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Stability of hypersurfaces with constant \((r+1)\)-th anisotropic mean curvature. (English) Zbl 1181.53052

Summary: Given a positive function \(F\) on \(S^n\) which satisfies a convexity condition, we define the \(r\)-th anisotropic mean curvature function \(HrF\) for hypersurfaces in \(\mathbb R^{n+1}\) which is a generalization of the usual \(r\)-th mean curvature function. Let \(X : M\rightarrow \mathbb R^{n+1}\) be an \(n\)-dimensional closed hypersurface with \(Hr+1F=\mathrm{constant}\), for some \(r\) with \(0\leq r\leq n - 1\), which is a critical point for a variational problem. We show that \(X(M)\) is stable if and only if \(X(M)\) is the Wulff shape.

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
49Q10 Optimization of shapes other than minimal surfaces
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