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

Suppose $F$ is an integrand associated with a uniformly convex $\mathscr{C}^{3}$-norm, and $V$ is a $n$-dimensional varifold in an open subset of $\mathbf{R}^{n+1}$ such that $\mathscr{H}^n \llcorner \operatorname{spt} \| V \|$ is absolutely continuous with respect to $\| V \|$ 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 $\operatorname{spt} \| V \|$ is $\mathscr{C}^{2}$-rectifiable and the $\mathscr{C}^{1}$-regular part $M$ of $\operatorname{spt} \| V \|$ coincides $\mathscr{H}^n$ almost everywhere with the unit-density stratum of $V$. In this paper we prove that $\mathbf{h}_{F}(V,a) \in \operatorname{Nor}(M,a)$ for $\mathscr{H}^n$ a.e.\ $a \in M$ and that $\mathbf{h}_{F}(V, \cdot)$ agrees with the approximate mean $F$-curvature coming from the $\mathscr{C}^{2}$-rectifiable covering of $M$. These results provide anisotropic extensions of well known theorems in the Euclidean setting by Brakke, Sch\"atzle and Ambrosio-Masnou.