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

Abstract: Given a uniformly convex norm $ \phi $ on $ \mathbf{R}^{n+1} $ and a unit-density $ n $-dimensional varifold $ V $ in an open subset of $ \mathbf{R}^{n+1} $ with bounded mean $ \phi $-curvature, under the hypothesis $$ \textrm{$\mathcal{H}^n \llcorner {\rm spt}| V | $ is absolutely continuous with respect to $ | V | $},$$ we prove that there exists an open dense $ \mathscr{C}^{1, \alpha}$-regular part in $ {\rm spt} | V | $. Moreover we prove that the $ \mathscr{C}^{1, \alpha}$-regular part of $ V $ is $ \mathcal{H}^n $-almost equal to $({\rm spt}| V |)^\ast $, the subset of $ {\rm spt}| V | $ where at least one blow up (in Hausdorff distance) of $ {\rm spt}| V | $ is not equal to $ \mathbf{R}^{n+1} $. These results are consequences of a quadratic flatness theorem asserting that if $ V $ is a general -- not necessarily rectifiable -- varifold $ V $ with bounded mean $ \phi $-curvature and locally $ \mathcal{H}^n $-finite support, then the set of points $ R $ where $ {\rm spt}| V |$ admits (exactly) two mutually tangent balls satisfies $ \mathcal{H}^n(({\rm spt}| V |)^\ast \setminus R) =0 $ and meets each relatively open set of $ {\rm spt}| V | $ on a set of positive $ \mathcal{H}^n $-measure. Indeed, quadratic flatness at a point guarantees that the Allard anisotropic regularity theorem can be applied, hence proving that $ {\rm spt}| V |$ is regular around that point.