加载中…汉密尔顿赞中国数学家对证明庞加莱猜想的贡献
(2010-11-05 01:02:12)
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分类： 数学 |
中国数学家朱熹平１１日在香港http://weather.qq.com/images/endnew/weather_icon.gif中文大学就破解世界数学难题－－“庞加莱猜想”与记者见面。当今世界上有七大数学难题﹐其中悬而未决逾百年的就是“庞加莱猜想”。庞加莱猜想是法国数学家亨利﹑庞加莱１９０４年提出的拓扑学难题﹐百余年来吸引世界无数数学家钻研﹐近年成为世界七大数学难题之一。中国数学家朱熹平﹑曹怀东教授运用俄罗斯数学家佩雷曼的理论﹐利用两年时间为庞加莱猜想作出完全的证明﹐结果对物理和工程学都有深远影响。中新社发宋吉河 摄
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中新社香港六月十一日电 (记者邓卓明)着名华裔数学家、香港中文大学数学科学研究所所长丘成桐教授今日下午向记者公布了汉密尔顿教授关于“庞加莱猜想”的最新谈话录像。
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据悉，这应是汉密尔顿自近期中国出现“庞加莱猜想”热之后，首度公开表示自己的看法。这也是自丘成桐宣布中国教授朱熹平与曹怀东破解猜想后，国际数学界首位外方人士对此发表谈话。
国际着名数学家、Ricci流理论之父汉密尔顿教授近日在中国清华大学与中国科学院晨兴数学中心访问曹怀东教授。六月五日，汉密尔顿通过一段长达六分半钟的录像，谈到破解“庞加莱猜想”中的一些重要发展经过。
汉密尔顿表示，着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很长的历史。丘成桐教授最早提示他，三维流形上的Ricci流将会产生瓶颈现象，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，特别是最近佩雷尔曼的重大突破。他使Ricci流证明庞加莱猜想的整个纲领的可行性。
汉密尔顿指出，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。从一九七零年开始，丘成桐证明了几个重大的猜想，包括卡拉比猜想等。这为他赢得了众多国际数学界的大奖和崇高的学术声望，包括菲尔兹奖等。
在九十年代，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。
汉密尔顿说，曹怀东与朱熹平最近在佩雷尔曼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。他很高兴这两位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领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佩雷尔曼的工作对于证明庞加莱猜测所起的重要作用，同样，佩雷尔曼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。
理查德·汉密尔顿回应《纽约客》关于丘成桐的文章
数学家汉密尔顿致信丘成桐的律师Howard M Cooper
亲爱的Cooper先生：
《纽约客》杂志以不公平的方式描写了丘成桐，我对此深感不安。我在此写
出我的想法，以正视听。如果这能对丘有什么帮助的话，我授权你把这封信提供
给《纽约客》杂志和公众。
20 世纪80年代早期，当我的第一篇关于正Ricci曲率的三维流形的Ricci流
的论文完成之后不久，丘立刻认识到了它的重要性。尽管我证明了他那时正在研
究的最小曲面的一个结果，他并没有表现出丝毫的嫉妒，而是成为了我的最坚定
的支持者。当时他向我指出Ricci 流可以形成neck pinch奇点，解开connected
sum decomposition，这可能导致庞加莱猜想的证明。1985年，他把我、Rick
Schoen 和Gerhard Huisken一起带到了加州大学圣地亚哥分校，我们组成了一个
非常令人兴奋而多产的几何分析研究组。Huisken当时正在研究超曲面的平均曲
率流，它非常类似于Ricci流。平均曲率流和Ricci流分别是外曲率和内曲率的最
简单的流。丘不断敦促我们研究这些抛物面方程奇点的blow-up，使用与研究类
椭圆方程的最小曲面方程类似的方法。而丘和Rick是这方面的专家。没有丘在这
个早期阶段的指导和支持，就不可能有供Perelman完成的 Ricci流的研究纲要。
