Abstract
In this paper, we revisit the problem of characterize (r, s)-stable closed hypersurfaces immersed in a Riemannian space form, which was firstly established in Velásquez et al. (J Math Anal Appl 406:134–146, 2013). With a different approach of that used in the proof of the main theorem of Velásquez et al. (J Math Anal Appl 406:134–146, 2013), we complete its program showing that a closed hypersurface contained in the Euclidian space \({\mathbb {R}}^{n+1}\) and having higher order mean curvatures linearly related is (r, s)-stable if, and only if, it is a geodesic sphere of \({\mathbb {R}}^{n+1}\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alencar, H., do Carmo, M., Colares, A.G.: Stable hypersurfaces with constant scalar curvature. Math. Z. 213, 117–131 (1993)
Barbosa, J.L.M., Colares, A.G.: Stability of hypersurfaces with constant \(r\)-mean curvature. Ann. Glob. Anal. Geom. 15, 277–297 (1997)
Barbosa, J.L.M., do Carmo, M.: Stability of hypersurfaces with constant mean curvature. Math. Z. 185, 339–353 (1984)
Barbosa, J.L.M., do Carmo, M., Eschenburg, J.: Stability of hypersurfaces with constant mean curvature. Math. Z. 197, 123–138 (1988)
de Lima, H.F., Velásquez, M.A.L.: A new characterization of \(r\)-Stable hypersurfaces in space forms. Arch. Math. 47, 119–131 (2011)
Hardy, G., Littlewood, J.E., Polya, G.: Inequalities. Cambridge University, Cambridge (1983)
He, Y., Li, H.: Stability of area-preserving variations in space forms. Ann. Glob. Anal. Geom. 34, 55–68 (2008)
Hsiung, C.-C.: Some integral formulas for closed hypersurfaces. Math. Scand. 2, 286–294 (1954)
Reilly, R.: Variational properties of functions of the mean curvatures for hypersurfaces in space forms. J. Differ. Geom. 8, 465–477 (1973)
Reilly, R.: On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space. Comment. Math. Helv. 52, 525–533 (1977)
Rosenberg, H.: Hypersurfaces of constant curvature in space forms. Bull. Sci. Math. 117, 217–239 (1993)
Velásquez, M.A.L., de Sousa, A.F., de Lima, H.F.: On the stability of hypersurfaces in space forms. J. Math. Anal. Appl. 406, 134–146 (2013)
Acknowledgments
The second author is partially supported by CNPq, Brazil, grant 300769/2012-1. The third author is partially supported by CAPES/CNPq, Brazil, grant Casadinho/Procad 552.464/2011-2.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Constantin.
Rights and permissions
About this article
Cite this article
da Silva, J.F., de Lima, H.F. & Velásquez, M.A.L. The stability of hypersurfaces revisited. Monatsh Math 179, 293–303 (2016). https://doi.org/10.1007/s00605-015-0776-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-015-0776-x