the uniformization theorem
/ Koebe: KUR-buh /
The Riemann mapping theorem classifies simply connected pieces of the plane: every one (except the whole plane) is conformally the disk. The uniformization theorem is its breathtaking generalization, lifting the story off the plane and onto curved surfaces. It says that the entire universe of simply connected Riemann surfaces — abstract surfaces on which complex analysis makes sense — comes in exactly THREE conformal types. No matter how exotic the surface, after a conformal change of coordinates it must be one of just three model spaces.
Precisely: every simply connected Riemann surface is conformally equivalent (biholomorphic) to exactly one of three models — the Riemann sphere (the extended plane, the unique compact one, with positive curvature), the complex plane C (flat, zero curvature), or the open unit disk, equivalently the upper half-plane (negative, hyperbolic curvature). These three are mutually inequivalent: the sphere is compact while the others are not, and the plane and disk are distinguished by Liouville's theorem (there is no nonconstant bounded entire function, so the plane is not conformally the disk). The general (not simply connected) case follows by passing to the universal cover: every Riemann surface is a quotient of one of these three models by a group of automorphisms acting freely. The vast majority of surfaces — anything of genus 2 or more, and most of genus 1 and 0 with punctures — are covered by the disk, which is why hyperbolic geometry is the 'generic' geometry of Riemann surfaces.
The reach of this single theorem is enormous: it puts a canonical geometry (spherical, Euclidean, or hyperbolic) on every Riemann surface, underlies the theory of automorphic and modular forms, and connects complex analysis to topology (the trichotomy mirrors the curvature trichotomy of surfaces). It contains the Riemann mapping theorem as the special case of plane domains. Honest caveats. The proof is far deeper than the Riemann mapping theorem — it needs either potential theory (the Perron method for harmonic functions and Green's functions on surfaces) or the modern theory of PDE/curvature, and it is genuinely a 20th-century result (Koebe and Poincare). And it is a conformal classification, not a metric one: two surfaces can be the same conformal type yet carry wildly different shapes; uniformization picks out the one canonical constant-curvature metric in each conformal class.
Three concrete simply connected surfaces, one of each type: the Riemann sphere itself (compact, spherical); the complex plane C (the entire plane, flat); and the unit disk (hyperbolic). A more surprising example of the third type: the universal cover of a torus-with-one-point-removed is the disk, so a punctured torus is intrinsically hyperbolic, not flat.
Sphere, plane, disk — the only three conformal types, carrying spherical, flat, and hyperbolic geometry.
The classification is conformal, not metric, and is sharply three-fold: sphere, plane, disk, distinguished by compactness and (for plane vs disk) Liouville. The proof is far deeper than the Riemann mapping theorem, requiring potential theory or curvature/PDE methods.