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Stability of the Wulff shape. (English) Zbl 0924.53009

Let \(M^n\) be a compact oriented \(n\)-manifold and let \(x\colon M^n\to \mathbb{R}^{n+1}\) be an immersion of \(M^n\) into Euclidean space \(\mathbb{R}^{n+1}\). Let \(\nu\colon M^n\to S^n\subset \mathbb{R}^{n+1}\) be the Gauss map of \(x\), and let \(F\colon S^n\to \mathbb{R}^+\) 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 \[ J_F(x) =\int_MF(\gamma) dM \] under such variations; for \(F\equiv 1\), 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 \[ \phi\colon S^n\to \mathbb{R}^{n+1} \] be given by \(\phi(\nu)=F_\nu + \nabla F\), where \(\nabla F\) is the gradient of \(F\) in the standard metric of \(S^n\). The Wulff shape \(W_F\) is the hypersurface given by \(W_F=\phi(S^n)\).
If \(F\equiv 1\), 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 \(\mathbb{R}^{n+1}\) is stable iff it is a round sphere.

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
52A15 Convex sets in \(3\) dimensions (including convex surfaces)
49Q05 Minimal surfaces and optimization

Citations:

Zbl 0529.53006
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