Characterization of John domains via weak tangents

TL;DR

This work characterizes bounded simply connected planar John domains through the infinitesimal boundary structure captured by weak tangents. By developing a framework of local Hausdorff and Gromov–Hausdorff convergence, it shows that $D$ is a John domain if and only if every $D$-component of every weak tangent $Y$ of $\partial D$ is a John domain (with a uniform constant equivalent to the global John constant when present). The authors also derive structural properties of weak tangents, including finiteness bounds on components and a decomposition of $Y$ into component boundaries, strengthening the connection between global distortion properties and boundary infinitesimal geometry. These results refine a prior necessary condition by Kinneberg into a complete, if not always quantitative, characterization and illuminate how weak tangents govern planar quasiconformal distortion.

Abstract

We characterize simply connected John domains in the plane with the aid of weak tangents of the boundary. Specifically, we prove that a bounded simply connected domain $D$ is a John domain if and only if, for every weak tangent $Y$ of $\partial D$, every connected component of the complement of $Y$ that ``originates" from $D$ is a John domain, not necessarily with uniform constants. Our main theorem improves a result of Kinneberg (arXiv:1507.04698), who obtains a necessary condition for a John domain in terms of weak tangents but not a sufficient one. We also establish several properties of weak tangents of John domains.

Theorems & Definitions (38)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Theorem 1.4
• Definition 2.1
• Definition 2.2
• Remark 2.3
• Lemma 2.4
• proof
• Proposition 2.5