Hypersurfaces with Null Higher Order Anisotropic Mean Curvature

Abstract EN

Given a positive function F on ?n which satisfies a convexity condition, for 1≤r≤n, we define for hypersurfaces in ℝn+1 the rth anisotropic mean curvature function Hr;F, a generalization of the usual rth mean curvature function.

We call a hypersurface 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 nonempty, then x(M) is a part of a hyperplane of ℝn+1.

We also prove that if an oriented hypersurface M is anisotropic r-minimal and its rth anisotropic mean curvature Hr;F is nonzero everywhere, and the set W is open and nonempty, then M has anisotropic relative nullity n−r.