佩雷尔曼关于庞加莱猜想的论文0211159
庞加莱猜想
庞加莱猜想百科名片庞加莱猜想电脑三维模型庞加莱猜想是法国数学家提出的一个猜想，是悬赏的（七个千年大奖问题）之一。
2006年被确认由俄罗斯数学家最终证明，但将解题方法公布到网上之后，佩雷尔曼便拒绝接受马德里国际数学联合会声望颇高的。
目录展开庞加莱猜想图示令人头疼的世纪难题缘起如果我们伸缩围绕一个表面的橡皮带，那么我们可以既不扯断它，也不让它离开表面，使它慢慢移动收缩为一个点。
另一方面，如果我们想象同样的橡皮带以适当的方向被伸缩在一个轮胎面上，那么不扯断橡皮带或者轮胎面，是没有办法把它收缩到一点的。
我们说，苹果表面是“的”，而轮胎面不是。
大约在一百年以前，已经知道，球面本质上可由单连通性来刻画，他提出(中与原点有单位距离的点的全体)的对应问题。
这个问题立即变得无比困难，从那时起，数学家们就在为此奋斗。
一位史家曾经如此形容1854年出生的（Henri Poincare）:“有些人仿佛生下来就是为了证明天才的存在似的，每次看到亨利，我就会听见这个恼人的声音在我耳边响起。
”庞加莱作为的伟大，并不完全在于他解决了多少问题，而在于他曾经提出过许多具有开创意义、奠基性的大问题。
庞加莱猜想，就是其中的一个。
1904年，庞加莱在一篇论文中提出了一个看似很简单的的：在一个中，假如每一条封闭的都能收缩到一点，那么这个空间一定是一个三维的圆球。
但1905年发现提法中有错误，并对之进行了修改，被推广为：“任何与n维球面的n维封闭流形必定于n维球面。
”后来，这个猜想被推广至三维以上空间，被称为“高维庞加莱猜想”。
猜想的简单比喻如果你认为这个说法太抽象的话，我们不妨做这样一个想象：我们想象这样一个房子，这个空间是一个球。
或者，想象一只巨大的足球，里面充满了气，我们钻到里庞加莱猜想面看，这就是一个球形的房子。
我们不妨假设这个球形的房子墙壁是用钢做的，非常结实，没有窗户没有门，我们现在在这样的球形房子里。
拿一个气球来，带到这个球形的房子里。
庞加莱猜想浅谈
庞加莱猜想浅谈庞加莱猜想，故名思意，最早是由法国数学家庞加莱提出的，这是克雷数学研究所悬赏的数学方面七大千禧年难题之一。
2006年确认由俄罗斯数学家格里戈里•佩雷尔曼（俄语：ГригорийЯковлевичПерельман，1966年6月13日出生）完成了最终证明，他也因此在同年获得菲尔兹奖，但可以，佩雷尔曼在颁奖典礼上并未现身领奖。
猜想是庞加莱在1904年发表的一组论文中提出，猜想本身并不复杂：任一单连通的、封闭的三维流形与三维球面同胚。
解释来说就是：每一个没有破洞的封闭三维物体，都拓扑等价于三维的球面。
粗浅的比喻以下，如果我们伸缩围绕一个苹果表面的橡皮带，那么我们可以既不扯断它，也不让它离开表面，使它慢慢移动收缩为一个点；另一方面，如果我们想象同样的橡皮带以适当的方向被伸缩在一个轮胎面上，那么不扯断橡皮带或者轮胎面，是没有办法把它不离开表面而又收缩到一点的。
我们说，苹果表面是“单连通的”，而防真轮胎面不是。
该猜想是一个属于代数拓扑学领域的具有基本意义的命题，对“庞加莱猜想”的证明及其带来的后果将会加深数学家对流形性质的认识，甚至会对人们用数学语言描述宇宙空间产生影响。
对于猜想的破解，前后经历了近100年的时间：20世纪这个问题曾经被搁置了很长时间，直到1930年怀特海（J. H. C. Whitehead）首先宣布已经证明然而又收回，才再次引起了人们的兴趣。
怀特海提出了一些有趣的三流形实例，其原型现在称为怀特海流形。
1950和1960年代，又有许多著名的数学家包括R·H·宾（R. H. Bing）、沃夫冈·哈肯（Wolfgang Haken）、爱德华·摩斯（Edwin E. Moise）和Christos Papakyriakopoulos声称得到了证明，但最终都发现证明存在致命缺陷。
1961年，美国数学家史提芬·斯梅尔采用十分巧妙的方法绕过三、四维的困难情况，证明了五维以上的庞加莱猜想。
拓扑学难以证明庞加莱猜想
拓扑学难以证明庞加莱猜想你知道庞加莱猜想吗？要是知道的话，想必你也听说过这个让无数数学家抓破头的难题。
简直像是数学界的“定海神针”，没有它，整个拓扑学的世界都不完整。
可是呢，庞加莱猜想不仅难，还能让你看得眼花缭乱，摸不着头脑。
你要问我为什么？哎，别急，咱慢慢聊。
庞加莱猜想的意思其实很简单：假设你有一个三维空间中闭合的、没有边界的物体，如果它满足某些条件，那它一定是一个“球体”。
看起来是不是简单得不能再简单了？就像告诉你，世界上所有的圆都是圆，所有的苹果都是苹果似的。
但问题就是，数学家们花了大半辈子也没法证明这件事。
再聪明的头脑也没有办法直接去验证这条猜想，甚至有的数学家，搞得差点晕头转向。
想想看吧，拓扑学这玩意儿，讲的就是形状和空间的“变形”游戏。
你拿个橡皮泥捏捏，挤挤，捏成一个大球、一个小球，或者把它搓成个玩具车形状，理论上它们的拓扑结构是一样的。
奇怪吧？但这就是拓扑学的精髓——不管你怎么捏，最终只要不撕裂、拉伸，它们的“本质”是一样的。
你说，这种东西搞不定也就算了，结果庞加莱猜想竟然和这个有关系，难度可想而知。
更离谱的是，大家都知道，数学家要解决问题，得有证据、有论证，不能靠想象和猜测。
庞加莱猜想要么是对的，要么是错的。
你让我用普通的眼光去看，感觉它简直就像在拿一个几百公斤的铁锤砸一颗坚硬的石头，想要准确打个小洞。
看似简单的“球体”到底是啥？它该怎么在无数变形中依然“保形”？这个问题，别说当时的数学家，就是现在的很多人也解不开。
最奇葩的是，这个猜想让数学家们陷入了“循环困境”。
他们发现，证明这个猜想并不是直接找个三维空间球体来“验证”。
实际上，要证明庞加莱猜想需要用到一个叫做“高维空间”的概念。
就是你想要解决一个三维问题，结果发现自己跑到了四维、五维的世界里去。
就像是你跑去买米，结果被人推到厨房做饭，最后还要自己烧火做饭那种尴尬。
你明明想解决一个问题，结果每一步都被带到更难的地方去。
但是呢，奇迹也会发生！有一天，2003年，数学家佩雷尔曼终于给出了一种方法，这个方法的背后就是“理性与智慧的结晶”。
庞加莱猜想证明概述
庞加莱猜想证明概述在庞加莱猜想提出后，很多数学家对其展开了探索和研究，但一直没有找到一个确凿的证明或反例。
直到2003年，俄罗斯数学家格雷戈里·佩雷尔曼通过利用里奇流理论和梯度流的理论等一系列数学方法，证明了庞加莱猜想。
这篇文章将介绍庞加莱猜想的历史背景和相关概念，然后详细描述佩雷尔曼的证明过程和相关数学原理，最后分析庞加莱猜想对数学和科学领域的重要意义。
一、庞加莱猜想的历史背景庞加莱猜想的提出可以追溯到19世纪末的数学发展。
当时，数学家们已经开始探讨对多维几何空间的研究，如三维流形的性质和拓扑结构等。
此时，亨利·庞加莱成为了这一领域的先驱者，他提出了著名的庞加莱猜想，引发了数学界对于三维空间性质的深入思考和研究。
庞加莱猜想的提出也在一定程度上推动了数学领域的发展，为拓扑学和几何学等领域的研究提供了新的动力和方向。
然而，长期以来，庞加莱猜想一直未能找到确凿的证明，成为数学界的一个难题。
二、庞加莱猜想的相关概念1. 流形：在数学领域，流形是指一个局部与欧氏空间同胚的空间。
在庞加莱猜想中，主要讨论的是三维紧致的无边界的连通流形。
2. 欧氏空间：欧氏空间指的是平凡的三维空间，即我们所生活的空间。
在庞加莱猜想中，研究的对象是三维欧氏空间中的环流变形问题。
3. 拓扑结构：拓扑结构是指一个空间的结构，它并不依赖于空间的具体度量，而仅仅与空间的连通性和邻域关系有关。
在庞加莱猜想中，研究的就是流形的拓扑结构和性质。
三、佩雷尔曼的证明过程2003年，俄罗斯数学家格雷戈里·佩雷尔曼通过利用里奇流理论和梯度流的理论，证明了庞加莱猜想。
他的证明过程可以概括为以下几个步骤：1. 利用几何流的理论，建立了三维流形的梯度不等式，从而引入了里奇流的概念。
2. 利用里奇流的理论，证明了当流形上的里奇曲率为正时，流形是球面的概率。
3. 利用梯度流的理论，证明了当流形上的梯度不等式成立时，流形是球面的概率。
佩雷尔曼庞加莱定理
佩雷尔曼庞加莱定理佩雷尔曼和庞加莱定理，说起来就好像一个超级大难题，能让数学爱好者头痛不已。
但是别怕，今天我尽量用一种大家都能理解的方式，来聊聊这个话题。
说到佩雷尔曼，你可能觉得他是不是某个深藏不露的数学天才？嗯，确实，他的名字在数学界可算是响当当的。
不过，他也并不是什么高高在上的神秘人物，相反，他更像是一个生活中不怎么起眼的普通人。
有一个细节特别值得一提。
曾经有个数学大会上，佩雷尔曼的名气已经传开了，但他并没有像大部分的数学界大佬一样在台上发表演讲，而是穿着一件很普通的外套，安安静静地坐在观众席上。
很多人都没注意到他，就是这么低调，甚至不怎么和人打交道。
他就像是那种在学校里总能给人留下深刻印象，但永远不会主动找人搭话的“冷门高手”。
不过你要知道，正是这个“冷门高手”，在2003年突然用他自己的方式，解决了一个困扰了数学界几十年的问题——庞加莱猜想。
说回庞加莱猜想，这个问题一开始听上去就像个谜题：它讲的是一个几乎没人能搞清楚的三维空间问题。
简单说，庞加莱猜想就是猜测，如果一个三维物体没有“洞”，它是不是可以通过某些方式变成一个球体？在几百年前，有个叫庞加莱的数学家就提出了这个问题，但后来大部分人都认为，没人能解决它。
直到佩雷尔曼的出现，他利用了一种叫做“里奇流”的方法，终于让庞加莱猜想从谜题变成了定理。
现在，很多人可能会想，佩雷尔曼到底是怎么做到的呢？你不妨想象一下，解开庞加莱猜想就像是在拼一个特别复杂的拼图。
这个拼图里，既有难度极高的数学公式，也有无数个看似不相关的概念和工具。
佩雷尔曼就像是一个擅长拼图的人，他把一个个散乱的部分巧妙地组合在了一起，最终拼出了一个完美的结果。
但问题来了，佩雷尔曼做出的这个成就，并没有让他在数学界大肆宣扬自己，反而，他选择了远离公众视野，甚至拒绝了很多奖项和荣誉。
比如，他拒绝了2006年普林兹奖的100万美元奖金。
这个举动让很多人都傻眼了。
你说，别人拿到奖项会开心得不行，佩雷尔曼怎么就这么“淡定”呢？其实，这就像我前段时间看的一件事：有个朋友参加一个小型比赛，虽然奖品并不丰厚，但是他还是非常认真地准备了很久，最后得了第一名。
庞加莱猜想前言
庞加莱猜想-前言Wir m\"ussen wissen! Wir werden wissen!(我们必须知道！我们必将知道！)—— David Hilbert两年前科学版举行过一次版聚，我报告了低维拓扑里面的一些问题和进展，其中有一半篇幅是关于Poincar\'e 猜想。
版聚后，flyleaf 要求大家回去后把自己所讲的内容发在版上。
当时我甚至已经开始写了一两段，但后来又搁置了。
主要是因为自己对于低维拓扑还是一个门外汉，写出来的东西难免有疏漏之处，不敢妄下笔。
两年过去，我对低维拓扑这门学科的了解比原先多了，说话的底气也就比原先足了。
另外，由于Clay 研究所的百万巨赏，近年来Poincar\'e 猜想频频在媒体上曝光；而且Perelman 最近的工作使数学家们有理由相信我们已经充分接近于这一猜想的最后解决。
所以大概会有很多人对Poincar\'e 猜想的来龙去脉感兴趣，我也好借机一偿两年来的宿愿。
现代科学的高速发展使各学科之间的鸿沟加大，不同学科之间难以互相理解，所以非数学专业的读者在阅读本文时可能会遇到一些困难。
但限于篇幅和文章的形式，我也不可能对很多东西详细解释。
一些最基本的拓扑概念如“流形”，我将在本文的附录中解释。
还有一些“同调群”、“基本群”之类的名词，读者见到时大可不去理会它们的确切含义。
我将尽量避免使用这一类的专业术语。
作者并非拓扑方面的专家，对下面要说的很多内容都是道听途说，只知其然而不知其所以然；作者更不善于写作，写出来的东东总会枯燥无味，难登大雅之堂。
凡此种种，还请读者诸君海涵。
问题的由来Consid\'erons maintenant une vari\'et\'e [ferm\'ee] $V$ \`a trois dimensions ... Est-il possible que le groupe fondamental de $V$ ser\'eduise \`a la substitution identique, et que pourtant $V$ ne soit pas simplement connexe?—— Henri Poincar\'e在拓扑学家的眼里，篮球、排球和乒乓球并没有什么不同，它们都同胚于三维空间中的球面S^2. (我们把n+1维欧氏空间中到原点距离为1的点的集合记作S^n，称为n维球面(sphere)。
佩雷尔曼关于庞加莱猜想的论文0303109
Proof. A gradient shrinking soliton gij(t), −∞ < t < 0, satisfies the equation
庞加莱猜想
庞加莱猜想【摘要】庞加莱是法国著名数学家，他提出的“庞加莱猜想”引起极大轰动，后人为证明此猜想而不懈努力，经过一个世纪的钻研，终于对此猜想给出完整证明。
本文就是通过对庞加莱猜想及后人对此做的努力做出叙述，以此让大家体会数学家们的事业热情和博大胸怀。
【关键词】庞加莱猜想、代数拓扑学、证明、萨密尔、瑟斯顿、米歇尔、汉密尔顿、佩雷尔曼、朱熹平、曹怀东、伟大贡献、宽阔胸怀、钻研精神、敬佩庞加莱是法国数学家，被称为是19世纪最后四分之一和20世纪初期的数学界的领袖人物，是对数学和它的应用具有全面了解、能够雄观全局的最后一位大师。
他的研究和贡献涉及数学的各个分支，例如函数论、代数拓扑学、阿贝尔函数和代数几何学、数论、代数学、微分方程、数学基础、非欧几何、渐近级数、概率论等当代数学不少研究课题，都溯源于他的工作。
在他留下的巨大科学遗产中，有一个属于代数拓扑学中带有基本意义的命题，这就是困扰了数学家整整一个世纪的“庞加莱猜想”。
1904 年，庞加莱提出有关空间几何结构的猜想：在一个三维空间中，假如每一条封闭的曲线都能收缩成一点，那么这个空间一定是一个三维的圆球。
