2011/12/09 by Yijun He, He, Yijun
Mathematics · #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities #math.DG
paper · pdf · doi:10.48550/arxiv.1112.2231
This paper has been withdrawn by the author
openalex publication_date 2011/12/09 · arxiv created 2013/06/20 · arxiv updated 2013/06/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a positive function F on \mathbb Sn which satisfies a convexity condition, for 1≤ r≤ n, we define for hypersurfaces in ℝn+1 the r-th anisotropic mean curvature function Hr; F, a generalization of the usual r-th mean curvature function. We call a hypersurface is anisotropic minimal if HF=H1; F=0, and anisotropic r-minimal if Hr+1; F=0. Let W be the set of points which are omitted by the hyperplanes tangent to M. We will prove that if an oriented hypersurface M is anisotropic minimal, and the set W is open and non-empty, then x(M) is a part of a hyperplane of \mathbb Rn+1. We also prove that if an oriented hypersurface M is anisotropic r-minimal and its r-th anisotropic mean curvature Hr; F is nonzero everywhere, and the set W is open and non-empty, then M has anisotropic relative nullity n-r.