the Riemann mapping theorem
/ REE-mahn /
The Riemann mapping theorem is the crown jewel of conformal geometry. It says that an enormous variety of differently shaped regions are, from the angle-preserving point of view, all the SAME region. Take any simply connected domain — a region with no holes — that is not the entire plane: a square, the inside of a heart-shaped curve, an infinite strip, a slit plane, anything topologically like a disk. The theorem promises a single conformal (angle-preserving, holomorphic, one-to-one) map carrying it perfectly onto the standard unit disk. One shape, one model, no matter how jagged or strange the original boundary.
Precisely: if D is a simply connected open subset of the complex plane and D is not all of C, then there is a biholomorphic map f : D -> the open unit disk (holomorphic, bijective, with holomorphic inverse). To pin it down uniquely you add a normalization: pick a basepoint z_0 in D and require f(z_0) = 0 and f'(z_0) > 0 (a positive real number); these three real conditions determine f exactly. The standard proof is a triumph of the compactness machinery of this chapter. You consider the family of all injective holomorphic maps from D into the disk sending z_0 to 0; this family is nonempty and locally bounded, hence normal by Montel. You then solve the extremal problem of maximizing |f'(z_0)| over the family; a maximizing sequence has a locally uniformly convergent subsequence (Montel), its limit is holomorphic (Weierstrass) and still injective (Hurwitz), and a Schwarz-lemma argument shows the maximizer must actually be ONTO the whole disk — that surjectivity is the heart of the proof.
The consequences are sweeping: every simply connected domain (other than C) inherits the disk's rich structure — its automorphisms, its hyperbolic metric, its boundary theory — and any analysis problem posed on a complicated region can be transplanted to the disk and solved there. Honest caveats that matter. First, the theorem is an EXISTENCE statement; the proof is non-constructive and gives no formula for f (explicit maps, when available, come from special tools like Schwarz-Christoffel). Second, the exclusion of C is unavoidable: by Liouville's theorem there is no bounded nonconstant entire function, so the whole plane is NOT conformally equivalent to the disk. Third, 'simply connected' is essential — an annulus has a hole and is not conformally a disk, and indeed two annuli are equivalent only when their radius ratios match.
The upper half-plane H = { z : Im z > 0 } is simply connected and not all of C, so it is conformally a disk. Here the map is even explicit: the Cayley transform f(z) = (z - i)/(z + i) sends H biholomorphically onto the unit disk, with the basepoint i going to 0. Most domains have no such tidy formula, but the theorem guarantees the map exists all the same.
The half-plane is conformally a disk via the Cayley transform — a rare case where the guaranteed map is explicit.
Two non-negotiable exclusions: the domain must be simply connected (no holes), and it must not be all of C (ruled out by Liouville). The theorem only asserts existence; it provides no construction or formula for the map.