这就是著名的“庞加莱猜想”庞加莱是在1904年发表的一组论文中提出这一猜想的：“单连通的三维闭流形同胚于三维球面。
”它后来被推广为：“任何与n维球面同伦的n维闭流形必定同胚于n维球面。
”粗浅的比喻为：如果我们伸缩围绕一个苹果表面的橡皮带，那么我们可以既不扯断它，也不让它离开表面，使它慢慢移动收缩为一个点。
另一方面，如果我们想象同样的橡皮带以适当的方向被伸缩在一个轮胎面上，那么不扯断橡皮带或者轮胎面，是没有办法把它收缩到一点的。
我们说，苹果表面是“单连通的”，而轮胎面不是。
大约在一百年以前，庞加莱已经知道，二维球面本质上可由单连通性来刻画，他提出三维球面(四维空间中与原点有单位距离的点的全体)的对应问题。
这个问题立即变得无比困难，从那时起，数学家们就在为此奋斗。
百年未解的谜题：庞加莱猜想
百年未解的谜题：庞加莱猜想《三联生活周刊》2006年8月17日《百年未解的谜题：庞加莱猜想》十几年来，没有哪一届国际数学家大会，能像8月22日将在西班牙马德里召开的2006年国际数学家大会(ICM2006)这样引人注目。
早在几个月前，ICM2006的网站上，就贴出了这样的消息:“一个有100年历史的数学难题的证明，将在本届大会上宣布。
”尽管做出欲说还休的姿态，但看一眼会议的日程表——8月22日17:15至18:15，里查德·汉密尔顿(Richard Hamilton)，题目:庞加莱猜想。
答案，已经无需再言。
一位数学史家曾经如此形容1854年出生的亨利·庞加莱(Henri Poincare):“有些人仿佛生下来就是为了证明天才的存在似的，每次看到亨利，我就会听见这个恼人的声音在我耳边响起。
”庞加莱作为数学家的伟大，并不完全在于他解决了多少问题，而在于他曾经提出过许多具有开创意义、奠基性的大问题。
庞加莱猜想，就是其中的一个。
1904年，庞加莱在一篇论文中提出了一个看似很简单的拓扑学猜想:在一个三维空间中，假如每一条封闭的曲线都能收缩到一点，那么这个空间一定是一个三维的圆球。
提出这个猜想后，庞加莱一度认为，自己已经证明了它。
但没过多久，证明中的错误就被暴露了出来。
于是，拓扑学家们开始了证明它的努力。
20世纪30年代以前，庞加莱猜想的研究只有零星几项。
但突然，英国数学家怀特黑德(Whitehead)对这个问题产生了浓厚兴趣。
他一度声称自己完成了证明，但不久就撤回了论文。
失之桑榆、收之东隅的是，在这个过程中，他发现了三维流形的一些有趣的特例，而这些特例，现在被统称为怀特黑德流形。
50年代到60年代之间，又有一些著名的数学家宣称自己解决了庞加莱猜想，著名的宾(R.Bing)、哈肯(Haken)、莫伊泽(Moise)和帕帕奇拉克普罗斯(Papa-kyriakopoulos)均在其中。
帕帕奇拉克普罗斯是1964年的维布伦奖得主，一名希腊数学家。
攻克庞加莱猜想的数学隐士——佩雷尔曼
攻克庞加莱猜想的数学隐士——佩雷尔曼2002年11月在数学预印本网站上，佩雷尔曼贴出了仅仅只有三页非常简洁的论文，目标直指百年数学难题庞加莱猜想（实际上，他描述的是更一般的瑟斯顿几何化猜想，庞加莱猜想是它的一个特例）。
这个消息不胫而走，随即引发了数学界的轩然大波，更多的人认为这是一个“恶作剧”，就像曾经有很多人宣称自己证明了哥特巴赫猜想一样。
但四个月后，佩雷尔曼又上传了一份更加详细的论文，补充了很多证明的细节，并以邮件的形式和少数数学界内他信得过的人讨论交流。
第二年，佩雷尔曼再上传了一篇论文，给出了更多相关细节。
这三篇论文的正确与否直接关系着一个百年数学难题的命运。
实际上，佩雷尔曼为公众所知，更多的原因在于他拒绝了数学最高奖菲尔兹奖与千禧百万美元数学大奖，至于他到底做了哪些贡献，是一个什么样的数学家，一般人还是不太了解。
在他发布论文以后，了解佩雷尔曼的人都对其正确性持乐观态度，因为在此之前，他已经是一位卓有成就，天赋异禀且严谨谦虚的数学家，如果没有把握，他是不会轻易发布这样的论文。
那么，佩雷尔曼到底是一位什么样的数学家？与庞加莱猜想又有怎样的渊源？佩雷尔曼佩雷尔曼（全名为Grigori Yakovlevich Perelman ，俄语为: Григо́рийЯ́ковлевичПерельма́н），1966年6月13日出生于圣彼得堡（也即苏联时期的列宁格勒），是著名的俄罗斯籍犹太裔数学家，他的父亲是电路工程师而母亲则为当地小学数学教师。
佩雷尔曼从小就对玩乐没什么兴趣，宁愿呆在家看书或者下棋。
母亲发现他的数学天才之后，送他去了一个数学培训班，之后又让他去了以数学和物理见长的239中学。
仅仅三个月之后，佩雷尔曼代表苏联在国际中学生数学奥林匹克上获得满分金牌，而且据传，佩雷尔曼还在另一个更难的几何竞赛中拿下赛史上第一个满分。
在同时代的人中，佩雷尔曼的数学天才几乎找不出对手。
佩雷尔曼这样非凡的天才立即引来了美国顶级名校耶鲁大学的极大兴趣，并以巨额奖学金吸引他来就学，但佩雷尔曼毫不犹豫地拒绝了。
用拓扑方法重证庞加莱猜想
用拓扑方法重证庞加莱猜想哎呀，说到庞加莱猜想，这个名字一听就让人有点头大，尤其是对我们这些普通人来说。
说实话，一开始我也觉得这东西离我挺远的。
不过，随着时间一长，你就发现，哦，原来这不是天上的星星，也不是遥不可及的高楼大厦，它离我们其实近得很，甚至就藏在我们生活中的某个小细节里。
所以今天我们聊聊，怎么用拓扑的方式重证庞加莱猜想，大家别紧张，听着听着就能明白，咱不一定要拿着公式和复杂的图形去证明什么，生活中也有它的影子呢！首先啊，庞加莱猜想的核心其实就是想说：“如果一个三维的空间，看上去像是‘球’一样的，那它其实就是一个‘三维球’。
”这句话说的简单，但要证明它可不简单。
我们生活中的球呢，通常是光滑的，拿在手里可以随便摆弄一下，但想想，要是这个球的每个点都可以自由地扭曲、拉伸、折叠，甚至压扁，结果还是保持‘球’的特征，那可真是头疼。
说白了，就是一堆看似复杂的形状，但是你站在它面前，能认出来它“顶多”就是一个球的形状。
你可能会想，“诶，这个问题咋这么简单？不就是找一个球，看看它是不是三维的吗？”哦，绝不是这么简单。
你得想象一下，如果让这个球表面有无数个小洞或者复杂的缝隙，咱能不能依然觉得它是一个球？这就涉及到拓扑学了，拓扑学就是研究物体形状和空间结构的一门学问，重点是“形状”能不能通过拉伸、弯曲来变，而不管它是怎么扭曲的。
想象你捏捏一个橡皮泥球，把它拉成各种奇形怪状的东西，不管怎么变，它还是那个橡皮泥对吧？只要它不撕裂或者破坏，拓扑上它还是“球”嘛。
懂了吗？要是你脑袋里还在“球”来“球”去，那就不如想想我们常见的那个例子：咖啡杯和甜甜圈。
嘿，别笑，别觉得这俩不搭。
拓扑学有一个经典的比喻，就是咖啡杯和甜甜圈其实在拓扑上是等价的，咋理解呢？想象你把一个杯子和一个甜甜圈都拿到手里，把它们的形状“拉扯”一下，逐渐可以看到它们之间的“相似点”——咖啡杯有一个把手，甜甜圈也有一个孔。
通过拉伸、弯曲，甚至捏合，它们的结构不变，这就是拓扑学的精髓。
数学毕业论文庞加莱以及庞加莱猜想
庞加莱以及庞加莱猜想庞加莱以及庞加莱猜想摘要: 本文首先介绍了109世纪法国数学家庞加莱的生平，接着对＂庞加莱猜想＂的背景及解决过程进行了较为详细的介绍.最后从数学家们在整个庞加莱猜想的证明过程中所做出的努力说明了从事科学研究所必须具备的学术素养和精神品质.关键词：庞加莱；庞加莱猜想；庞加莱猜想证明过程的人文意义.Poincaré and Poincaré Conjecture Abstact : This paper introduces the 19th-century French mathematician Poincarés whole life, and then the background of the " Poincaré Conjecture" and the process of settling are also interpretted in details. Lastly, from the efforts that mathematicians made in proving the Poincaré Conjecture,suggest the importance of the academic qualities and mental quality in scientificresearch.Keywords : Poincaré; the Poincaré Conjecture; humanistic significance of proving the Poinca ré Conjecture.目录中英文摘要 (11)引言……………………………………………………………………………22庞加莱其人…………………………………………………………………32.1 生平……………………………………………………………………32.2 多产的数学天才………………………………………………………42.3 科学探险者……………………………………………………………63 庞加莱猜想…………………………………………………………………63.1 什么是『庞加莱猜想』? ...................................................63.2 百年证明之路..................................................................83.3 看见曙光--「Racci Flow瑞奇流」..........................................103.4 突破瓶颈.....................................................................113.5 世纪猜想变定理 (124)科学的本意…………………………………………………………………13致谢词....................................................................................15参考文献 (1)6【包括：毕业论文、开题报告、任务书】【说明：论文中有些数学符号是编辑器编辑而成，网页上无法显示或者显示格式错误，给您带来不便请谅解。
佩雷尔曼关于庞加莱猜想的论文0307245
a rXiv:mat h .DG /37245v117J ul23Finite extinction time for the solutions to the Ricci flow on certain three-manifolds Grisha Perelman ∗May 21,2006In our previous paper we constructed complete solutions to the Ricci flow with surgery for arbitrary initial riemannian metric on a (closed,oriented)three-manifold [P,6.1],and used the behavior of such solutions to classify three-manifolds into three types [P,8.2].In particular,the first type consisted of those manifolds,whose prime factors are diffeomorphic copies of spherical space forms and S 2×S 1;they were characterized by the property that they admit metrics,that give rise to solutions to the Ricci flow with surgery,which become extinct in finite time.While this classification was sufficient to answer topological ques-tions,an analytical question of significant independent interest remained open,namely,whether the solution becomes extinct in finite time for every initial metric on a manifold of this type.In this note we prove that this is indeed the case.Our argument (in con-junction with [P,§1-5])also gives a direct proof of the so called ”elliptization conjecture”.It turns out that it does not require any substantially new ideas:we use only a version of the least area disk argument from [H,§11]and a regu-larization of the curve shortening flow from [A-G].1Finite time extinction 1.1Theorem.Let M be a closed oriented three-manifold,whose prime decom-position contains no aspherical factors.Then for any initial metric on M thesolution to the Ricci flow with surgery becomes extinct in finite time.Proof for irreducible M .Let ΛM denote the space of all contractible loops in C 1(S 1→M ).Given a riemannian metric g on M and c ∈ΛM,define A (c,g )to be the infimum of the areas of all lipschitz maps from D 2to M,whose restriction to ∂D 2=S 1is c.For a family Γ⊂ΛM let A (Γ,g )be the supremum of A (c,g )over all c ∈Γ.Finally,for a nontrivial homotopy class α∈π∗(ΛM,M )let A (α,g )be the infimum of A (Γ,g )over all Γ∈α.Since M is not aspherical,it follows from a classical (and elementary)result of Serre that such a nontrivial homotopy class exists.1.2Lemma.