Perpendicularity and Locality for Codimension-One Varifolds with Bounded Anisotropic Mean Curvature

Abstract

Suppose $F$ is an integrand associated with a uniformly convex ${\mathcal{C}}^3$-norm, and $V$ is a $n$-dimensional varifold in an open subset of ${\mathbf{R}}^{n+1}$ such that ${\mathcal{H}}^n└\mathrm{spt}\Vert V\Vert$ is absolutely continuous with respect to $\Vert V\Vert$ and the mean $F$-curvature ${\mathbf{h}}_F(V,\cdot )$ is bounded in ${\mathbf{L}}^{\infty }$. In our previous result arXiv:2507.18357 we prove that $\mathrm{spt}\Vert V\Vert$ is ${\mathcal{C}}^2$-rectifiable and the ${\mathcal{C}}^1$-regular part $M$ of $\mathrm{spt}\Vert V\Vert$ coincides ${\mathcal{H}}^n$ almost everywhere with the unit-density stratum of $V$. In this paper we prove that ${\mathbf{h}}_F(V,a)\in \mathrm{Nor}(M,a)$ for ${\mathcal{H}}^n$ a.e.\ $a\in M$ and that ${\mathbf{h}}_F(V,\cdot )$ agrees with the approximate mean $F$-curvature coming from the ${\mathcal{C}}^2$-rectifiable covering of $M$. These results provide anisotropic extensions of well known theorems in the Euclidean setting by Brakke, Schätzle and Ambrosio-Masnou.