Uniformization theorem

Statement

Any surface (viz, two-dimensional differential manifold) admits a Riemannian metric of constant Gaussian curvature (which, for a surface, is the same as a constant-curvature metric). More strongly, given a surface, and a conformal class of Riemannian metrics on that surface, there exists a constant-curvature metric in that conformal class.

Relation with other results

Geometrization conjecture

Further information: Geometrization conjecture