Submitted Paper

Inserted: 30 jul 2025
Last Updated: 16 dec 2025

Year: 2025

Abstract:

We prove that a $ n $-dimensional unit-density varifold $ V $ in an open subset of $ \mathbf{R}^{n+1} $ with bounded anisotropic mean curvature is $ C^{1, \alpha} $-regular almost everywhere, provided $\mathcal{H}^{n} \llcorner {\rm spt} \
V \
$ is absolutely continuous with respect to the weight measure $ \
V \
$. This result answers a natural question arising from the work of Allard on the regularity of varifolds with bounded anisotropic mean curvature (1986). Indeed, we prove a quadratic flatness theorem for general varifolds of bounded anisotropic mean curvature, that can be combined with Allard's local anisotropic regularity theorem to infer almost everywhere regularity. The quadratic flatness theorem is of independent interest, since it provides the first result in the literature dealing with the full support of a varifold with bounded anisotropic mean curvature.