Worked solution: Perelman's Ricci flow with surgery (2002–2003), condensed

Hamilton (1982) proposed evolving a Riemannian metric $g_{ij}$ on a closed manifold by $\partial_t g_{ij} = -2R_{ij}$. After pulling back by the time-dependent diffeomorphisms generated by a solution of the harmonic-map heat equation (de Turck's trick), the system becomes strictly parabolic — its leading term is the rough Laplacian $\Delta g_{ij}$ — which yields short-time existence and uniqueness on closed manifolds. Curvature quantities evolve by reaction–diffusion equations; for the scalar curvature, $\partial_t R = \Delta R + 2|\operatorname{Ric}|^2$, so the maximum principle makes $\min_M R$ nondecreasing along the flow.

In dimension 3 Hamilton applied the tensor maximum principle to the evolution of $\mathrm{Rm}$ and proved that positivity of the Ricci tensor is preserved, and moreover that the trace-free part is pinched relative to $R$: $|\mathring{\operatorname{Ric}}| \le C R^{1-\delta}$ for some $\delta>0$. Regions of large scalar curvature therefore become asymptotically round, and combined with Myers's diameter bound the volume-normalised flow on a $\operatorname{Ric}>0$ 3-manifold exists for all time and converges smoothly to a metric of constant positive sectional curvature — so $M^3$ is a quotient of the round $S^3$.

Hamilton's programme was to apply the same flow to an arbitrary metric on any closed 3-manifold and read off Thurston's geometric decomposition from its long-time behaviour, with the Poincaré conjecture as the simply connected special case. The obstruction is that for a general initial metric $|\mathrm{Rm}|$ can blow up before any nice limit is reached; understanding what the flow converges to therefore requires, in order, blow-up analysis (next step), a mechanism ruling out degenerate limits, and a way to continue the flow past singular times.