(cf.[H,§11])If g t is a smooth solution to the Ricciflow,then for anyαthe rate of change of the function A t=A(α,g t)satisfies the estimatedR t min A t2(in the sense of the lim sup of the forward difference quotients),where R t min denotes the minimum of the scalar curvature of the metric g t.A rigorous proof of this lemma will be given in§3,but the idea is simple and can be explained here.Let us assume that at time t the value A t is attained by the familyΓ,such that the loops c∈Γwhere A(c,g t)is close to A t are embedded and sufficiently smooth.For each such c consider the minimal disk D c with boundary c and with area A(c,g t).Now let the metric evolve by the Ricciflow and let the curves c evolve by the curve shorteningflow(which moves every point of the curve in the direction of its curvature vector at this point) with the same time parameter.Then the rate of change of the area of D c can be computed as(−Tr(Ric T))+ c(−k g)D cwhere Ric T is the Ricci tensor of M restricted to the tangent plane of D c,and k g is the geodesic curvature of c with respect to D c(cf.[A-G,Lemma3.2]).In three dimensions thefirst integrand equals−1R−K)+ c(−k g)= D c(−12the conclusion of the lemma above holds for the solutions to the Ricciflow with surgery as well.Now recall that the evolution equation for the scalar curvatured3R2+2|Ric◦|2implies the estimate R t min≥−3t+const.It follows thatˆA t=A tdtˆA t≤−2πdt c t(x)=H t(x),where x is the parameter on S1,and H t is the curvaturevectorfield of c t with respect to g t.It is known[G-H]that for any smoothly immersed initial curve c the solution c t exists on some time interval[t0,t′1),each c t for t>t0is an analytic immersed curve,and either t′1=t1,or the curvature k t=g t(H t,H t)1Denote by X t the tangent vectorfield to c t,and let S t=g t(X t,X t)−1g(X,X)=−2Ric(X,X)−2g(X,X)k2,(1)dtwhich implies[H,S]=(k2+Ric(S,S))S(2) Now we can computedk≤k′′+k3+const·(k+1)(4)dtNow it follows from(1)and(4)that the length L and the total curvature Θ= kds satisfy dΘ≤ const·(k+1)ds(6)dtIn particular,both quantities can grow at most exponentially in t(they would be non-increasing in aflat manifold).2.2In general the curvature of c t may concentrate near certain points,cre-ating singularities.However,if we know that this does not happen at some time t∗,then we can estimate the curvature and higher derivatives at times shortly thereafter.More precisely,there exist constantsǫ,C1,C2,...(which may depend on the curvatures of the ambient space and their derivatives,but are independent of c t),such that if at time t∗for some r>0the length of c t is at least r and the total curvature of each arc of length r does not exceedǫ,then for every t∈(t∗,t∗+ǫr2)the curvature k and higher derivatives satisfy the estimates k2=g(H,H)≤C0(t−t∗)−1,g(∇S H,∇S H)≤C1(t−t∗)−2,... This can be proved by adapting the arguments of Ecker and Huisken[E-Hu]; see also[A-G,§4].2.3Now suppose that our manifold(M,g t)is a metric product(¯M,¯g t)×S1λ, where the second factor is the circle of constant lengthλ;let U denote the unit tangent vectorfield to this factor.Then u=g(S,U)satisfies the evolution equationdAssume that u was strictly positive everywhere at time t0(in this case the curve is called a ramp).Then it will remain positive and bounded away from zero as long as the solution exists.Now combining(4)and(7)we can estimate the right hand side of the evolution equation for the ratio kµt≤(2n−1)|Rm t|µt,(8)dtwhere|Rm t|denotes a bound on the absolute value of sectional curvatures of g t.Indeed,the curves c t1and c t2,being ramps,are embedded and without substantial loss of generality we may assume them to be disjoint.In this case the results of Morrey[M]and Hildebrandt[Hi]yield an analytic minimal annulus A, immersed,except at mostfinitely many branch points,with prescribed boundary and with areaµ.The rate of change of the area of A can be computed as(−Tr(Ric T))+ ∂A(−k g)≤ A(−Tr(Ric T)+K)A≤ A(−Tr(Ric T)+Rm T)≤(2n−1)|Rm|µ,where thefirst inequality comes from the Gauss-Bonnet theorem,with possible contribution of the branch points,and the second one is due to the fact that a minimal surface has nonpositive extrinsic curvature with respect to any normal vector.2.5The estimate(8)implies thatµt can grow at most exponentially;in particular,if c t1and c t2were very close at time t0,then they would be close for all t∈[t0,t1]in the sense of minimal annulus area.In general this does not imply that the lengths of the curves are also close.However,an elementary argument shows that ifǫ>0is small then,given any r>0,one canfind¯µ, depending only on r and on upper bound for sectional curvatures of the ambient space,such that if the length of c t1is at least r,each arc of c t1with length r has total curvature at mostǫ,andµt≤¯µ,then L(c t2)≥(1−100ǫ)L(c t1).3Proof of lemma1.23.1In this section we prove the following statementLet M be a closed three-manifold,and let(M,g t)be a smooth solution to the Ricciflow on afinite time interval[t0,t1].Suppose thatΓ⊂ΛM is a compact family.Then for anyξ>0one can construct a continuous deformationΓt,t∈[t0,t1],Γt0=Γ,such that for each curve c∈Γeither the value A(c t1,g t1)is bounded from above byξplus the value at t=t1of the solution to the ODE5d2R t min w(t)with the initial data w(t0)=A(c t0,g t0),orL(c t1)≤ξ;moreover,if c was a constant map,then all c t are constant maps.It is clear that our statement implies lemma1.2,because a family consisting of very