|
Hi, I・m here in Beijing, visiting Professor Cao Huai-dong at Tsinghua university and Morningside institute.
Professor Yau ask me to say a few words about Poincare conjecture, stating that any compact simply connected manifold is the same as the three sphere.
The idea of proving the Poincare conjecture with analysis have a long history starting with Yamabe, to introduce the idea of trying to put a good metric on the manifold.
The next developing of the program was to use the Ricci flow. The Ricci flow is the heat equation for a Riemannian metric, which is a very good way to take any metric on a manifold and improve it, so the curvature spreads out indefinitely over the manifold.
Prof. Yau was the first to suggest to me that this would produce the phenomena of nechpinches on the three manifold and break up the three manifold into it・s connected-sum pieces, and hence could be used to prove the Poincare conjecture.
Many people have worked on this problem for the last twenty years, cultimating in the recent breakthrough by Grisha Perelman. Previous work at completed analysis of the possible singularities in the equation showing neckpinches and degenerate neckpinches all of which could be removed by surgery. But there was one remaining possibility, which was something that when the possible removed by surgery, but would be collapse in parts compare to the curvature. The achievements of Perelman was to prove an uncollapasing result for the Ricci flow, thus rolls out the singularity and makes it possible to complete the program. Chinese mathematicians have played a very important part in this development. First Prof. Chern and then Prof. Yau build up a terrifically strong school of Chinese mathematics in differential geometry. Starting in the seventies, Yau proved a number of spectacular results, including the Calabi conjecture in the existence of Calabi-Yau manifolds in string theory, the positive mass conjecture with Rick Schoen in relativity, the Frankel conjecture with Siu in Kahler geometry, and Severi conjecture in algebraic geometry. For these results he won a number of prizes and distinctions, including the Fields medal, the Crafoord prize, the national medal of science and McCarty prize. In the 90s, Yau trained a number of brilliant Chinese young mathematicians who have done major work in Ricci flow. Cao Huai-dong proved long time existence for the Ricci flow in Kahler manifolds and convergence in the case of zero and negative Chern class. He also proved Harnack estimates for positive bisectional holomorphic curvature and is the major worker in the Ricci-Kahler flow today. Shi Wanxiong proved the local derivative estimates for the Ricci flow, which are basic to many arguments in the Ricci flow, including all of the blowup arguments in Perelman・s paper. And Ben Chow completed the proof of the Ricci flow on surfaces.
A major influence on the whole theory of geometric flows was the proof by Yau and Peter Li in 1982 the Harnack estimate for heat equation. This led to the development of whole theory of Harnack estimate for geometric flows, including the Harnack estimate for the Ricci flow, which is absolutely essential in the classification of ancient solutions for the Ricci flow.
It also led to Perelman・s Harnack inequality for solutions of the Ricci flow. It also shows up in Perelman・s Harnack estimate for adjoint solutions of the heat equation on a Ricci flow manifold, which leads directly by integration to the entropy formula. And the Li-Yau method leads to the Riesz function of the reduce volume. These are the two methods that Perelman uses to prove his brilliant and very important result on non-collapsing of Ricci flow.
The work of Yau and others on minimal surface theory also plays an important role in the Ricci flow in proving the extinction of Ricci flow in finite time on manifolds of finite fundamental group.
Cao huai-dong and Zhu Xiping have recently given a complete and detailed account of the proof of Poincare conjecture based on the work of Perelman and earlier work of others. It・s very nice to have such an account written by two outstanding people in the field of Ricci flow. They also introduced ideas of their own which makes the proof easier to understand. This includes a new proof for the uniqueness of solutions on complete manifolds, and different idea for doing the backwards blowup in time and proof of the canonical neiborhood theorem based on results of Zhu and Chen on expanding solitons.
They fully acknowledge Perelman・s role in the completion of the proof of Poincare conjecture and likewise Perelman has acknowledged the work of previous researchers on which it・s based. All Chinese can be proud of the achievements of their mathematicians in differential geometry and their contributions to the completion of the proof of Poincare conjecture.
I・m here in Beijing discussing the details of the proof with Huai-dong and I・ll talk about that work with Huisken and Ilmanen when I got in Zurich next week. We want to be complete certain that everything in the proof is beyond question before making a formal announcement, because many researchers will base their work on it.
|