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Stability of hypersurfaces with constant \(r\)-mean curvature. (English) Zbl 0891.53044

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 J. L. Barbosa, M. P. do Carmo and J.-H. Eschenburg [Math. Z. 197, 123-138 (1988; Zbl 0653.53045)] resp. H. Alencar, M. P. do Carmo and H. Rosenberg [Ann. Global Anal. Geom. 11, 387-395 (1993; Zbl 0816.53031)] resp. H. Alencar, M. P. do Carmo and A. G. Colares [Math. Z. 213, 117-131 (1993; Zbl 0792.53057)]. They use variational properties of functions of the mean curvatures derived by R. C. Reilly [J. Differ. Geom. 8, 465-477 (1973; Zbl 0277.53030)].

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
49K10 Optimality conditions for free problems in two or more independent variables
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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