A Commentary on Teichm{ü}ller's paper ``Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Fl{ä}chen''

Abstract: This is a mathematical commentary on Teichm{\"u}ller's paper Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Fl{\"a}chen'' (Determination of extremal quasiconformal maps of closed oriented Riemann surfaces). This paper is among the last (and may be the last one) that Teichm{\"u}ller wrote on the theory of moduli. It contains the proof of the so-called Teichm{\"u}ller existence theorem for a closed surface of genus at least 2. For this proof, the author defines a mapping between a space of equivalence classes of marked Riemann surfaces (the Teichm{\"u}ller space) and a space of equivalence classes of certain Fuchsian groups (the so-called Fricke space). After that, he defines a map between the latter and the Euclidean space of dimension 6g-6 Using Brouwer's theorem of invariance of domain, he shows that this map is a homeomorphism. This involves in particular a careful definition of the topologies of Fricke space, the computation of its dimension, and comparison results between hyperbolic distance and quasiconformal dilatation. The use of the invariance of domain theorem is in the spirit of Poincar{\'e} and Klein's use of the so-calledcontinuity principle'' in their attempts to prove the uniformization theorem.