A Commentary on Teichm\"uller's paper "\"Uber Extremalprobleme der konformen Geometrie" (On extremal problems in conformal geometry)

TL;DR

This paper comments on Teichmüller's 1941 work on extremal problems in conformal geometry, highlighting its broad generalizations across function theory, topology, and algebra, and its significance in Teichmüller theory.

Contribution

It provides a detailed commentary on Teichmüller's generalization of extremal problems, connecting it to broader mathematical fields and its role in the development of Teichmüller theory.

Findings

Highlights Teichmüller's generalization of extremal problems

Connects conformal geometry with topology and algebra

Prepares for inclusion in the Handbook of Teichmüller theory

Abstract

We comment on the paper \"Uber Extremalprobleme der konformen Geometrie (On extremal problems in conformal geometry) by Teichm\"uller, published in 1941. This paper contains ideas on a wide generalization of his previous work on the solution of extremal problems in conformal geometry. The generalization concerns at the same time the fields of function theory, topology and algebra. The final version of this paper will appear as a chapter in Volume VI of the Handbook of Teichm\"uller theory. This volume is dedicated to the memory of Alexander Grothendieck.