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

Abstract

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 $\mathrm{spt}\Vert V\Vert$ has locally finite ${\mathcal{H}}^n$-measure, then $\mathrm{spt}\Vert V\Vert$ 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 $\mathrm{spt}\Vert V\Vert$ 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 ${\mathcal{H}}^n└\mathrm{spt}\Vert V\Vert$ is absolutely continuous with respect to $\Vert V\Vert$, then $\mathrm{spt}\Vert V\Vert$ is $C^{1,\hat{I}\pm }$-regular around ${\mathcal{H}}^n$ almost every point of density $1$.

Employing some modifications of the arguments already employed in previous versions, we are able to show in v4 quadratic flatness of $\mathrm{spt}\Vert V\Vert$ for a codimension one varifold $V$ of bounded anisotropic mean curvature in ${\mathbf{R}}^{n+1}$, provided $\mathrm{spt}\Vert V\Vert$ has locally finite ${\mathcal{H}}^n$ measure