A commentary on Teichm\"uller's paper "Extremale quasikonforme Abbildungen und quadratische Differentiale"

TL;DR

This paper provides a detailed commentary on Teichmüller's 1940 work on extremal quasiconformal mappings and quadratic differentials, highlighting its main results, ideas, and influence on subsequent research in Teichmüller theory.

Contribution

It offers the first comprehensive analysis and contextualization of Teichmüller's seminal 1940 paper, clarifying its significance and impact.

Findings

Main results of Teichmüller's paper are highlighted.

Key ideas behind extremal quasiconformal mappings are explained.

The influence of Teichmüller's work on later developments is described.

Abstract

We provide a commentary on Teichm{\"u}ller's paper "Extremale quasikonforme Abbildungen und quadratische Differentiale" (Extremal quasiconformal mappings of closed oriented Riemann surfaces), Abh. Preuss. Akad. Wiss., Math.-Naturw. Kl. 1940, No.22, 1-197 (1940). The paper is quoted in several works, although it was read by very few people. Some of the results it contains were rediscovered later on and published without any reference to Teichm{\"u}ller.In this commentary, we highlight the main results and the main ideas contained in that paper and we describe some of the important developments they gave rise to.The final version of this paper, together with the English translation of Teichm{\"u}ller's paper, will apper in Volume V of the \emph{Handbook of Teichm{\"u}ller theory} (European Mathematical Society Publishing House, 2015).