Ricci Flow and the Geometrization of $3$-Manifolds

Hamilton introduced the Ricci flow $\partial_t g_{ij} = -2R_{ij}$ as a tool for deforming Riemannian metrics toward canonical geometric structures. We review the basic theory: short-time existence, maximum principles, and Li-Yau gradient estimates. The key difficulty is the formation of singularities \in finite time. Perelman's breakthrough was the introduction of the $\mathcal{W}$-entropy $\mathcal{W}(g, f, \tau) = \int_M left[\tau(|\nabla f|^2 + R) + f - nright] \frac{e^{-f}}{(4pitau)^{n/2}} dV$ and the non-collapsing theorem, which together enable a controlled surgery procedure. We outline how this yields the geometrization of closed $3$-manifolds.