A Commentary on Teichm{\"u}ller's paper ``Vollst{\"a}ndige L{\"o}sung einer Extremalaufgabe der quasikonformen Abbildung'' (Complete solution of an extremal problem of the quasiconformal mapping)

We comment on Teichm{"u}ller 's paper ‘‘Vollst{"a}ndige L{"o}sung einer Extremalaufgabe der quasikonformen Abbildung’’ (Complete solution of an ex-tremal problem of the quasiconformal mapping), published in 1941. In this paper, Teichm{"u}ller gives a proof of the existence of extremal quasiconformal mappings in the case of the pentagon (disc with five distinguished points on the boundary). The final version of this paper will appear as a chapter in Volume VI of the Handbook of Teichm{"u}ller theory.