Introductory Remarks

Refer to Section 0.3 for the statement and sketch of the proof of the Riemann mapping theorem. The Riemann mapping function is the solution to a certain extremal problem: to find a map of the given domain U into the disc D which is one-to-one, maps a given point P to 0, and has derivative of greatest possible modulus λP at P. The existence of the extremal function, which also turns out to be one-to-one, is established by normal families arguments; the fact that the extremal function is onto is established by an extra argument which is in fact the only step of the proof where the topological hypotheses about U are used.

The point of the present discussion is to observe that the scheme we just described can be applied even if U is not topologically equivalent to the disc. Constantin Carathéodory's brilliant insight was that the quantity λP can be used to construct a metric, now called the Carathéodory metric. We maximize the derivative at P of maps π of U into D such that π(P) = 0 but we no longer require the maps to be injective. Of course the proof of the Riemann mapping theorem will break down at the stage where we attempt to show that the limit map is surjective; we will also be unable to prove that it is injective. All other steps, including the existence of the extremal function, are correct and give rise to a metric (as in Section 3.1 following) on the domain.