Rectifiable Reifenberg and uniform positivity under almost calibrations
The Reifenberg theorem \cite{reif_orig} tells us that if a set $S\subseteq B_2\subseteq \mathbb R^n$ is uniformly close on all points and scales to a $k$-dimensional subspace, then $S$ is H\"older homeomorphic to a $k$-dimensional Euclidean ball. In general this is sharp, for instance such an $S$ may have infinite volume, be fractal in nature, and have no rectifiable structure. The goal of this note is to show that we can improve upon this for an almost calibrated Reifenberg set, or more generally under a positivity condition in the context of an $\epsilon$-calibration $\Omega$ . An $\epsilon$-calibration is very general, the condition holds locally for all continuous $k$-forms such that $\Omega[L]\leq 1+\epsilon$ for all $k$-planes $L$. We say an oriented $k$-plane $L$ is $\alpha$-positive with respect to $\Omega$ if $\Omega[L]>\alpha>0$. If $\Omega[L]>\alpha> 1-\epsilon$ then we call $L$ an $\epsilon$-calibrated plane. The main result of this paper is then the following. Assume at all points and scales $B_r(x)\subseteq B_2$ that $S$ is $\delta$-Hausdorff close to a subspace $L_{x,r}$ which is uniformly positive $\Omega[L_{x,r}]>\alpha $ with respect to an $\epsilon$-calibration. Then $S$ is $k$-rectifiable with uniform volume bounds.