Conformal dimension and boundaries of planar domains

TL;DR

This paper proves that the conformal dimension of boundaries of John domains in the Riemann sphere is either 0 or 1, using a new discretized condition called the discrete UWS property.

Contribution

It introduces the discrete UWS property and shows that boundaries of John domains have this property, leading to the conformal dimension result.

Findings

Boundaries of John domains have the discrete UWS property.

Spaces with the discrete UWS property have conformal dimension 0 or 1.

The paper establishes connectivity properties of weak tangents of these spaces.

Abstract

Building off of techniques that were recently developed by M. Carrasco, S. Keith, and B. Kleiner to study the conformal dimension of boundaries of hyperbolic groups, we prove that uniformly perfect boundaries of John domains in the Riemann sphere have conformal dimension equal to 0 or 1. Our proof uses a discretized version of Carrasco's "uniformly well-spread cut point" condition, which we call the discrete UWS property, that is well-suited to deal with metric spaces that are not linearly connected. More specifically, we prove that boundaries of John domains have the discrete UWS property and that any compact, doubling, uniformly perfect metric space with the discrete UWS property has conformal dimension equal to 0 or 1. In addition, we establish other geometric properties of metric spaces with the discrete UWS property, including connectivity properties of their weak tangents.