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 . 1. Introduction Let be a smooth function which satisfies the following convexity condition: where is the standard unit sphere in , denotes the intrinsic Hessian of on , denotes the identity on , and >0 means that the matrix is positive definite. We consider the map its image is a smooth, convex hypersurface in called the Wulff shape of (see [1–9]). When , the Wulff shape is just . Now let be a smooth immersion of an oriented hypersurface. Let denote its Gauss map. The map is called the anisotropic Gauss map of . Let . is called the -Weingarten operator, and the eigenvalues of are called anisotropic principal curvatures. Let be the elementary symmetric functions of the anisotropic principal curvatures : We set . The th anisotropic mean curvature is defined by , also see Reilly [10]. is called the anisotropic mean curvature. When , is just the Weingarten operator of hypersurfaces, and is just the th mean curvature of hypersurfaces which has been studied by many authors (see [11–14]). Thus, the th anisotropic mean curvature generalizes the th mean curvature of hypersurfaces in the -dimensional Euclidean space . We say that is anisotropic -minimal if . For , we define . We call the anisotropic relative nullity; it generalized the usual relative nullity. For a smooth immersion of a hypersurface into an -dimensional space form with constant sectional curvature , we denote by where for every , is the totally geodesic hypersurface of tangent to at . So, in the case of , is the set of points which are omitted by the hyperplanes tangent to . We will study immersion with nonempty. In this direction, Hasanis and Koutroufiotis (see [15]) proved the following. Theorem 1. Let be a complete minimal immersion with . If is nonempty, then is totally geodesic. Later, in [16], Alencar and Frensel extended the result above assuming an extra condition. They proved the following.