Calibrated Reifenberg With Holes

Abstract

In this article, we study a calibrated version of Reifenberg theorem "with holes". In particular we study sets that are suitably approximable at all points and scales by calibrated planes and show that, without any additional hypotheses on $\beta$-numbers, this implies measure upper bounds and rectifiability. This article follows the main techniques introduced in a previous article, but it allows for holes in the sets under consideration, and is more self-contained.