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 $\mathrm{s}\mathrm{p}\mathrm{t}\mathrm{\parallel }V\mathrm{\parallel }$ has locally finite ${\mathcal{H}}^n$-measure, then $\mathrm{s}\mathrm{p}\mathrm{t}\mathrm{\parallel }V\mathrm{\parallel }$ can be touched by two mutually tangent balls at ${\mathcal{H}}^n$ almost all points. In particular, this result implies that ${\mathcal{H}}^n$ almost all of $\mathrm{s}\mathrm{p}\mathrm{t}\mathrm{\parallel }V\mathrm{\parallel }$ 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 ${\mathcal{H}}^n\mathrm{s}\mathrm{p}\mathrm{t}\mathrm{\parallel }V\mathrm{\parallel }$ is absolutely continuous with respect to $\mathrm{\parallel }V\mathrm{\parallel }$, then $\mathrm{s}\mathrm{p}\mathrm{t}\mathrm{\parallel }V\mathrm{\parallel }$ is $C^{1,\alpha }$-regular around ${\mathcal{H}}^n$ almost every point of density $1$.