平面中的统一化问题：综述
Uniformization problems in the plane: A survey
AI总结：
综述探讨了复平面上几个基础的（准）共形统一化问题的历史与最新进展，重点讨论了Koebe猜想的开放性问题及相关唯一性问题，以及Schottky集的映射问题。
摘要：
本文综述了复平面上几个基础的（准）共形统一化问题的历史和最新进展。统一化指的是通过一种良好的变换将空间映射到标准模型，该变换保持几何特性并以受控方式扭曲形状。该领域的一个核心问题是Koebe猜想，该猜想在近120年后仍保持开放，预测每个平面域可以共形映射到圆域——即其互补部分为点或闭合圆盘。我们追溯了该猜想的历史，概述了最新进展，并检查了相关唯一性问题。我们还讨论了变种问题，特别强调了是否可以通过平面的准共形自映射将紧集映射到Schottky集——即其补集为不相交的开放圆盘的集合。
英文摘要：
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.
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