Hypersurfaces with null higher order anisotropic mean curvature

Abstract

Given a positive function $F$ on ${\mathbb{S}}^n$ which satisfies a convexity condition, for $1\le r\le n$, we define for hypersurfaces in ${\mathbb{R}}^{n+1}$ the $r$-th anisotropic mean curvature function $H_{r;F}$, a generalization of the usual $r$-th mean curvature function. We call a hypersurface is 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 non-empty, then $x(M)$ is a part of a hyperplane of ${\mathbb{R}}^{n+1}$. 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 non-empty, then $M$ has anisotropic relative nullity $n-r$.

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