Quadratic flatness and Regularity for Codimension-One Varifolds with Bounded Anisotropic Mean Curvature

We prove that if $V$ is a $n$-dimensional varifold in an open subset of $\mathbf{R}^{n+1}$ with bounded anisotropic mean curvature such that ${\rm spt} \| V \|$ has locally finite $\mathscr{H}^n$-measure, then ${\rm spt} \| V \|$ can be touched by two mutually tangent balls at $\mathscr{H}^n$ almost all points. In particular, this result implies that $\mathscr{H}^n$ almost all of ${\rm spt} \| V \|$ can be covered by the union of countably many $C^2$-regular $n$-dimensional submanifolds of $\mathbf{R}^{n+1}$. Moreover, combined with Allard's local anisotropic regularity theorem, it implies that if $V$ is an integral varifold with bounded anisotropic mean curvature and if $\mathscr{H}^n \llcorner {\rm spt} \| V \|$ is absolutely continuous with respect to $\| V \|$, then ${\rm spt} \| V \|$ is $C^{1, \alpha}$-regular around $\mathscr{H}^n$ almost every point of density $1$.