The Ricci flow (RF) introduced by Hamilton in $1982$ is a $1$-parameter family of Riemannian metrics $g(t)$ on a manifold $M$ that satisfies the equation: $$\frac{\partial}{\partial t}g(t)=-2\ \text{Ric}_{g(t)}.$$ It is the geometric evolution equation on the space of Riemannian metrics on $M$. Starting with a given initial metric $g_{0}$ on $M$, the goal of the Ricci flow evolution is to possibly flow towards a metric on $M$ that is more natural to its underlying topology. For example, the most natural metric on $\Bbb H^2$ $($resp. $\Bbb S^2)$ is the one that has constant curvature $-1$ $($resp. $1)$. In this lecture, we are only going to talk about Ricci flow on surfaces, and we will give a sketch of the proof of uniformization theorem using some PDE theory.

Riemannian metrics, Basics of PDE theory