Hypersurfaces with Null Higher Order Anisotropic Mean Curvature

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 .