Hamilton’s Talk about Poincare conjecture in Beijing

June 8, 2005

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.

国际著名数学家，Ricci流理论之父Richard Hamilton教授近日在清华大学与晨兴数学中心访问曹怀东教授。以下是他的谈话记录。

著名的庞加莱猜想是说，每个单连通紧致三维流形都同胚于球面。

用分析方法研究庞加莱猜想有着很长的历史，起源于Yamabe，他试图在流形上赋加一些好的度量。接下来的发展则是Ricci流理论的出现。“Ricci流”是指黎曼流形上的一类热方程。在流形上给定一个度量，用Ricci流发展方程加以改进，流形的曲率也随之伸展。

丘成桐教授最早提示我，三维流形上的Ricci流将会产生瓶颈（neckpinch）现象，并把流形分解为一些连通的片，所以可以用来证明庞加莱猜想。

过去二十年中许多学者都在研究这个问题，特别最近Perelman的重大突破。

对Ricci流奇点的详细分析告诉我们，瓶颈和退化瓶颈都可以用拓扑手术的方法消除。但是还有一种可能性无法排除，当某种瓶颈用拓扑手术消除以后，会产生曲率塌陷。而Perelman的关键工作就是证明了非塌陷（non-collapsing）定理，从而排除了这种奇点的可能性，这就得到Ricci流证明庞加莱猜想的整个纲领的可行性。

中国数学家在这一发展中作出了非常重要的贡献。陈省身，丘成桐建立了非常了不起的微分几何中国学派。从1970年开始，丘证明了几个重大的猜想。包括卡拉比猜想，弦论中卡拉比－丘度量的存在性，广义相对论中的正质量猜想（与Schoen合作），凯勒几何中的Frankel猜想（与萧荫堂合作）以及代数几何中的Severi猜想。这为他赢得了众多国际数学界的大奖和崇高的学术声望，包括菲尔兹奖，Crafoord奖，美国国家科学奖和McCarty奖等。

在80-90年代，丘培养了好几位出色的学生，在Ricci流理论中作出了重要的贡献。曹怀东证明了凯勒流形上长时间解的存在性，并证明了第一陈类为零或负时解的收敛性。他还证明了正双截曲率流形的Harnack估计，他是当今Ricci－凯勒流理论的国际权威。施皖雄证明了Ricci流的局部导数估计，在Ricci流理论中具有基本的重要性，包括Perelman工作中所有与爆破（blowup）有关的论证。周培能证明了曲面上Ricci流的收敛性。对几何流理论产生重大的影响的一项工作，当数丘成桐与李伟光在1982年证明了热方程的Harnack估计，这导致了几何流Harnack估计的发展，包括Ricci流的Harnack估计，这对Ricci流古典解的分类是绝对重要的。

这也导致Perelman的Harnack不等式，同样也出现在Perelman，积分直接得到Entropy公式。Perelman用丘成桐－李伟光的方法得到约化体积的Riesz函数。这些都对Perelman证明非塌陷定理起了关键的作用。

丘成桐等许多学者在极小曲面理论上的工作也在Ricci流理论中扮演了重要的角色，比如证明具有有限基本群的流形上Ricci流的有限消亡时间。

曹怀东与朱熹平最近在Perelman与前人的工作基础上，给出了关于庞加莱猜想证明的一个完整与详细的描述。我很高兴这两位Ricci流领域里的杰出学者所写的这篇文章。他们引入了自己的新思想，使得证明变得更容易理解，包括完备流形上解的唯一性，用新的方法研究典则邻域定理证明中的反向爆破，这是基于朱熹平与陈兵龙关于孤立子扩张的工作。

曹怀东与朱熹平的文章中充分肯定了Perelman的工作对于证明庞加莱猜测所起的重要作用，同样，Perelman在文章中也明确指出他的工作是建构于前人的众多贡献基础上的。所有中国人都应该为中国数学家在微分几何领域所取得的成就，和对庞加莱猜测的贡献而感到骄傲。