Weak geometric lemma for ADR boundary of a uniform domain implies uniform rectifiability

Abstract

We resolve a conjecture of Guy David and Svitlana Mayboroda. We show that if $\Omega \subset \mathbb{R}^n$ is a uniform domain with $(n-1)$-dimensional Ahlfors David regular boundary $E = \partial \Omega$, and if $E$ satisfies the Weak Geometric Lemma, then $E$ satisfies the Bilateral Weak Geometric Lemma. In particular, $E$ is uniformly rectifiable.