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

TL;DR

This paper establishes regularity and touching properties of codimension-one varifolds with bounded anisotropic mean curvature, showing they are mostly covered by smooth submanifolds and are regular at almost all points of density 1.

Contribution

It proves that varifolds with bounded anisotropic mean curvature are almost everywhere touchable by tangent balls and can be approximated by smooth manifolds, extending regularity results.

Findings

Varifolds can be touched by tangent balls at almost all points.

Most of the support can be covered by countably many $C^2$-regular submanifolds.

Under additional conditions, the support is $C^{1,\alpha}$-regular at almost all points of density 1.

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 $.