for . Since must contain a closed disk with centre and radius , no points of can lie in . Let be the unique Möbius transformation taking onto with the normalization and . By construction is in , so that is non-empty. The method of Koebe is to use an extremal function to produce a conformal mapping solving the problem: in this situation it is often called the Ahlfors function of , after Ahlfors. Let be the supremum of for . Pick with tending to . By Montel's theorem, passing to a subsequence if necessary, tends to a holomorphic function uniformly on compacta. By Hurwitz's theorem, is either univalent or constant. But has and . So is finite, equal to and . It remains to check that the conformal mapping takes onto . If not, take in and let be a holomorphic square root of on . The function is univalent and maps into . Let where . Then and a routine computation shows that This contradicts the maximality of , so that must take all values in . Remark. As a consequence of the Riemann mapping theorem, every simply connected domain in the plane is homeomorphic to the unit disk. If points are omitted, this follows from the theorem. For the whole plane, the homeomorphism gives a homeomorphism of onto . Parallel slit mappings Koebe's uniformization theorem for normal families also generalizes to yield uniformizers for multiply-connected domains to finite parallel slit domains, where the slits have angle to the -axis. Thus if is a domain in ℂ ∪ {∞} containing and bounded by finitely many Jordan contours, there is a unique univalent function on with = near , maximizing and having image a parallel slit domain with angle to the -axis. The first proof that parallel slit domains were canonical domains for in the multiply connected case was given by David Hilbert in 1909. , on his book on univalent functions and conformal mappings, gave a treatment based on the work of Herbert Grötzsch and René de Possel from the early 1930s; it was the precursor of quasiconformal mappings and quadratic differentials, later developed as the technique of extremal metric due to Oswald Teichmüller. Menahem Schiffer gave a treatment based on very general variational principles, summarised in addresses he gave to the International Congress of Mathematicians in 1950 and 1958. In a theorem on "boundary variation" (to distinguish