Caratheodory convergence and harmonic measure

TL;DR

This paper explores new ways to characterize Caratheodory convergence of simply connected domains and examines how these definitions extend to multiply-connected domains.

Contribution

It introduces several novel characterizations of Caratheodory convergence and analyzes their generalizations to more complex multiply-connected domains.

Findings

New characterizations of Caratheodory convergence for simply connected domains

Extension of convergence definitions to multiply-connected domains

Insights into the behavior of harmonic measure in different convergence scenarios

Abstract

We give several new characterizations of Caratheodory convergence of simply connected domains. We then investigate how different definitions of convergence generalize to the multiply-connected case.