the extremal problem for conformal maps

How do you conjure a conformal map onto the disk out of thin air, when there is no formula? The standard proof of the Riemann mapping theorem does it by an optimization trick: among all the 'allowed' maps, single out the one that spreads the domain as widely as possible at the basepoint, and show that the greediest such map is exactly the conformal map you wanted. The greed is measured by the size of the derivative at the basepoint, |f'(z_0)| — large derivative means the map magnifies aggressively right there, and intuitively the map that magnifies the most must fill the whole disk.

Here is the construction in plain steps. Fix a basepoint z_0 in the simply connected domain D (not the whole plane). Let F be the family of all injective holomorphic maps f : D -> unit disk with f(z_0) = 0. First one checks F is nonempty (a logarithm-and-Mobius construction builds one such map, using that D misses some value). The family is bounded by 1, so by Montel it is normal. Now consider the number sup over F of |f'(z_0)|; this supremum is finite (Cauchy estimate) and positive. Take a sequence f_n in F with |f_n'(z_0)| approaching the sup. By Montel, extract a locally uniformly convergent subsequence; its limit f is holomorphic (Weierstrass) with |f'(z_0)| equal to the sup, so f attains the maximum. Hurwitz's theorem guarantees this limit f is still injective (a locally uniform limit of injective holomorphic functions is injective or constant, and f' (z_0) > 0 rules out constant). The final, decisive step uses the Schwarz lemma: if the extremal f did NOT cover the whole disk, you could compose with a clever square-root-and-automorphism map to manufacture a competitor with a STRICTLY larger derivative at z_0, contradicting maximality. Hence f must be onto — and an onto, injective, holomorphic map to the disk is the Riemann map.

The beauty of this argument is that it converts an existence question into an optimization that compactness can solve: you never write the map down, you prove the maximizer exists and then show maximality forces every property you need. The same extremal philosophy — find the function that optimizes some functional and read off its properties from the optimality — recurs throughout geometric function theory (extremal length, the area theorem, Loewner theory). Honest caveat: the proof is genuinely non-constructive. It guarantees a maximizer exists and identifies it as the Riemann map, but it offers no recipe for computing it, and the clever square-root step requires the domain to be simply connected so that a single-valued square root of a non-vanishing function can be chosen.