Uniformization problems in the plane: A survey

TL;DR

This survey reviews the history and recent advances in uniformization problems in the complex plane, focusing on Koebe's conjecture and related conformal and quasiconformal mapping questions.

Contribution

It provides a comprehensive overview of the development, current status, and open problems in planar uniformization, highlighting recent progress and key variants.

Findings

Koebe's conjecture remains open after 120 years.

Recent progress has advanced understanding of conformal and quasiconformal mappings.

The survey discusses the uniqueness and variants of uniformization problems.

Abstract

In this survey we present the history and recent progress on several fundamental (quasi)conformal uniformization problems in the complex plane. Uniformization refers to the process of mapping a space to a canonical model by means of a well-behaved transformation that preserves the geometry and distorts shapes in a controlled fashion. A central problem in the area is Koebe's conjecture, which remains open after almost 120 years and predicts that each planar domain can be conformally mapped to a circle domain -- that is, a domain whose complementary components are points or closed disks. We trace the history of the conjecture, outline recent developments, and examine the associated uniqueness problem. We also discuss variants, with particular emphasis on the question whether a compact set can be mapped by a quasiconformal self-map of the plane to a Schottky set -- that is, a set in the plane whose complement is the union of disjoint open disks.