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Geometry
Volume 2013 (2013), Article ID 718272, 7 pages
http://dx.doi.org/10.1155/2013/718272
Research Article

Hypersurfaces with Null Higher Order Anisotropic Mean Curvature

1School of Mathematical Sciences, Shanxi University, Taiyuan 030006, China
2Research Institute of Mathematics and Applied Mathematics, Shanxi University, Taiyuan 030006, China

Received 18 April 2013; Accepted 11 June 2013

Academic Editor: Reza Saadati

Copyright © 2013 Hua Wang and Yijun He. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Given a positive function on which satisfies a convexity condition, for , we define for hypersurfaces in the th anisotropic mean curvature function , a generalization of the usual th mean curvature function. We call a hypersurface anisotropic minimal if , and anisotropic -minimal if . Let be the set of points which are omitted by the hyperplanes tangent to . We will prove that if an oriented hypersurface is anisotropic minimal, and the set is open and nonempty, then is a part of a hyperplane of . We also prove that if an oriented hypersurface is anisotropic -minimal and its th anisotropic mean curvature is nonzero everywhere, and the set is open and nonempty, then has anisotropic relative nullity .