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Compact embedded hypersurfaces with constant higher order anisotropic mean curvatures. (English) Zbl 1166.53035
Summary: Given a positive function F on S n which satisfies a convexity condition, for 1rn, we define for hypersurfaces in n+1 the r-th anisotropic mean curvature function H r F , a generalization of the usual r-th mean curvature function. We prove that a compact embedded hypersurface without boundary in n+1 with constant H r F is the Wulff shape, up to translations and homotheties. In the case r=1, our result is the anisotropic version of Alexandrov’s Theorem, which gives an affirmative answer to an open problem of F. Morgan.
53C40Global submanifolds (differential geometry)
53A10Minimal surfaces, surfaces with prescribed mean curvature
52A20Convex sets in n dimensions (including convex hypersurfaces)