We also verify several assertions related to Richard Hamilton's program for the proof of Thurston geometrization conjecture for closed three-manifolds, and give a sketch of an eclectic proof of this conjecture, making use of earlier results on collapsing with local lower curvature bound.
Ken also mentioned to me that Perelman is a well-respected mathematician, who was a post-doc at UC Berkeley when he was a graduate student. As Perelman says in the acknowledgements in his paper, he has supported himself for five years on earnings from post-docs at Courant, Stoneybrook, and Berkeley. This seemed exciting, for several reasons. There have been many claims to solving the Poincare conjecture, notably Dunwoody's recent claim and retraction which were posted on the web as a five page preprint (which seems to no longer be available). In the early '90's, Anderson had announced at conferences a proof of the hyperbolization conjecture, that irreducible compact 3-manifolds with infinite fundamental group and no Z+Z subgroup are hyperbolic, but subsequently retracted his claim (although he has papers establishing a program to prove this).
But Hamilton has been working on the Ricci flow approach to geometrization for many years now. He initially wanted to solve the Poincare conjecture, and succeeded in the case that the manifold has a metric of positive Ricci curvature, in 1982. Since then, he has made several contributions to understanding the Ricci flow, and in the process has established a program for trying to prove the geometrization conjecture for 3-manifolds using this approach. The idea is based on a technique from elliptic equations. The geometrization conjecture would imply that "most" 3-manifolds admit a hyperbolic metric, which can be expressed in three dimensions as being an Einstein metric with negative curvature, i.e. g=cRic(g), where g is a Riemannian metric, Ric(g) is the ricci curvature, and c is a negative constant. Thus, to try to prove geometrization, one could flow the metric on the manifold proportional to the Ricci curvature, and hope that the flow converges to a fixed point (up to scaling), which would be an Einstein metric. Of course, things are not this simple, and Hamilton first considered an equation g_t=2/3 R g-2 Ric(g), the traceless Ricci curvature, which was suggested by Eels and Sampson as the gradient flow of the total scalar curvature, but for which he couldn't even prove short-time existence. But this technique of solving equations by evolving an arbitrary solution to find a fixed point is a common technique in mathematics (indeed, Thurston's proof of geometrization for Haken 3-manifolds came down to solving a fixed point problem by iteration), and this technique works very well in elliptic PDE theory. Hamilton noticed that Ric(g) was similar to an elliptic operator, such as the Laplacian, and that the flow g_t=-2Ric(g) behaves like a parabolic equation. His program is then to analyze this equation. He showed short-time existence for arbitrary initial metrics, but noticed that the metric could blow up in finite time, and the solution would only exist for finite time until the curvature blew up. He proved eventually that if the solution exists for all time with certain bounds on the curvature, that the manifold satisfied the geometrization conjecture. The solution does not actually converge to a fixed point of the Ricci flow, i.e. an Einstein manifold, but it collapses along a characteristic submanifold, and the uncollapsed part is hyperbolic, while the collapsed part has one of Thurston's other geometries. I took a class from Hamilton on the Ricci flow when I was at UCSD, and he explained this program. His lectures were excellent, but I was intimidated by the analyis involved. He explained his program, and the difficulties left over. When singularities occured, he wants to show that the manifold pinches along a 2-sphere, that is the metric develops a neck. This sort of thing occured in a paper by Hamilton on 4-manifolds with positive isotropic curvature. The difficulties he had were to show that one could analyze the singularity by scaling the metric near the singularity to get a new solution to the Ricci flow, and showing that the scaled solution converges to a Ricci soliton, which is a solution where the isometry type of the manifold is fixed by the Ricci flow, but the metric is changed by a 1-parameter family of diffeomorphisms. To do this, he needed bounds on the injectivity radius, to prove a "little loop lemma", which he could do in the positively curved case. For the little loop lemma, he needed a "Harnack inequality", which he and Yau were working on but hadn't achieved. He also needed to classify Ricci solitons. Then he could analyze the singularities, and when a neck pinch occured, he could cut it off and cap off the spheres by balls, continuing the flow on the new manifold, which was similar again to what he did in the case of 4-manifolds with positive isotropic curvature. Then he would need to show that one need perform surgery only finitely many times, after which the flow would exist for all time. This sort of operation is necessary if one expects to prove geometrization using Ricci flow, since a 3-manifold may be a non-trivial connect sum of 3-manifolds which are not 3-spheres. Perelman's paper solves Hamilton's Harnack estimate conjecture, and therefore he is able to classify the limit solitons of singularities. He also extends Hamilton's result by showing that if the Ricci flow exists for all time, then the manifold satisfies geometrization. In the last section of Perelman's paper, he claims to have solved the finiteness of surgering necks as well. So this seems very promising, since it completes a well-established program of Hamilton, and has a nice conceptual outline, even if the details are extraordinarily difficult. But it remains to be seen whether Perelman has a complete proof.
Supposedly, many people are looking at Perelman's paper and Ricci flow - there are seminars being run by Kleiner and Lott at U. Michigan, Anderson and Sullivan at Stoneybrook, and here at UIC, and experts such as Chow, Hamilton and Yau are looking at the paper.
Because of its promise, I'm trying to understand Hamilton's program for geometrization by Ricci flow.