A Commentary on Teichm{\"u}ller's paper "Untersuchungen \"uber konforme und quasikonforme Abbildungen" (Investigations on conformal and quasiconformal mappings) (to appear in Vol. VII of the \emph{Handbook of Teichm\"uller theory}

TL;DR

This paper comments on Teichmüller's 1938 work, highlighting fundamental results in conformal geometry such as the Modulsatz and Main Lemma, which have significant applications in value distribution theory and conformal invariants.

Contribution

It provides a detailed analysis of Teichmüller's foundational lemmas and their impact on conformal geometry and quasiconformal mappings, enriching the understanding of conformal invariants.

Findings

Introduction of the Modulsatz for almost circularity of loci

Main Lemma ensuring circularity near infinity of images of circles

Development of conformal invariants of doubly connected domains

Abstract

This is a commentary on Teichm{\"u}ller's paper Unter-suchungen{\"u}ber konforme und quasikonforme Abbildungen (Inves-tigations on conformal and quasiconformal mappings) published in 1938. The paper contains fundamental results in conformal geometry , in particular a lemma, known as the Modulsatz, which insures the almost circularity of certain loci defined as complementary components of simply connected regions in the Riemann sphere, and another lemma, which we call the Main Lemma, which insures the circularity near infinity of the image of circles by a qua-siconformal map. The two results find wide applications in value distribution theory, where they allow the efficient use of moduli of doubly connected domains and of quasiconformal mappings. Te-ichm{\"u}ller's paper also contains a thorough development of the theory of conformal invariants of doubly connected domains.The final version of this paper will appear in Vol. VII of the \emph{Handbook of Teichm{\"u}ller theory} (European Mathematical Society Publishing House, 2020).