (2009), 85–97.CrossRefGoogle Scholar
dos Santos, F. R.. Rigidity of surfaces with constant extrinsic curvature in Riemannian product spaces. Bull. Braz. Math. Soc. New Series, 2020. https://doi.org/10.1007/s00574-020-00203-y.Google Scholar
Erbacher, J.. Isometric immersions of constant mean curvature and triviality of the normal connection. J. Nagoya Math. 45 (1972), 139–165.CrossRefGoogle Scholar
Eschenburg, J. H. and Tribuzy, R.. Existence and uniqueness of maps into affine homogeneous spaces. Rend. Sem. Mat. Univ. Padova 89 (1993), 11–18.Google Scholar
Fetcu, D., Oniciuc, C. and Rosenberg, H.. Biharmonic submanifolds with parallel mean curvature in $\mathbb {S}^n\times \mathbb {R}$. J. Geom. Anal. 23 (2013), 2158–2176.CrossRefGoogle Scholar
Fetcu, D. and Rosenberg, H.. On complete submanifolds with parallel mean curvature in product spaces. Rev. Mat. Iberoam. 29 (2013), 1283–1306.CrossRefGoogle Scholar
Grosjean, J. F.. Upper bounds for the first eigenvalue of the Laplacian on compact submanifolds. Pacific J. Math. 206 (2002), 93–112.CrossRefGoogle Scholar
Lawson, H. B.. Local rigidity theorems for minimal hypersurfaces. Ann. Math., 89 (1969), 187–197.CrossRefGoogle 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
Meeks, W. H. and Rosenberg, H.. Stable minimal surfaces in $M^n\times \mathbb {R}$. J. Differ. Geom. 68 (2004), 515–534.Google Scholar
Navarro, M., Ruiz-Hernández, G. and Solis, D. A.. Constant mean curvature hypersurfaces with constant angle in semi-Riemannian space forms. Differ. Geom. Appl. 49 (2016), 473–495.CrossRefGoogle Scholar
Nistor, A. I.. New developments on constant angle property in $\mathbb {S}^2\times \mathbb {R}$. Ann. Mat. Pura Appl. 196 (2017), 863–875.CrossRefGoogle Scholar
Nomizu, K. and Smyth, B.. A formula of Simons’ type and hypersurfaces with constant mean curvature. J Differ. Geom. 3 (1969), 367–377.CrossRefGoogle Scholar
Omori, H.. Isometric immersions of Riemannian manifolds. J Math. Soc. Japan 19 (1967), 205–214.CrossRefGoogle Scholar
O'Neill, B.. Semi-Riemannian Geometry, with Applications to Relativity (New York: Academic Press, 1983).Google Scholar
Pigola, S., Rigoli, M. and Setti, A. G.. Maximum principles on Riemannian manifolds and applications. Mem. Am. Math. Soc. 174, no. 822, (2005), x+99 pp.Google Scholar
Rosenberg, H.. Minimal surfaces in $M^2\times \mathbb {R}$. J. Math. 46 (2002), 1177–1195.Google Scholar
Santos, W.. Submanifolds with parallel mean curvature vector in spheres. J. Tohoku Math. 46 (1994), 403–415.CrossRefGoogle Scholar
Simons, J.. Minimal varieties in Riemannian manifolds. Ann. Math. 88 (1968), 62–105.CrossRefGoogle Scholar
Yau, S-T.. Submanifolds with constant mean curvature I. Am. J. Math. 96 (1975), 346–366.CrossRefGoogle Scholar
Yau, S-T.. Harmonic functions on complete Riemannian manifolds. Commun. Pure Appl. Math. 28 (1975), 201–228.CrossRefGoogle Scholar