Abstract
In a Riemannian space form, we define the (r, k, a, b)-stability concerning closed hypersurfaces, where r and k are entire numbers satisfying the inequality \(0\le k<r\le n-2\) and a and b are real numbers (at least one nonzero). In this context, when \(b=0\), we provide a characterization of the geodesic spheres as critical points of the Jacobi functional associated with the notion of (r, k, a, 0)-stability. Moreover, in the case \(b\not =0\), by supposing that a hypersurface \(\Sigma ^n\) is contained either in an open hemisphere of the Euclidean sphere or in the Euclidean space or in the hyperbolic space, and considering some appropriate restrictions on the constants a and b, we are able to show that \(\Sigma ^n\) is (r, k, a, b)-stable if, and only if, \(\Sigma ^n\) is a geodesic sphere.
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References
Alencar, H., Colares, A.G.: Integral formulas for the \(r\)-mean curvature linearized operator of a hypersurface. Ann. Glob. Anal. Geom. 16, 203–220 (1998)
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 in Riemannian manifolds. Math. Z 197, 123–138 (1988)
Chen, H., Wang, X.: Stability and eigenvalue estimates of linear Weingarten hypersurfaces in a sphere. J. Math. Anal. Appl. 397, 658–670 (2013)
Cheng, S.Y., Yau, S.T.: Hypersurfaces with constant scalar curvature. Math. Ann. 225, 195–204 (1977)
Cheng, X., Rosenberg, H.: Embedded positive constant \(r\)-mean curvature hypersurfaces in \(M^m\times {\mathbb{R}}\). Ann. Braz. Acad. Sci. 77, 183–199 (2005)
Colares, A.G., da Silva, J.F.: Stable hypersurfaces as minima of the integral of an anisotropic mean curvature preserving a linear combination of area and volume. Math. Z 1–2, 595–623 (2013)
da Silva, J.F., de Lima, H.F., Velásquez, M.A.L.: The stability of hypersurfaces revisited. Monatshefte für Mathematik 179(2), 293–303 (2016)
de Lima, H.F., Velásquez, Marco A.L.: A new characterization of \(r\)-stable hypersurfaces in space forms. Arch. Math. 47, 119–131 (2011)
Elbert, M.F.: Contant positive \(2\)-mean curvature hypersurfaces. Illinois J. Math. 46, 247–267 (2002)
Gärding, L.: An inequality for hyperbolic polynomials. J. Math. Mech. 8, 957–965 (1959)
Hardy, G., Littlewood, J.E., Pólya, G.: Inequalities, 2nd edn. Cambridge Mathematical Library, Cambridge (1989)
He, Y., Li, H.: Stability of area-preserving variations in space forms. Ann. Glob. Anal. Geom. 34, 55–68 (2008)
Koh, S.E.: A characterization of round spheres. Proc. Am. Math. Soc. 126, 3657–3660 (1998)
Reilly, R.: Variational properties of functions of the mean curvatures for hypersurfaces in space forms. J. Differ. Geom. 8, 465–477 (1973)
Rosenberg, H.: Hypersurfaces of constant curvature in space forms. Bull. Sci. Math. 117, 217–239 (1993)
Velásquez, M.A., de Sousa, A.F., de Lima, H.F.: On the stability of hypersurfaces in space forms. J. Math. Anal. Appl. 406, 134–146 (2013)
Xin, Y.: Minimal submanifolds and related topics. World Scientific Publishing, Singapore (2003)
Acknowledgements
The authors would like to thank the referee for his/her valuable suggestions and useful comments which enable them to improve this paper. The first author is partially supported by CNPq, Brazil, grant 311224/2018-0. The second author is partially supported by CNPq, Brazil, grant 303977/2015-9.
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Velásquez, M.A.L., de Lima, H.F., da Silva, J.F. et al. (r, k, a, b)-Stability of hypersurfaces in space forms. Monatsh Math 192, 939–964 (2020). https://doi.org/10.1007/s00605-020-01428-1
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DOI: https://doi.org/10.1007/s00605-020-01428-1
Keywords
- Riemannian space forms
- Closed hypersurfaces
- Higher order mean curvatures
- (r, k, a, b)-Stability
- Geodesic spheres