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

Step 1 of 9: Hamilton's Ricci flow programme

In plain words

Heating an oddly shaped metal plate makes its temperature smooth out over time. In 1982 Richard Hamilton proposed doing something similar to the shape of space itself: let a metric $g_{ij}$ evolve by the heat-like equation $\partial_t g_{ij} = -2R_{ij}$ so that bumps of positive curvature shrink.

Hamilton proved that if a closed 3-dimensional shape starts out already curved positively everywhere, this flow smooths it all the way down to a perfectly round sphere. His hope was that running the same equation on any starting shape would eventually reveal the geometric building blocks that Thurston had conjectured every 3-manifold decomposes into.

\partial_t g_{ij} = -2R_{ij}, \qquad \partial_t R = \Delta R + 2|\operatorname{Ric}|^2

Detailed analysis

Hamilton (1982) proposed evolving an arbitrary Riemannian metric $g_{ij}$ on a closed 3-manifold $M^3$ by the equation $\partial_t g_{ij} = -2R_{ij}$ , where $R_{ij}$ is the Ricci curvature tensor. Under this flow the scalar curvature $R$ satisfies the reaction-diffusion equation $\partial_t R = \Delta R + 2|\operatorname{Ric}|^2$ , so by the maximum principle its minimum is nondecreasing in time.

Using a maximum principle for tensors, Hamilton showed the flow preserves positivity of the Ricci tensor in dimension 3, and that when $\operatorname{Ric} > 0$ holds initially, the (volume-normalised) flow exists for all time and converges smoothly to a metric of constant positive curvature, so $M^3$ is a quotient of the round sphere $S^3$ .

Hamilton's programme was to extend this convergence result to an arbitrary starting metric on any closed 3-manifold, which — combined with a topological classification of the possible outcomes — would prove Thurston's geometrization conjecture, and the Poincaré conjecture as the special simply connected case. The obstacle, addressed starting in the next step, is that for a general metric the flow can develop singularities in finite time before it reaches any such nice limit.

Terms in this step

Ricci flow: An evolution equation $\partial_t g_{ij} = -2R_{ij}$ that deforms a Riemannian metric in the direction of its own Ricci curvature, shrinking regions of high positive curvature the way heat flow smooths out temperature.
Ricci curvature ( $\operatorname{Ric}$ ): A symmetric tensor built from the curvature of a Riemannian manifold that measures, roughly, how the volume of a small ball deviates from its Euclidean value; its trace is the scalar curvature $R$ .

Knowledge used in this step

Differential geometry Riemannian geometry Smooth manifolds