short loops can not represent a nontrivial relative homotopy class.3.2As afirst step of the proof of the statement we can replaceΓby a family,which consists of piecewise geodesic loops with some largefixed number of vertices and with each segment reparametrized in some standard way to make the parametrizations of the whole curves twice continuously differentiable.Now consider the manifold Mλ=M×S1λ,0<λ<1,and for each c∈Γconsider the smooth embedded closed curve cλsuch that p1cλ(x)=c(x)and p2cλ(x)=λx modλ,where p1and p2are projections of Mλto thefirst and second factor respectively,and x is the parameter of the curve c on the standard circle of length ing2.3we can construct a solution c tλ,t∈[t0,t1]to the curve shorteningflow with initial data cλ.The required deformation will be obtained asΓt=p1Γtλ(whereΓtλdenotes the family consisting of c tλ)for certain sufficiently smallλ>0.We’ll verify that an appropriateλcan be found for each individual curve c,or for anyfinite number of them,and then show that if ourλworks for all elements of aµ-net inΓ,for sufficiently smallµ>0, then it works for all elements ofΓ.3.3In the following estimates we shall denote by C large constants that may depend on metrics g t,familyΓandξ,but are independent ofλ,µand a particular curve c.Thefirst step in3.2implies that the lengths and total curvatures of cλare uniformly bounded,so by2.1the same is true for all c tλ.It follows that the area swept by c tλ,t∈[t′,t′′]⊂[t0,t1]is bounded above by C(t′′−t′),and therefore we have the estimates A(p1c tλ,g t)≤C,A(p1c t′′λ,g t′′)−A(p1c t′λ,g t′)≤C(t′′−t′).3.4It follows from(5)that t1t0 k2dsdt≤C for any c tλ.Fix some large constant B,to be chosen later.Then there is a subset I B(cλ)⊂[t0,t1]of measure at least t1−t0−CB−1where k2ds≤B,hence kds≤ǫon any arc of length≤ǫ2B−1.Assuming that c tλare at least that long,we can apply2.2 and construct another subset J B(cλ)⊂[t0,t1]of measure at least t1−t0−CB−1, consisting offinitely many intervals of measure at least C−1B−2each,such that for any t∈J B(cλ)we have pointwise estimates on c tλfor curvature and higher derivatives,of the form k≤CB,...Nowfix c,B,and consider any sequence ofλ→0.Assume again that the lengths of c tλare bounded below byǫ2B−1,at least for t∈[t0,t2],where t2= t1−B−1.Then an elementary argument shows that we canfind a subsequence Λc and a subset J B(c)⊂[t0,t2]of measure at least t1−t0−CB−1,consisting offinitely many intervals,such that J B(c)⊂J B(cλ)for allλ∈Λc.It follows that on every interval of J B(c)the curve shorteningflows c tλsmoothly converge (asλ→0in some subsequence ofΛc)to a curve shorteningflow in M.Let w c(t)be the solution of the ODE d2R t min w c(t)withinitial data w c(t0)=A(c,g t0).Then for sufficiently smallλ∈Λc we have A(p1c tλ,g t)≤w c(t)+1disk argument as in 1.2,and this implies the corresponding estimate for p 1c t λif λ∈Λc is small enough,whereas for the intervals of the complement of J B (c )we can use the estimate in 3.3.On the other hand,if our assumption on the lower bound for lengths does not hold,then it follows from (5)that L (c t 2λ)≤CB−1≤12ξor L (ˆc t 2λ)≤12ξand µ≤C −1ξ,then A (p 1c t1λ,g t 1)≤w c (t 1)+ξ.Onthe other hand,if L (ˆc t2λ)≤14ξfor all t ∈[t 2,t 1];on the other hand,using (5)we can find a t ∈[t 2,t 1],such that k 2ds ≤CB for c t λ;hence,applying 2.5,we get L (ˆc t λ)>22ξ.The proof of the statement3.1is complete.References[A-G]S.Altschuler,M.Grayson Shortening space curves and flow through singularities.Jour.Diff.Geom.35(1992),283-298.[B]S.Bando Real analyticity of solutions of Hamilton’s equation.Math.Zeit.195(1987),93-97.[E-Hu]K.Ecker,G.Huisken Interior estimates for hypersurfaces moving by mean curvature.Invent.Math.105(1991),547-569.[G-H]M.Gage,R.S.Hamilton The heat equation shrinking convex plane curves.Jour.Diff.Geom.23(1986),69-96.[H]R.S.Hamilton Non-singular solutions of the Ricci flow on mun.Anal.Geom.7(1999),695-729.[Hi]S.Hildebrandt Boundary behavior of minimal surfaces.Arch.Rat.Mech.Anal.35(1969),47-82.[M]C.B.Morrey The problem of Plateau on a riemannian manifold.Ann.Math.49(1948),807-851.[P]G.Perelman Ricci flow with surgery on three-manifolds.arXiv:math.DG/0303109v17。
关于庞加莱(Poincare)猜想
关于庞加莱Poincare猜想你在一张平的橡皮膜上任意画一个圈——一条封闭的没有“８”字那样自交的曲线。
橡皮有弹性，你可以把这个圈变形，变为另一式样的圈，例如变形为一个正方形的边或更令人喜爱的图形——圆。
既然在橡皮的世界中圈与圆可以如此变来变去，我们就用圆做圈的代表。
变化中总有不变的东西，你发现没有，不管圈如何变，总是把平面分割成两部分这点不变。
反过来，如果橡皮平面上的一条闭曲线把平面分割成两部分，那它就是一个没有自交点的圈，方便地说，是一个圆了。
只有一个自交点的“８”字式闭曲线已把平面分成三部分，自交点越多分割出的部分也越多。
我们所用的橡皮品质似乎极好，不但能随意拉伸而且能随意压缩，除非动用刀剪之类工具，否则它永远不破。
当然，如果你肯用脑子想象，那就不需要去寻找这种世上并不存在的橡皮了。
现在，你想象在橡皮体即３维橡皮空间中作一个球面，这个球面可以变形为坑坑洼洼凹凸不平的闭曲面，所谓闭曲面即一个有界、无边缘的曲面。
与平面上的圈一样，球面无论如何变都把空间分割成两部分。
但是反过来就与平面圈不一样了，橡皮体中把空间分割成两部分的闭曲面却不一定是球面。
一个例子是轮胎面，它是闭曲面、把空间分割成两部分，但你没有办法把它变成球面，因为轮胎面有一个中通的“洞”。
不过，索性从“洞”这个东西出发到也可以建立一个判定方法：如果闭曲面没有轮胎面那样中通的“洞”，那它一定是球面。
这个办法看上去简单但有一个缺点，要站到闭曲面外面去看。
有没有办法直接在闭曲面上面找到判定的办法呢？你在球面上任意画一个圈，直观上看这个圈都能在球面上缩成一个点，这个性质叫做单连通。
球面的单连通性可以说得更严格一些：在球面上挖一个洞，就能把带洞球面变形为平面。
球面上任何圈不可能围住整个球面，你在圈外挖一个洞，把带洞球面展成平面，这个圈就变为平面上的圈了。
平面圈在平面上有很多办法收缩到一点，对应到球面上，就得到原球面圈收缩到一点的办法。
轮胎面不是单连通的。
写在尘埃落定时(庞加莱猜想的故事)
写在尘埃落定时──证明世纪难题Poincaré (庞加莱) 猜想的俄国数学家佩尔曼拒绝菲尔兹奖任平生２００６年８月２２日西班牙马德里世界数学大会，由西班牙国王颁奖，菲尔兹奖花落四家。
这一届的菲尔兹奖颇戏剧性：一如众人期待的那样，证明世纪难题Poincare猜想的俄国数学家佩尔曼拒绝菲尔兹奖，这在菲尔兹奖历史上还是第一次。
世界数学大会主席BALL 在仪式上说，“我对佩尔曼博士拒绝菲尔兹奖深表遗憾。
”菲尔兹奖是数学界的最高奖项，在四年一度的世界数学大会上颁发，是数学家们梦寐以求的最高荣誉。
设立菲尔兹奖的一个初衷是即表彰过去的成就又鼓励将来的研究，故规定只有40岁以下者才能得奖，使得获奖的难度更大。
象上个世纪末证明世纪难题费马大定理的Wiles，由于过了40大限，世界数学大会还专门为他设了特别奖。
抽象的数学象这一次这样吸引公众的眼球，尤其的是中国公众的眼球还不多见。
原因是在此之前丘成桐对北大学术腐败的指责和白热化论战，以及６月丘成桐宣布曹怀东朱熹平完成Poincare猜想的完整证明，为猜想的证明“封顶”。
加上一些中国权威数学家出来说中国人在Poincare猜想的贡献约有30%，而佩尔曼只有25%。
从现在权威媒体的报道，情形同丘先生的版本有点不一样。
世界各大媒体都开始报道此事，但以纽约客杂志给出的故事最为详细。
该文把丘成桐写成反角，说丘成桐出来抢功是搅局，而他对北大学术腐败的指责源于他对北大田刚的嫉妒。
其实，数学界的共识是把最高奖赏给予最后决定性一击的作者。
比如此次把最高荣誉给予佩尔曼，是因为他把通往目的地最后一座桥的蓝图绘好了，而在此只前众人只能站在河边看。
蓝图是有缺陷，但都不是致命的，都可以改好。
曹朱也是大家，可能也为蓝图的完备出了力，但毕竟是按佩尔曼指引的方向做出来的，况且文章发表也有一些仓促。
纽越客文章作者不愧是写《The Beautiful Mind》（美丽心灵）的高手，虽然真实性有待查证，但十几页的故事确实引人入胜，有兴趣者不妨一读。
庞加莱猜想
庞加莱猜想百科名片庞加莱猜想电脑三维模型庞加莱猜想是法国数学家庞加莱提出的一个猜想，是克雷数学研究所悬赏的世界七大数学难题（七个千年大奖问题）之一。
