Stability of hypersurfaces with constant \(r\)-mean curvature
The authors show that in the standard spaces \(\overline M^{n+1} (c)\) of constant curvature \(c\), the compact (closed) hypersurfaces of constant \(r\)-mean curvature, \(r\in \{1, \dots, n\}\), are critical points for some variational problem. In this class, the geodesic spheres are characterized as the stable ones, i.e., at these ``points'' the second derivative of the functional under consideration is \(\geq 0\) with respect to variations ``keeping the balance of volume zero''. The special situations \(r=1\) resp. \(c=0\) resp. \((r,c) =(2,1)\) were already investigated by \textit{J. L. Barbosa}, \textit{M. P. do Carmo} and \textit{J.-H. Eschenburg} [Math. Z. 197, 123-138 (1988; Zbl 0653.53045)] resp. \textit{H. Alencar}, \textit{M. P. do Carmo} and \textit{H. Rosenberg} [Ann. Global Anal. Geom. 11, 387-395 (1993; Zbl 0816.53031)] resp. \textit{H. Alencar}, \textit{M. P. do Carmo} and \textit{A. G. Colares} [Math. Z. 213, 117-131 (1993; Zbl 0792.53057)]. They use variational properties of functions of the mean curvatures derived by \textit{R. C. Reilly} [J. Differ. Geom. 8, 465-477 (1973; Zbl 0277.53030)].
hypersurfaces in standard spaces of constant curvature
scientific article; zbMATH DE number 1038403