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Compact hypersurfaces: The Alexandrov theorem for higher order mean curvatures. (English) Zbl 0723.53032
Differential geometry. A symposium in honour of Manfredo do Carmo, Proc. Int. Conf., Rio de Janeiro/Bras. 1988, Pitman Monogr. Surv. Pure Appl. Math. 52, 279-296 (1991).
[For the entire collection see Zbl 0718.00010.] In this clearly written article the authors discuss compact hypersurfaces of constant r-th order mean curvature in a space form. The second author [Rev. Mat. Iberoam. 3, No.3/4, 447-453 (1987; Zbl 0673.53003)] proved that the hypersphere is the only compact hypersurface with constant r-th mean curvature embedded in the Euclidean space. His proof uses results of {\it R. C. Reilly} [Indiana Univ. Math. J. 26, 459-472 (1977; Zbl 0391.53019)]. In the paper under review by using a more geometric approach and some ideas of {\it E. Heintze} and {\it H. Karcher} [Ann. Sci. Ec. Norm. Supér., IV. Sér. 11, 451-470 (1978; Zbl 0416.53027)] the authors prove: A compact hypersurface embedded into the Euclidean or hyperbolic space or into the open half-sphere with constant r-th mean curvature must be a geodesic hypersphere.

53C40Global submanifolds (differential geometry)