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Stability of the Wulff shape. (English) Zbl 0924.53009

Let ${M}^{n}$ be a compact oriented $n$-manifold and let $x:{M}^{n}\to {ℝ}^{n+1}$ be an immersion of ${M}^{n}$ into Euclidean space ${ℝ}^{n+1}$. Let $\nu :{M}^{n}\to {S}^{n}\subset {ℝ}^{n+1}$ be the Gauss map of $x$, and let $F:{S}^{n}\to {ℝ}^{+}$ 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

${J}_{F}\left(x\right)={\int }_{M}F\left(\gamma \right)dM$

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 $k{W}_{F}$, where $k$ is a (computable) constant and ${W}_{F}$ is the Wulff shape of $F$ defined as follows. Let

$\varphi :{S}^{n}\to {ℝ}^{n+1}$

be given by $\varphi \left(\nu \right)={F}_{\nu }+\nabla F$, 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 ${W}_{F}=\varphi \left({S}^{n}\right)$.

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 ${ℝ}^{n+1}$ is stable iff it is a round sphere.

##### MSC:
 53A10 Minimal surfaces, surfaces with prescribed mean curvature 52A15 Convex sets in 3 dimensions (including convex surfaces) 49Q05 Minimal surfaces (calculus of variations)