The Caratheodory Topology for Multiply Connected Domains I

TL;DR

This paper studies how multiply connected domains behave under Caratheodory convergence, focusing on hyperbolic geodesics called meridians, and establishes conditions linking geodesic behavior to domain convergence.

Contribution

It introduces a continuity result for hyperbolic geodesics in converging domains and relates Caratheodory convergence to Riemann mappings of standard slit domains.

Findings

Continuity of hyperbolic geodesics under domain convergence

Characterization of Caratheodory convergence via geodesic properties

Equivalence of convergence conditions using Riemann mappings

Abstract

We consider the convergence of pointed multiply connected domains in the Caratheodory topology. Behaviour in the limit is largely determined by the properties of the simple closed hyperbolic geodesics which separate components of the complement. Of particular importance are those whose hyperbolic length is as short as possible which we call meridians of the domain. We prove a continuity result on convergence of such geodesics for sequences of pointed hyperbolic domains which converge in the Caratheodory topology to another pointed hyperbolic domain. Using this we describe an equivalent condition to Caratheodory convergence which is formulated in terms of Riemann mappings to standard slit domains.