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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.
MSC:
53C40Global submanifolds (differential geometry)
53A10Minimal surfaces, surfaces with prescribed mean curvature
52A20Convex sets in n dimensions (including convex hypersurfaces)