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Stability of the Wulff shape. (English) Zbl 0924.53009

Let M n be a compact oriented n-manifold and let x:M n n+1 be an immersion of M n into Euclidean space n+1 . Let ν:M n S n n+1 be the Gauss map of x, and let F:S n + be a position function in the unit sphere S n .

Consider compactly supported variations of M that preserve volume and consider the critical points of

J F (x)= M F(γ)dM

under such variations; for F1, 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 M is stable iff it is kW F , where k is a (computable) constant and W F is the Wulff shape of F defined as follows. Let

ϕ:S n n+1

be given by ϕ(ν)=F ν +F, where F is the gradient of F in the standard metric of S n . The Wulff shape W F is the hypersurface given by W F =ϕ(S n ).

If F1, then W F =S n , 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 n+1 is stable iff it is a round sphere.

53A10Minimal surfaces, surfaces with prescribed mean curvature
52A15Convex sets in 3 dimensions (including convex surfaces)
49Q05Minimal surfaces (calculus of variations)