Koebe uniformization of nondegenerate domains with bounded gap-ratio

TL;DR

This paper proves that any nondegenerate, uncountably connected domain with a bounded gap-ratio can be conformally mapped to a circle domain using transboundary extremal length, advancing the understanding of Koebe uniformization.

Contribution

It establishes a new conformal uniformization result for domains with bounded gap-ratio using transboundary extremal length, filling a gap in complex analysis.

Findings

Domains with bounded gap-ratio are conformally equivalent to circle domains.

The method involves transboundary extremal length techniques.

The result applies to uncountably connected, nondegenerate domains.

Abstract

Koebe uniformization is a fundemental problem in complex analysis. In this paper, we use transboundary extremal length to show that every nondegenerate and uncountably connected domain with bounded gap-ratio is conformally homeomorphic to a circle domain.