Let be a compact oriented -manifold and let be an immersion of into Euclidean space . Let be the Gauss map of , and let be a position function in the unit sphere .
Consider compactly supported variations of that preserve volume and consider the critical points of
under such variations; for , these critical points are hypersurfaces of constant mean curvature. Of course, a notion of stability is naturally defined for the above variational problem.
The author proves that is stable iff it is , where is a (computable) constant and is the Wulff shape of defined as follows. Let
be given by , where is the gradient of in the standard metric of . The Wulff shape is the hypersurface given by .
If , then , hence the author’s result generalizes the theorem of J. L. Barbosa and M. P. do Carmo [Math. Z. 185, 339-353 (1984; Zbl 0529.53006)] that a compact hypersurface of constant mean curvature in is stable iff it is a round sphere.