Uniformization theorem

Statement

Given a surface (viz, two-dimensional differential manifold) and a conformal class of Riemannian metrics on that surface, there exists a constant-curvature metric in that conformal class. Note that for surfaces, having constant Gaussian curvature is equivalent to having constant sectional curvature, constant Ricci curvature or constant scalar curvature.

Proof

One proof of the uniformization conjecture relies on the volume-normalized Ricci flow in two dimensions, which is the same as the Yamabe flow. The idea is to show that starting with any Riemannian metric, we can, using the Yamabe flow, evolve it without changing the conformal class, to attain a constant-curvature metric in the limit.

Relation with other results/conjectures

Theorems on Ricci flows

Hamilton's theorem on Ricci flows along with Chow's theorem provide the Ricci flow-route to proving the uniformization theorem.

Yamabe conjecture

Further information: Yamabe conjecture

The Yamabe conjecture is an attempt to generalize the uniformization theorem to higher dimensions. Here, we use scalar curvature to generalize the Gaussian curvature for surfaces.

Geometrization conjecture

Further information: Geometrization conjecture