# The Uniformization theorem The Uniformization Theorem: any [[Simply-connected topological space|simply connected]] [[Riemann surface]] there is a holomorphic bijection map to one and only one of the spaces $\widehat{\mathbb C}, \mathbb C, \mathbb D$ . This means that there are only 3 simply-connected Riemann surfaces (up to biholomorphism). $\widehat{\mathbb C}, \mathbb C, \mathbb D$ can be given unique canonical metrics with constant curvature: +1, 0, -1 respectively. These metrics are called spherical, Euclidean and hyperbolic. ## Created 2026-05-29 13:40