Critical Allard regularity: pointwise tilt-excess estimates

Abstract

The main results of this paper provide VMO-type estimates for the quadratic tilt-excess on varifolds with critical generalized mean curvature. These estimates apply to varifolds with "almost-integral" density which are close to a multiplicity one $m$-disc in a ball in the usual senses. The class of almost-integral varifolds allows for varifolds with non-perpendicular mean curvature. Moreover, the estimates hold \emph{uniformly for every point} in a relatively open set in $\text{spt}||V||$ and naturally imply a Reifenberg-type parametrization. The proof relies upon generalizing the $Q$-valued Lipschitz approximation and Sobolev-Poincar\'e estimates of arXiv:0808.3660 to almost-integral rectifiable varifolds.