Hypersurfaces with Null Higher Order Anisotropic Mean Curvature. (24th June 2013)

• Record Type: Journal Article
• Authors: Wang, Hua; He, Yijun
• Other Names: Saadati Reza Academic Editor.
• Is Part Of: Geometry. Volume 2013(2013)
• Publication Date: 2013-06-24
• Subjects: Geometry -- Periodicals; Geometry; Periodicals; 516
• DOI: 10.1155/2013/718272
• Languages: English
• ISSNs: 2314-422X

Abstract

Abstract : Given a positive function F on 𝕊 n which satisfies a convexity condition, for 1 ≤ r ≤ n, we define for hypersurfaces in ℝ n + 1 the r th anisotropic mean curvature function H r ; F, a generalization of the usual r th mean curvature function. We call a hypersurface anisotropic minimal if H F = H 1 ; F = 0, and anisotropic r -minimal if H r + 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 nonempty, then x ( M ) is a part of a hyperplane of ℝ n + 1 . We also prove that if an oriented hypersurface M is anisotropic r -minimal and its r th anisotropic mean curvature H r ; F is nonzero everywhere, and the set W is open and nonempty, then M has anisotropic relative nullity n − r .