2006年被确认由俄罗斯数学家格里戈里·佩雷尔曼最终证明，但将解题方法公布到网上之后，佩雷尔曼便拒绝接受马德里国际数学联合会声望颇高的菲尔兹奖。
目录[隐藏]令人头疼的世纪难题艰难的证明之路早期的证明柳暗花明的突破最后的决战破解与争议破解解题者佩雷尔曼庞加莱猜想的意义其他难题的解决情况令人头疼的世纪难题艰难的证明之路早期的证明柳暗花明的突破最后的决战破解与争议破解解题者佩雷尔曼庞加莱猜想的意义其他难题的解决情况[编辑本段]令人头疼的世纪难题前言：如果我们伸缩围绕一个苹果表面的橡皮带，那么我们可以既不扯断它，也不让它离开表面，使它慢慢移动收缩为一个点。
另一方面，如果我们想象同样的橡皮带以适当的方向被伸缩在一个轮胎面上，那么不扯断橡皮带或者轮胎面，是没有办法把它收缩到一点的。
我们说，苹果表面是“单连通的”，而轮胎面不是。
大约在一百年以前，庞加莱已经知道，二维球面本质上可由单连通性来刻画，他提出三维球面(四维空间中与原点有单位距离的点的全体)的对应问题。
这个问题立即变得无比困难，从那时起，数学家们就在为此奋斗。
一位数学史家曾经如此形容1854年出生的亨利·庞加莱（Henri Poincare）:“有些人仿佛生下来就是为了证明天才的存在似的，每次看到亨利，我就会听见这个恼人的声音在我耳边响起。
”庞加莱作为数学家的伟大，并不完全在于他解决了多少问题，而在于他曾经提出过许多具有开创意义、奠基性的大问题。
庞加莱猜想，就是其中的一个。
1904年，庞加莱在一篇论文中提出了一个看似很简单的拓扑学的猜想：在一个三维空间中，假如每一条封闭的曲线都能收缩到一点，那么这个空间一定是一个三维的圆球。
但1905年发现提法中有错误，并对之进行了修改，被推广为：“任何与n维球面同伦的n维封闭流形必定同胚于n维球面。
陶哲轩评述佩雷尔曼的庞加莱猜想证明
1This hypothesis is in fact quite natural, because the arguments also show conversely that finite time extinction for Ricci flow is only possible when the fundamental group is a free product of finite groups and infinite simple groups. For the purposes of proving just the Poincar´e conjecture, though, it is possible to work entirely in the category of simply connected manifolds throughout, although this only simplifies the argument at one small point (in the proof of finite time extinction). We thank John Morgan for clarifying this point.
庞加莱猜想难题poincare
THE POINCAR´E CONJECTUREJOHN MILNOR1.IntroductionThe topology of two-dimensional manifolds or surfaces was well understood in the19th century.In fact there is a simple list of all possible smooth compact orientable surfaces.Any such surface has a well-defined genus g≥0,which can be described intuitively as the number of holes;and two such surfaces can be put into a smooth one-to-one correspondence with each other if and only if they have the same genus.1The corresponding question in higher dimensions is much moreFigure1.Sketches of smooth surfaces of genus0,1,and2.difficult.Henri Poincar´e was perhaps thefirst to try to make a similar study of three-dimensional manifolds.The most basic example of such a manifold is the three-dimensional unit sphere,that is,the locus of all points(x,y,z,w)in four-dimensional Euclidean space which have distance exactly1from the origin: x2+y2+z2+w2=1.He noted that a distinguishing feature of the two-dimensional sphere is that every simple closed curve in the sphere can be deformed continuously to a point without leaving the sphere.In1904,he asked a corresponding question in dimension3.In more modern language,it can be phrased as follows:2 Question.If a compact three-dimensional manifold M3has the property that every simple closed curve within the manifold can be deformed continuously to a point, does it follow that M3is homeomorphic to the sphere S3?He commented,with considerable foresight,“Mais cette question nous entraˆıne-rait trop loin”.Since then,the hypothesis that every simply connected closed 3-manifold is homeomorphic to the3-sphere has been known as the Poincar´e Con-jecture.It has inspired topologists ever since,and attempts to prove it have led to many advances in our understanding of the topology of manifolds.1For definitions and other background material,see,for example,[21]or[29],as well as[48].2See[36,pages498and370].To Poincar´e,manifolds were always smooth or polyhedral,so that his term“homeomorphism”referred to a smooth or piecewise linear homeomorphism.12JOHN MILNOR2.Early MisstepsFrom thefirst,the apparently simple nature of this statement has led mathe-maticians to overreach.Four years earlier,in1900,Poincar´e himself had been the first to err,stating a false theorem that can be phrased as follows.False Theorem.Every compact polyhedral manifold with the homology of an n-dimensional sphere is actually homeomorphic to the n-dimensional sphere.But his1904paper provided a beautiful counterexample to this claim,based on the concept of fundamental group,which he had introduced earlier(see[36, pp.189–192and193–288]).This example can be described geometrically as fol-lows.Consider all possible regular icosahedra inscribed in the two-dimensional unit sphere.In order to specify one particular icosahedron in this family,we must provide three parameters.For example,two parameters are needed to specify a single vertex on the sphere,and then another parameter to specify the direction to a neighboring vertex.Thus each such icosahedron can be considered as a single “point”in the three-dimensional manifold M3consisting of all such icosahedra.3 This manifold meets Poincar´e’s preliminary criterion:By the methods of homology theory,it cannot be distinguished from the three-dimensional sphere.However,he could prove that it is not a sphere by constructing a simple closed curve that cannot be deformed to a point within M3.The construction is not difficult:Choose some representative icosahedron and consider its images under rotation about one vertex through angles0≤θ≤2π/5.This defines a simple closed curve in M3that cannot be deformed to a point.Figure2.The Whitehead linkThe next important false theorem was by Henry Whitehead in1934[52].As part of a purported proof of the Poincar´e Conjecture,he claimed the sharper state-ment that every open three-dimensional manifold that is contractible(that can be continuously deformed to a point)is homeomorphic to Euclidean space.Following in Poincar´e’s footsteps,he then substantially increased our understanding of the topology of manifolds by discovering a counterexample to his own theorem.His counterexample can be briefly described as follows.Start with two disjoint solidtori T0and T1in the3-sphere that are embedded as shown in Figure2,so that each one individually is unknotted,but so that the two are linked together withlinking number zero.Since T1is unknotted,its complement T1=S3 interior( T1)3In more technical language,this M3can be defined as the coset space SO(3)/I60where SO(3) is the group of all rotations of Euclidean3-space and where I60is the subgroup consisting of