Stable hypersurfaces with constant scalar curvature.

*(English)*Zbl 0792.53057It is well known that hypersurfaces with constant mean curvature in Riemannian spaces are solutions to the variational problem of minimizing area for volume-preserving variations. Less known is the fact that hypersurfaces with constant scalar curvature in Riemannian spaces of constant sectional curvature are also solutions to a variational problem, namely, that of minimizing the integral of the mean curvature for volume preserving variations. Thus questions of stability can be posed for hypersurfaces with constant scalar curvature. The following result is proved. Let \(M\) be a compact immersed hypersurface of the euclidean space, or of a sphere, with constant scalar curvature. In the case the ambient space is a sphere, assume, in addition, that \(M\) is contained in an open hemisphere. Then \(M\) is stable if and only if \(M\) is a geodesic sphere. The question envolves the study of a second order differential operator related to the Laplacian and the second fundamental form of the hypersurface.

Reviewer: M.do Carmo (Rio de Janeiro)

##### MSC:

53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |

##### References:

[1] | [BdC] Barbosa, J.L., do Carmo, M.: Stability of hypersurfaces with constant mean curvature. Math. Z.185, 339–353 (1984) · Zbl 0529.53006 · doi:10.1007/BF01215045 |

[2] | [BdCE] Barbosa, J.L., do Carmo, M., Eschenburg, J.: Stability of hypersurfaces of constant mean curvature in Riemannian manifolds. Math. Z.197, 123–138 (1988). · Zbl 0653.53045 · doi:10.1007/BF01161634 |

[3] | [CY] Cheng, S.Y., Yau, S.Y.: Hypersurfaces with constant scalar curvature. Math. Ann.225, 195–204 (1977) · Zbl 0349.53041 · doi:10.1007/BF01425237 |

[4] | [H] Hsiung, C.C.: Some integral formulas for closed hypersurfaces. Math. Scand.2, 286–294 (1954) · Zbl 0057.14603 |

[5] | [MR] Montiel, S., Ros, A.: Compact hypersurfaces: The Alexandrov theorem for higher order mean curvatures. In: Lawson, B., Tenenblat, K. (eds.) Differential Geometry, a symposium in honor of Manfredo do Carmo. (Pitman Monogr., vol. 52, pp. 279–296) Essex: Longman Scientific and Technical 1991 |

[6] | [R] Reilly, R.C.: Variational properties of functions of the mean curvatures for hypersurfaces in space forms. J. Differ. Geom.8 465–477 (1973) · Zbl 0277.53030 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.