A Commentary on Teichm{\"u}ller's paper ``Vollst{\"a}ndige L{\"o}sung einer Extremalaufgabe der quasikonformen Abbildung'' (Complete solution of an extremal problem of the quasiconformal mapping)

TL;DR

This paper provides a commentary on Teichmüller's 1941 work, discussing the existence of extremal quasiconformal mappings for pentagons, which is a foundational result in Teichmüller theory.

Contribution

It offers an analysis of Teichmüller's proof of extremal quasiconformal mappings for pentagons, highlighting its significance in the development of Teichmüller theory.

Findings

Confirmed existence of extremal quasiconformal mappings for pentagons

Clarified the proof techniques used by Teichmüller

Connected the results to broader Teichmüller theory

Abstract

We comment on Teichm{\"u}ller 's paper ''Vollst{\"a}ndige L{\"o}sung einer Extremalaufgabe der quasikonformen Abbildung'' (Complete solution of an ex-tremal problem of the quasiconformal mapping),, published in 1941. In this paper, Teichm{\"u}ller gives a proof of the existence of extremal quasiconformal mappings in the case of the pentagon (disc with five distinguished points on the boundary). The final version of this paper will appear as a chapter in Volume VI of the Handbook of Teichm{\"u}ller theory.