Quadratic flatness and Regularity for Codimension-One Varifolds with Bounded Anisotropic Mean Curvature
Sławomir Kolasiński, Mario Santilli
TL;DR
The paper develops a quadratic flatness framework for codimension-one varifolds with bounded anisotropic mean curvature, measured with respect to a uniformly convex norm $\phi$. By combining barrier-principle techniques and a detailed analysis of the unit-normal bundle and $(n,h)$-sets, it first proves that the support of such varifolds is $(\mathscr{H}^n,n)$-rectifiable and, away from a $\mathscr{H}^n$-null set, contained in a countable union of $C^2$ submanifolds; moreover, a two-ball touching property holds at almost every point. This quadratic flatness result, together with Allard’s local anisotropic regularity, yields $C^{1,\alpha}$ regularity around density-one points for integral varifolds with bounded anisotropic mean curvature when $\mathscr{H}^n\llcorner\mathrm{spt}\|V\|$ is absolutely continuous with respect to $\|V\|$. Collectively, these results extend the classical isotropic regularity theory to anisotropic variational problems, enabling a comprehensive rectifiability and regularity framework for codimension-one varifolds under anisotropic curvature bounds.
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, α} $-regular around $ \mathscr{H}^n $ almost every point of density $ 1 $.