the60 rotations that carry a standard icosahedron to itself.The fundamental groupπ1(M3),consisting of all homotopy classes of loops from a point to itself within M3,is a perfect group of order120.THE POINCAR ´E CONJECTURE 3is another unknotted solid torus that contains T 0.Choose a homeomorphism h of the 3-sphere that maps T 0onto this larger solid torus T 1.Then we can inductively construct solid toriT 0⊂T 1⊂T 2⊂···in S 3by setting T j +1=h (T j ).The union M 3= T j of this increasing sequence isthe required Whitehead counterexample,a contractible manifold that is not home-omorphic to Euclidean space.To see that π1(M 3)=0,note that every closed loop in T 0can be shrunk to a point (after perhaps crossing through itself)within the larger solid torus T 1.But every closed loop in M 3must be contained in some T j ,and hence can be shrunk to a point within T j +1⊂M 3.On the other hand,M 3is not homeomorphic to Euclidean 3-space since,if K ⊂M 3is any compact subset large enough to contain T 0,one can prove that the difference set M 3 K is not simply connected.Since this time,many false proofs of the Poincar´e Conjecture have been proposed,some of them relying on errors that are rather subtle and difficult to detect.For a delightful presentation of some of the pitfalls of three-dimensional topology,see [4].3.Higher DimensionsThe late 1950s and early 1960s saw an avalanche of progress with the discovery that higher-dimensional manifolds are actually easier to work with than three-dimensional ones.One reason for this is the following:The fundamental group plays an important role in all dimensions even when it is trivial,and relations between generators of the fundamental group correspond to two-dimensional disks,mapped into the manifold.In dimension 5or greater,such disks can be put into general position so that they are disjoint from each other,with no self-intersections,but in dimension 3or 4it may not be possible to avoid intersections,leading to serious difficulties.Stephen Smale announced a proof of the Poincar´e Conjecture in high dimensions in 1960[41].He was quickly followed by John Stallings,who used a completely different method [43],and by Andrew Wallace,who had been working along lines quite similar to those of Smale [51].Let me first describe the Stallings result,which has a weaker hypothesis and easier proof,but also a weaker conclusion.He assumed that the dimension is seven or more,but Christopher Zeeman later extended his argument to dimensions 5and 6[54].Stallings–Zeeman Theorem.If M n is a finite simplicial complex of dimension n ≥5that has the homotopy type 4of the sphere S n and is locally piecewise linearly homeomorphic to the Euclidean space R n ,then M n is homeomorphic to S n under a homeomorphism that is piecewise linear except at a single point.In other words,the complement M n (point )is piecewise linearly homeomorphic to R n .The method of proof consists of pushing all of the difficulties offtoward a single point;hence there can be no control near that point.4In order to check that a manifold M n has the same homotopy type as the sphere S n ,we must check not only that it is simply connected,π1(M n )=0,but also that it has the same homology as the sphere.The example of the product S 2×S 2shows that it is not enough to assume that π1(M n )=0when n >3.4JOHN MILNORThe Smale proof,and the closely related proof given shortly afterward by Wal-lace,depended rather on differentiable methods,building a manifold up inductively, starting with an n-dimensional ball,by successively adding handles.Here a k-handle can be added to a manifold M n with boundary byfirst attaching a k-dimensional cell,using an attaching homeomorphism from the(k−1)-dimensional boundary sphere into the boundary of M n,and then thickening and smoothing corners so as to obtain a larger manifold with boundary.The proof is carried out by rearranging and canceling such handles.(Compare the presentation in[24].)Figure3.A three-dimensional ball with a1-handle attachedSmale Theorem.If M n is a differentiable homotopy sphere of dimension n≥5, then M n is homeomorphic to S n.In fact,M n is diffeomorphic to a manifold obtained by gluing together the boundaries of two closed n-balls under a suitable diffeomorphism.This was also proved by Wallace,at least for n≥6.(It should be noted that thefive-dimensional case is particularly difficult.)The much more difficult four-dimensional case had to wait twenty years,for the work of Michael Freedman[8].Here the differentiable methods used by Smale and Wallace and the piecewise linear methods used by Stallings and Zeeman do not work at all.Freedman used wildly non-differentiable methods,not only to prove the four-dimensional Poincar´e Conjecture for topological manifolds,but also to give a complete classification of all closed simply connected topological4-manifolds.The integral cohomology group H2of such a manifold is free abelian.Freedman needed just two invariants:The cup productβ:H2⊗H2→H4∼=Z is a symmetric bilinear form with determinant±1,while the Kirby–Siebenmann invariantκis an integer mod2that vanishes if and only if the product manifold M4×R can be given a differentiable structure.Freedman Theorem.Two closed simply connected4-manifolds are homeomor-phic if and only if they have the same bilinear formβand the same Kirby–Sieben-mann invariantκ.Anyβcan be realized by such a manifold.Ifβ(x⊗x)is odd for some x∈H2,then either value ofκcan be realized also.However,ifβ(x⊗x) is always even,thenκis determined byβ,being congruent to one eighth of the signature ofβ.THE POINCAR´E CONJECTURE5 In particular,if M4is a homotopy sphere,then H2=0andκ=0,so M4 is homeomorphic to S4.It should be noted that the piecewise linear or differen-tiable theories in dimension4are much more difficult.It is not known whether every smooth homotopy4-sphere is diffeomorphic to S4;it is not known which4-manifolds withκ=0actually possess differentiable structures;and it is not known when this structure is essentially unique.The major results on these questions are due to Simon Donaldson[7].As one indication of the complications,Freedman showed,using Donaldson’s work,that R4admits uncountably many inequivalent differentiable structures.