Can Ricci flow be used to prove Poincaré’s conjecture for $n=2$?

Question

This question is concerning the conjecture described in this Wikipedia article. The conjecture has been proved for dimension two ($n = 2$). For $n=3$, conjecture is proved using Ricci flow. My question is, can Ricci flow be used to solve the conjecture for any $n$ or for $n=2$? Can Ricci flow defined for any $n$?

Answer 1

The fact that simply-connected closed surfaces $S$ is diffeomorphic to $\mathbb S^2$ is well-known, indeed the uniformization theorem states that any metric on $S$ is conformal to the standard metric on $\mathbb S^2$.

There is, indeed, a proof of this fact using the Ricci flow by Hamilton and Chow. You may read this survey, in particular section 3 for more information.

Let me point out that Ricci flow in dimension $n\ge 4$ is much more difficult, in dimension 2, the only curvature is the Gauss curvature (a scalar), while in dimension 3, the Ricci curvature determine the whole Riemann curvature tensor $R$ (see here). In dimension $n\ge 4$, it is much harder to control the whole $R$ along the Ricci flow (which is defined using only the Ricci curvature).