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.
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
Further information: Yamabe conjecture
Further information: Geometrization conjecture