在加州大学圣地亚哥分校，丘还有一些非常优秀的学生，这些学生是和他一
起从普林斯顿来的，特别是曹怀东、周培能和施皖雄。丘鼓励他们研究Ricci流，
他们都对这个领域做出了非常重要的贡献。曹怀东证明了规范Kaehler情况下的
正规化Ricci流总是存在，且收敛于零或负陈示性类。曹的结果成为了Perelman
关于Kaehler Ricci流的令人兴奋的研究的基础，Perelman证明了对于正陈示性
类，直径和标量曲率是有界的。周培能除了在其他的流方面做出了卓越的贡献外，
他还把我的关于二维球面Ricci流的工作扩展到了不同符号的曲率的情况下。施
皖雄开创了完备非紧流形的Ricci流的工作。除了许多漂亮的论证，他还证明了
Ricci流的局部导数估计。奇点的blow-up常常产生非紧的解，而关于收敛到
blow-up极限的证明总是依赖于施的导数估计。因此施的工作对于 Perelman和我
使用的所有极限论证都至关重要。
1982年，丘成桐和李伟光撰写了一篇极其重要的论文，为线性热方程提出了
一个逐点微分不等式，它可以沿曲线积分，给出经典的Harnack不等式。丘不断
敦促我研究这篇论文，根据他们的方法，我证明了Ricci流和平均曲率流的
Harnack不等式。由李-丘的研究一般化而得到Harnack不等式，构成了我所开始
研究的ancient solutions的基础，Perelman完成了它们，并把它们作为他的规
范邻域定理的一个基本工具。曹怀东证明了Kahler情况下Ricci流的 Harnack估
计，而施皖雄对Yamabe流和高斯曲率流做出了同样的证明。
但是这个故事还没有完。Perelman最重要的成果是他的关于Ricci流非塌陷
的结果，这在所有的维度都成立，而不仅仅是在三维情况成立。它对未来的重要
性远远超过了庞加莱猜想。对于庞加莱猜想，它是消除 cigars奇点的工具，而
我无法消除这类奇点。这个结果有两个证明，一个是使用逆向标量热方程的熵，
另外一个方法是使用路径积分。熵估计来自对共轭热方程做李-丘型微分Harnack
不等式的积分，另一个是对同样的Harnack不等式做最优李-丘路径积分。正如
Perelman在他的第一篇论文7.4 中承认的，他写道：“一个更接近的参考是[李-
丘]，他们使用“长度”与线性抛物面方程关联，这与我们的这个问题非常相
同”。
多年来，丘一直支持Ricci流和整个几何流领域的研究，在这个领域还有其
他重要的成果，例如最近Huisken 和Ilmanen证明了彭罗斯猜想，这是广义相对
论领域的一个非常重要的结果。除了丘成桐，我无法想象还有其他任何著名的数
学权威会对我们的研究领域给予密切的支持。
丘成桐建立的是一群天才的群体，而不是一个权力帝国。人们被他的精力、
他的超群思想以及他对一流数学的不懈支持所吸引。丘成桐把他们集合在一起，
共同研究最困难的问题。在过去的许多年中，丘和我花了无数时间一起研究
Ricci流和其他问题，常常工作到深夜。从观察到neck pinch奇点问题开始，他
总是慷慨地与我分享他的建议，但是从未要求分享荣誉。
事实上，当去年冬天我最终努力证明了Ricci流的一个局部型Harnack不等式
的时候——我们一起研究这个问题已经很多年了——我说我应该把他的名字加在
论文上，他谦虚地拒绝了。（《纽约客》）这样严重地歪曲他的人格，这真是不
幸。据我所知，他从未提出关于（解决庞加莱猜想的功劳的）百分比，他也没说
过Perelman应该只与我分享（解决）庞加莱猜想的荣誉。这是合情合理的，因为
事实上除了Perelman本人，没有任何人比他更加慷慨地归功于我的工作。丘成桐
根本没有偷窃Perelman的成果，正相反，他赞扬了Perelman的工作，并与我一起
支持Perelman获得菲尔茨奖。Perelman借助Ricci流的研究纲要获得了菲尔茨奖，
而事实上，丘成桐正是建立这一纲要的人。
谨启
Richard S Hamilton
哥伦比亚大学数学教授
Dear Mr. Cooper
I am very disturbed by the unfair manner in which Yau Shing-Tung has been portrayed in the New Yorker article. I am providing my thoughts below to set the record straight. I authorize you to share this letter with the New Yorker and the public if that will be helpful to Yau.
