English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

This is a mathematical commentary on Teichm{\"u}ller's paper ``Bestimmung derextremalen quasikonformen Abbildungen bei geschlossenen orientiertenRiemannschen Fl{\"a}chen'' (Determination of extremal quasiconformal maps ofclosed oriented Riemann surfaces). This paper is among the last (and may be thelast one) that Teichm{\"u}ller wrote on the theory of moduli. It contains theproof of the so-called Teichm{\"u}ller existence theorem for a closed surfaceof genus at least 2. For this proof, the author defines a mapping between aspace of equivalence classes of marked Riemann surfaces (the Teichm{\"u}llerspace) and a space of equivalence classes of certain Fuchsian groups (theso-called Fricke space). After that, he defines a map between the latter andthe Euclidean space of dimension 6g-6 Using Brouwer's theorem of invariance ofdomain, he shows that this map is a homeomorphism. This involves in particulara careful definition of the topologies of Fricke space, the computation of itsdimension, and comparison results between hyperbolic distance andquasiconformal dilatation. The use of the invariance of domain theorem is inthe spirit of Poincar{\'e} and Klein's use of the so-called ``continuityprinciple'' in their attempts to prove the uniformization theorem.

A Commentary on Teichm{\"u}ller's paper ``Bestimmung der extremalen  quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen  Fl{\"a}chen''