structural:expansion, information:extraction, stylistic:style_transfer
Ricci flow with surgery

Hamilton's program for proving the Poincaré conjecture involves first putting a Riemannian metric on the unknown simply connected closed 3-manifold. The basic idea is to try to "improve" this metric; for example, if the metric can be improved enough so that it has constant positive curvature, then according to classical results in Riemannian geometry, it must be the 3-sphere. Hamilton prescribed the " Ricci flow equations" for improving the metric;

where g is the metric and R its Ricci curvature, and one hopes that, as the time t increases, the manifold becomes easier to understand. Ricci flow expands the negative curvature part of the manifold and contracts the positive curvature part.

In some cases, Hamilton was able to show that this works; for example, his original breakthrough was to show that if the Riemannian manifold has positive Ricci curvature everywhere, then the above procedure can only be followed for a bounded interval of parameter values, t ∈ [ 0, T) {\displaystyle t\in [0,T)} with T < ∞ {\displaystyle T<\infty }, and more significantly, that there are numbers c t {\displaystyle c_{t}} such that as t ↗ T {\displaystyle t\nearrow T}, the Riemannian metrics c t g (t) {\displaystyle c_{t}g(t)} smoothly converge to one of constant positive curvature. According to classical Riemannian geometry, the only simply-connected compact manifold which can support a Riemannian metric of constant positive curvature is the sphere. So, in effect, Hamilton showed a special case of the Poincaré conjecture: if a compact simply-connected 3-manifold supports a Riemannian metric of positive Ricci curvature, then it must be diffeomorphic to the 3-sphere.

If, instead, one only has an arbitrary Riemannian metric, the Ricci flow equations must lead to more complicated singularities. Perelman's major achievement was to show that, if one takes a certain perspective, if they appear in finite time, these singularities can only look like shrinking spheres or cylinders. With a quantitative understanding of this phenomenon, he cuts the manifold along the singularities, splitting the manifold into several pieces and then continues with the Ricci flow on each of these pieces. This procedure is known as Ricci flow with surgery.

Perelman provided a separate argument based on curve shortening flow to show that, on a simply-connected compact 3-manifold, any solution of the Ricci flow with surgery becomes extinct in finite time. An alternative argument, based on the min-max theory of minimal surfaces and geometric measure theory, was provided by Tobias Colding and William Minicozzi. Hence, in the simply-connected context, the above finite-time phenomena of Ricci flow with surgery is all that is relevant. In fact, this is even true if the fundamental group is a free product of finite groups and cyclic groups.

This condition on the fundamental group turns out to be necessary and sufficient for finite time extinction. It is equivalent to saying that the prime decomposition of the manifold has no acyclic components and turns out to be equivalent to the condition that all geometric pieces of the manifold have geometries based on the two Thurston geometries S × R and S. In the context that one makes no assumption about the fundamental group whatsoever, Perelman made a further technical study of the limit of the manifold for infinitely large times, and in so doing, proved Thurston's geometrization conjecture: at large times, the manifold has a thick-thin decomposition, whose thick piece has a hyperbolic structure, and whose thin piece is a graph manifold. Due to Perelman's and Colding and Minicozzi's results, however, these further results are unnecessary in order to prove the Poincaré conjecture.