On the Modulus of a Union of Nested Annuli

TL;DR

This paper establishes conditions under which a union of nested conformal annuli has bounded modulus, providing estimates and a method to construct quasiconformal maps between annuli.

Contribution

It introduces a new theorem with sufficient conditions and estimates for the modulus of unions of nested conformal annuli, using an interpolation approach.

Findings

Provides sufficient conditions for bounded modulus of union of annuli

Offers estimates for the modulus of such unions

Develops a quasiconformal interpolation method

Abstract

We prove a result which gives sufficient conditions for a conformal annulus which is a countable union of nested conformal annuli to have bounded modulus. Our theorem also gives estimates for the modulus of such an annulus and is proved using an interpolation result which constructs a quasiconformal map between two annuli given a smooth identification of their boundaries.