Carathéodory's extension theorem
/ ka-ra-the-o-DOH-ree /
Carathéodory's theorem is the precise answer to a natural worry about the Riemann mapping theorem: yes, the conformal map exists between interiors, but does it behave at the boundary? For nicely-shaped regions the answer is a satisfying yes — the map reaches the boundary, stays continuous and one-to-one there, and matches the edge of your region cleanly onto the unit circle. This theorem says exactly which regions are 'nicely shaped': those bounded by a Jordan curve.
Precisely: let D be a Jordan domain — the interior of a simple (non-self-intersecting) closed curve in the plane — and let f : D -> unit disk be a Riemann map. Then f extends to a homeomorphism from the closure of D (region plus boundary curve) onto the closed unit disk. In particular the boundary Jordan curve is carried continuously, bijectively, and order-preservingly onto the unit circle, and the inverse map is continuous too. The proof revolves around showing that as an interior point approaches the boundary, its image approaches the circle without 'oscillating' between different circle points — the key technical tools are an estimate on how the image of a small circular crosscut shrinks (a length-area argument) and the fact that a Jordan curve is locally connected, which prevents the boundary from being approached in genuinely different ways.
Its importance is entirely practical: it is the license to transport boundary data through a conformal map, which is the whole point of using conformal mapping to solve boundary-value problems in physics. Without a boundary correspondence, a map between interiors could not match boundary conditions. Honest caveats. The hypothesis is exactly 'Jordan curve' — a simple CLOSED curve; remove the simplicity (let the boundary cross itself or have slits) and the homeomorphic extension can fail. The conclusion is a homeomorphism, i.e. a continuous bijection with continuous inverse; it does NOT claim differentiability at the boundary, which requires extra smoothness of the curve. For domains whose boundary is not a Jordan curve, the correct generalization replaces boundary points with Carathéodory's notion of prime ends.
A square is a Jordan domain. The Riemann map from the square onto the unit disk extends, by Carathéodory's theorem, to a homeomorphism of the closed square onto the closed disk; the four edges and four corners of the square correspond continuously to four arcs and four marked points on the unit circle. (Schwarz-Christoffel gives the map explicitly.)
A square's edges and corners map homeomorphically onto arcs and points of the circle.
The exact hypothesis is a Jordan (simple closed) boundary curve, and the exact conclusion is a homeomorphism of closures — continuity, not differentiability. For non-Jordan boundaries one must use prime ends instead of ordinary boundary points.