Hamilton's theorem on Ricci flows

Statement

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

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