(Compare[12].)In dimension3,the discrepancies between topological,piecewise linear,and dif-ferentiable theories disappear(see[18],[28],and[26]).However,difficulties with the fundamental group become severe.4.The Thurston Geometrization ConjectureIn the two-dimensional case,each smooth compact surface can be given a beauti-ful geometrical structure,as a round sphere in the genus zero case,as aflat torus in the genus1case,and as a surface of constant negative curvature when the genus is2 or more.A far-reaching conjecture by William Thurston in1983claims that some-thing similar is true in dimension3[46].This conjecture asserts that every compact orientable three-dimensional manifold can be cut up along2-spheres and tori so as to decompose into essentially unique pieces,each of which has a simple geometri-cal structure.There are eight possible three-dimensional geometries in Thurston’s program.Six of these are now well understood,5and there has been a great deal of progress with the geometry of constant negative curvature.6The eighth geometry, however,corresponding to constant positive curvature,remains largely untouched. For this geometry,we have the following extension of the Poincar´e Conjecture. Thurston Elliptization Conjecture.Every closed3-manifold withfinite funda-mental group has a metric of constant positive curvature and hence is homeomorphic to a quotient S3/Γ,whereΓ⊂SO(4)is afinite group of rotations that acts freely on S3.The Poincar´e Conjecture corresponds to the special case where the groupΓ∼=π1(M3)is trivial.The possible subgroupsΓ⊂SO(4)were classified long ago by [19](compare[23]),but this conjecture remains wide open.5.Approaches through Differential Geometryand Differential Equations7In recent years there have been several attacks on the geometrization problem (and hence on the Poincar´e Conjecture)based on a study of the geometry of the infinite dimensional space consisting of all Riemannian metrics on a given smooth three-dimensional manifold.5See,for example,[13],[3],[38,39,40],[49],[9],and[6].6See[44],[27],[47],[22],and[30].The pioneering papers by[14]and[50]provided the basis for much of this work.7Added in20046JOHN MILNORBy definition,the length of a path γon a Riemannian manifold is computed,in terms of the metric tensor g ij ,as the integral γds = γ g ij dx i dx j .From the first and second derivatives of this metric tensor,one can compute the Ricci curvature tensor R ij ,and the scalar curvature R .(As an example,for the flat Euclidean space one gets R ij =R =0,while for a round three-dimensional sphere of radius r ,one gets Ricci curvature R ij =2g ij /r 2and scalar curvature R =6/r 2.)One approach by Michael Anderson,based on ideas of Hidehiko Yamabe [53],studies the total scalar curvature M 3R dV as a functional on the space of all smooth unit volume Riemannian metrics.The critical points of this functional are the metrics of constant curvature (see [1]).A different approach,initiated by Richard Hamilton studies the Ricci flow [15,16,17],that is,the solutions to the differential equationdg ij dt=−2R ij .In other words,the metric is required to change with time so that distances de-crease in directions of positive curvature.This is essentially a parabolic differential equationa and behaves much like the heat equation studied by physicists:If we heat one end of a cold rod,then the heat will gradually flow throughout the rod until it attains an even temperature.Similarly,a naive hope for 3-manifolds with finite fundamental group might have been that,under the Ricci flow,positive curvature would tend to spread out until,in the limit (after rescaling to constant size),the manifold would attain constant curvature.If we start with a 3-manifold of posi-tive Ricci curvature,Hamilton was able to carry out this program and construct a metric of constant curvature,thus solving a very special case of the Elliptization Conjecture.However,in the general case,there are very serious difficulties,since this flow may tend toward singularities.8I want to thank many mathematicians who helped me with this report.May 2000,revised June 2004References[1]M.T.Anderson,Scalar curvature,metric degenerations and the static vacuum Einstein equa-tions on 3-manifolds ,Geom.Funct.Anal.9(1999),855–963and 11(2001)273–381.See also:Scalar curvature and the existence of geometric structures on 3-manifolds ,J.reine angew.Math.553(2002),125–182and 563(2003),115–195.[2]M.T.Anderson,Geometrization of 3-manifolds via the Ricci flow ,Notices AMS 51(2004),184–193.[3]L.Auslander and F.E.A.Johnson,On a conjecture of C.T.C.Wall ,J.Lond.Math.Soc.14(1976),331–332.[4]R.H.Bing,Some aspects of the topology of 3-manifolds related to the Poincar´e conjecture ,in Lectures on Modern Mathematics II (T.L.Saaty,ed.),Wiley,New York,1964.[5]J.Birman,Poincar´e ’s conjecture and the homeotopy group of a closed,orientable 2-manifold ,J.Austral.Math.Soc.17(1974),214–221.8Grisha Perelman,in St.Petersburg,has posted three preprints on which go a long way toward resolving these difficulties,and in fact claim to prove the full geometrization conjecture[32,33,34].These preprints have generated a great deal of interest.(Compare [2]and [25],as well as the website /research/ricciflow/perelman.html organized by B.Kleiner and J.Lott.)However,full details have not appeared.THE POINCAR´E CONJECTURE7 [6] A.Casson and D.Jungreis,Convergence groups and Seifertfibered3-manifolds,Invent.Math.118(1994),441–456.[7]S.K.Donaldson,Self-dual connections and the topology of smooth4-manifolds,Bull.Amer.Math.Soc.8(1983),81–83.[8]M.H.Freedman,The topology of four-dimensional manifolds,J.Diff.Geom.17(1982),357–453.[9] D.Gabai,Convergence groups are Fuchsian groups,Ann.Math.136(1992),447–510.