Conformal contractions and lower bounds on the density of harmonic measure

TL;DR

This paper establishes a sufficient condition for simply-connected domains to be conformally mapped from the unit disk with nonexpansive maps, linking domain density with harmonic measure and extending the concept to higher dimensions.

Contribution

It introduces a concrete criterion connecting harmonic measure density with nonexpansive conformal maps, and explores higher-dimensional analogues.

Findings

Domains with dense harmonic measure can be characterized by nonexpansive conformal maps.

A higher-dimensional analogue of the problem is addressed.

The paper provides a new perspective on the relationship between harmonic measure and conformal mappings.

Abstract

We give a concrete sufficient condition for a simply-connected domain to be the image of the unit disk under a nonexpansive conformal map. This class of domains is also characterized by having sufficiently dense harmonic measure. The relation with the harmonic measure provides a natural higher-dimensional analogue of this problem, which is also addressed.