A commentary on Teichm\"uller's paper "Verschiebungssatz der quasikonformen Abbildung" (A displacement theorem of quasiconformal mapping)

TL;DR

This paper provides a detailed commentary on Teichmüller's 1944 work, explaining his solution to a specific extremal quasiconformal mapping problem and discussing its implications and open questions.

Contribution

It offers an in-depth explanation of Teichmüller's method for solving a particular extremal quasiconformal mapping problem and explores its consequences.

Findings

Teichmüller solved the extremal quasiconformal mapping problem for the unit disc.

The solution involves mapping 0 to a negative real point while fixing the boundary.

The paper discusses implications and open questions related to Teichmüller's work.

Abstract

This is a commentary on Teichm{\"u}ller's paper Ein Verschiebungssatz der quasikonformen Abbildung (A displacement theorem of quasiconformal mapping), published in 1944. We explain in detail how Teichm{\"u}ller solves the problem of finding the quasiconformal mapping from the unit disc to itself, sending 0 to a strictly negative point on the real line, holding the boundary of the disc pointwise fixed and with the smallest quasiconformal dilatation. We mention also some consequences of this extremal problem and we ask a question.