Hypersurfaces with null higher order anisotropic mean curvature

TL;DR

This paper studies hypersurfaces with null higher order anisotropic mean curvature, proving they are parts of hyperplanes or have specific nullity properties under certain conditions.

Contribution

It introduces a generalization of mean curvature for hypersurfaces with a convex function on the sphere and characterizes hypersurfaces with null higher order anisotropic mean curvature.

Findings

Anisotropic minimal hypersurfaces with certain tangent hyperplane conditions are hyperplanes.

Hypersurfaces with nonzero r-th anisotropic mean curvature have anisotropic relative nullity.

The paper extends classical results to anisotropic curvature settings.

Abstract

Given a positive function $F$ on $\mathbb S^n$ which satisfies a convexity condition, for $1\leq r\leq n$, 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 $n-r$.