A new characterization of chord-arc domains

TL;DR

This paper proves that uniform rectifiability of a domain's boundary ensures it is a chord-arc domain with both interior and exterior Corkscrew conditions, linking geometric boundary properties to domain regularity.

Contribution

It establishes that uniform rectifiability implies the existence of exterior Corkscrew points, characterizing chord-arc domains within uniform domains.

Findings

Uniform rectifiability implies exterior Corkscrew points at all scales.

Chord-arc domains are characterized by boundary regularity and Corkscrew conditions.

Results have implications for F. and M. Riesz theorems and free boundary problems.

Abstract

We show that if $\Omega \subset \mathbb{R}^{n+1}$, $n\geq 1$, is a uniform domain (aka 1-sided NTA domain), i.e., a domain which enjoys interior Corkscrew and Harnack Chain conditions, then uniform rectifiability of the boundary of $\Omega$ implies the existence of exterior Corkscrew points at all scales, so that in fact, $\Omega$ is a chord-arc domain, i.e., a domain with an Ahlfors-David regular boundary which satisfies both interior and exterior Corkscrew conditions, and an interior Harnack Chain condition. We discuss some implications of this result, for theorems of F. and M. Riesz type, and for certain free boundary problems.