As soon as my first paper on the Ricci Flow on three dimensional manifolds with positive Ricci curvature was complete in the early '80's, Yau immediately recognized its importance; and although I had proved a result on which he had been working with minimal surfaces, rather than exhibit any jealosy he became my strongest supporter. He pointed out to me way back then that the Ricci Flow would form the neck pinch singularities, undoing the connected sum decomposition, and that this could lead to a proof of the Poincare conjecture. In 1985 he brought me to UC San Diego together with Rick Schoen and Gerhard Huisken, and we had a very exciting and productive group in Geometric Analysis. Huisken was working on the Mean Curvature Flow for hypersurfaces, which closely parallels the Ricci low, being the most natural flows for intrinsic and extrinsic curvature respectively. Yau repeatedly urged us to study the blow-up of singularities in these parabolic equations using techniques parallel to those developed for elliptic equations like the minimal surface equation, on which Yau and Rick are experts. Without Yau's guidance and support at this early stage, there would have been no Ricci Flow program for Perelman to finish.
Yau also had some outstanding students at San Diego who had come with him from Princeton, in particular Cao Huai-Dong, Ben Chow and Shi Wan-Xiong. Yau encouraged them to work on the Ricci Flow, and all made very important contributions to the field. Cao proved existence for all time for the normalized Ricci Flow in the canonical Kaehler case, and convergence for zero or negative Chern class. Cao's results form the basis for Perelman's exciting work on the Kaehler Ricci Flow, where he shows for positive Chern class that the diameter and scalar curvature are bounded. Ben Chow, in addition to excellent work on other flows, extended my work on the Ricci Flow on the two dimensional sphere to the case of curvature of varying sign. Shi Wan-Xiong pioneered the study of the Ricci Flow on complete noncompact manifolds, and in addition to many beautiful arguments he proved the local derivative estimates for the Ricci Flow. The blow-up of singularities usually produces noncompact solutions, and the proof of convergence to the blow-up limit always depends on Shi's derivative estimates; so Shi's work is central to all the limit arguments Perelman and I use.
In '82 Yau and Peter Li wrote an exceedingly important paper giving a pointwise differential inequality for linear heat equations which can be integrated along curves to give classic Harnack inequalities. Yau repeatedly urged me to study this paper, and based on their approach I was able to prove Harnack inequalities for the Ricci Flow and for the Mean Curvature Flow. This Harnack inequality, generalized from Li-Yau, forms the basis for the analysis of ancient solutions which I started, and which Perelman completed and uses as the basic tool in his canonical neighborhood theorem. Cao Huai-Dong proved the Harnack estimate for the Ricci Flow in the Kahler case, and Ben Chow did the same for the Yamabe Flow and the Gauss Curvature Flow.
But there is more to this story. Perelman's most important is his noncollapsing result for Ricci Flow, valid in all dimensions, not just three, and thus one whose importance for the future extends well beyond the Poincare conjecture, where it is the tool for ruling out cigars, the one part of the singularity classification I could not do. This result has two proofs, one using an entropy for a backward scalar heat equation, and one using a path integral. The entropy estimate comes from integrating a Li-Yau type differential Harnack inequality for the adjoint heat equation, and the other is the optimal Li-Yau path integral for the same Harnack inequality; as Perelman acknowledges in 7.4 of his first paper, where he writes "an even closer reference is [L-Y],where they use "length" associated to a linear parabolic equation, which is pretty much the same as in our case".
Over the years Yau has consistently supported the Ricci Flow and the whole field of Geometric Flows, which has other important successes as well, such as the recent proof of the Penrose Conjecture by Huisken and Ilmanen, a very important result in General Relativity. I cannot think of any other prominent leader who gave nearly support to our field as Yau has.
Yau has built is an assembly of talent, not an empire of power, people attracted by his energy, his brilliant ideas, and his unflagging support for first rate mathematics, people whom Yau has brought together to work on the hardest problems. Yau and I have spent innumerable hours over many years working together on the Ricci Flow and other problems, often even late at night. He has always generously shared his suggestions with me, starting with the observation of neck pinches, never asking for credit. In fact just last winter when I finally managed to prove a local version of the Harnack inequality for the Ricci Flow, a problem we had worked on together for many years, and I said I ought to add his name to the paper, he modestly declined. It is unfortunate that his character has been so badly misrepresented. He has never to my knowledge proposed any percentages of credit, nor that Perelman should share credit for the Poincare conjecture with anyone but me; which is reasonable, as indeed no one has been more generous in crediting my work than Perelman himself. Far from stealing credit for Perelman's accomplishment, he has praised Perelman's work and joined me in supporting him for the Fields Medal. And indeed no one is more responsible than Yau for creating the program on Ricci Flow which Perelman used to win this prize.
Sincerely yours,
Richard S Hamilton
Professor of Mathematics,
Columbia University
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