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 $ {\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 $.