# Hamilton's theorem on Ricci flows

From Diffgeom

## Statement

Let be a compact oriented 2-dimensional differential manifold and be a Riemannian metric on . Then the following are true:

- If is not diffeomorphic to the 2-sphere, then converges to a constant-curvature metric under the Ricci flow
- If is diffeomorphic to the 2-sphere and has positive Gaussian curvature everywhere, then converges to a constant-curvature metric under the Ricci flow