[10] D.Gabai,Valentin Poenaru’s program for the Poincar´e conjecture,in Geometry,topology,&physics,Conf.Proc.Lecture Notes Geom.Topology,VI,Internat.Press,Cambridge,MA, 1995,139–166.[11] D.Gillman and D.Rolfsen,The Zeeman conjecture for standard spines is equivalent to thePoincar´e conjecture,Topology22(1983),315–323.[12]R.Gompf,An exotic menagerie,J.Differential Geom.37(1993)199–223.[13] C.Gordon and W.Heil,Cyclic normal subgroups of fundamental groups of3-manifolds,Topology14(1975),305–309.[14]W.Haken,¨Uber das Hom¨o omorphieproblem der3-Mannigfaltigkeiten I,Math.Z.80(1962),89–120.[15]R.S.Hamilton,Three-manifolds with positive Ricci curvature,J.Differential Geom.17(1982),255–306.[16]R.S.Hamilton,The formation of singularities in the Ricciflow,in Surveys in differentialgeometry,Vol.II(Cambridge,MA,1993),Internat.Press,Cambridge,MA,1995,7–136. [17]R.S.Hamilton,Non-singular solutions of the Ricciflow on three-manifolds Comm.Anal.Geom.7(1999),695–729.[18]M.Hirsch,Obstruction theories for smoothing manifolds and maps,Bull.Amer.Math.Soc.69(1963),352-356.[19]H.Hopf,Zum Clifford–Kleinschen Raumproblem,Math.Ann.95(1925-26)313-319.[20]W.Jakobsche,The Bing-Borsuk conjecture is stronger than the Poincar´e conjecture,Fund.Math.106(1980),127–134.[21]W.S.Massey,Algebraic Topology:An Introduction,Harcourt Brace,New York,1967;Springer,New York1977;or A Basic Course in Algebraic Topology,Springer,New York, 1991.[22] C.McMullen,Riemann surfaces and geometrization of3-manifolds,Bull.Amer.Math.Soc.27(1992),207–216.[23]nor,Groups which act on S n withoutfixed points,Amer.J.Math.79(1957),623–630.[24]nor(with L.Siebenmann and J.Sondow),Lectures on the h-Cobordism Theorem,Princeton Math.Notes,Princeton University Press,Princeton,1965.[25]nor,Towards the Poincar´e conjecture and the classification of3-manifolds,NoticesAMS50(2003),1226–1233.[26] E.E.Moise,Geometric Topology in Dimensions2and3,Springer,New York,1977.[27]J.Morgan,On Thurston’s uniformization theorem for three-dimensional manifolds,in TheSmith Conjecture(H.Bass and J.Morgan,eds.),Pure and Appl.Math.112,Academic Press, New York,1984,37–125.[28]J.Munkres,Obstructions to the smoothing of piecewise-differentiable homeomorphisms,Ann.Math.72(1960),521–554.[29]J.Munkres,Topology:A First Course,Prentice–Hall,Englewood Cliffs,NJ,1975.[30]J.-P.Otal,The hyperbolization theorem forfibered3-manifolds,translated from the1996French original by Leslie D.Kay,SMF/AMS Texts and Monographs7,American Mathemat-ical Society,Providence,RI;Soci´e t´e Mathatique de France,Paris,2001.[31] C.Papakyriakopoulos,A reduction of the Poincar´e conjecture to group theoretic conjectures,Ann.Math.77(1963),250–305.[32]G.Perelman,The entropy formula for the Ricciflow and its geometric applications,arXiv:math.DG/0211159v1,11Nov2002.[33]G.Perelman,Ricciflow with surgery on three-manifolds,arXiv:math.DG/0303109,10Mar2003.[34]G.Perelman,Finite extinction time for the solutions to the Ricciflow on certain three-manifolds,arXiv:math.DG/0307245,17Jul2003.8JOHN MILNOR[35]V.Po´e naru,A program for the Poincar´e conjecture and some of its ramifications,in Topicsin low-dimensional topology(University Park,PA,1996),World Sci.Publishing,River Edge, NJ,1999,65–88.[36]H.Poincar´e,Œuvres,Tome VI,Gauthier–Villars,Paris,1953.[37] C.Rourke,Algorithms to disprove the Poincar´e conjecture,Turkish J.Math.21(1997),99–110.[38]P.Scott,A new proof of the annulus and torus theorems,Amer.J.Math.102(1980),241–277.[39]P.Scott,There are no fake Seifertfibre spaces with infiniteπ1,Ann.Math.117(1983),35–70.[40]P.Scott,The geometries of3-manifolds,Bull.Lond.Math.Soc.15(1983),401–487.[41]S.Smale,Generalized Poincar´e’s conjecture in dimensions greater than four,Ann.Math.74(1961),391–406.(See also:Bull.Amer.Math.Soc.66(1960),373–375.)[42]S.Smale,The story of the higher dimensional Poincar´e conjecture(What actually happenedon the beaches of Rio),Math.Intelligencer12,no.2(1990),44–51.[43]J.Stallings,Polyhedral homotopy spheres,Bull.Amer.Math.Soc.66(1960),485–488.[44] D.Sullivan,Travaux de Thurston sur les groupes quasi-fuchsiens et sur les vari´e t´e s hyper-boliques de dimension3fibr´e es sur le cercle,S´e m.Bourbaki554,Lecture Notes Math.842, Springer,New York,1981.[45]T.L.Thickstun,Open acyclic3-manifolds,a loop theorem and the Poincar´e conjecture,Bull.Amer.Math.Soc.(N.S.)4(1981),192–194.[46]W.P.Thurston,Three dimensional manifolds,Kleinian groups and hyperbolic geometry,inThe Mathematical heritage of Henri Poincar´e,Proc.Symp.Pure Math.39(1983),Part1.(Also in Bull.Amer.Math.Soc.6(1982),357–381.)[47]W.P.Thurston,Hyperbolic structures on3-manifolds,I,deformation of acyclic manifolds,Ann.Math.124(1986),203–246[48]W.P.Thurston,Three-Dimensional Geometry and Topology,Vol.1,ed.by Silvio Levy,Princeton Mathematical Series35,Princeton University Press,Princeton,1997.[49]P.Tukia,Homeomorphic conjugates of Fuchsian groups,J.Reine Angew.Math.391(1988),1–54.[50] F.Waldhausen,On irreducible3-manifolds which are sufficiently large,Ann.Math.87(1968),56–88.[51] A.Wallace,Modifications and cobounding manifolds,II,J.Math.Mech10(1961),773–809.[52]J.H.C.Whitehead,Mathematical Works,Volume II,Pergamon Press,New York,1962.(Seepages21-50.)[53]H.Yamabe,On a deformation of Riemannian structures on compact manifolds,Osaka Math.J.12(1960),21–37.[54] E.C.Zeeman,The Poincar´e conjecture for n≥5,in Topology of3-Manifolds and RelatedTopics Prentice–Hall,Englewood Cliffs,NJ,1962,198–204.(See also Bull.Amer.Math.Soc.67(1961),270.)(Note:For a representative collection of attacks on the Poincar´e Conjecture,see [31],[5],[20],[45],[11],[10],[37],and[35].)。
庞加莱猜想的由来与解决
Ｐｏｉｎｃａｒｅ´ 猜测漫谈 梅加强 （南京大学数学系） 一、 引言 Ｐｏｉｎｃａｒｅ′（庞加莱），法国人，伟大的数学家。
他 １８５４ 年出生于 Ｎａｎｃｙ，１９１２ 年 逝世于 Ｐａｒｉｓ（巴黎）。
Ｐｏｉｎｃａｒｅ′ 对好几个科学分支，例如分析、数论、拓扑以及天体物理都有重要贡献，代数拓扑学也是他发明的。
（Ｐｏｉｎｃａｒｅ′， １８５４－１９１２） １９０４ 年，Ｐｏｉｎｃａｒｅ′ 在他的著作 Cinqui`eme Compl e´ment `a L’Analysis Situs提出了下面的问题：单连通的闭的３维流形是否同胚于３维球面？ 这个问题后来被人们称为 Ｐｏｉｎｃａｒｅ′ 猜测，这个猜测直到１００年以后的今天才被俄罗斯数学家Ｐｅｒｅｌｍａｎ 解决。
在本次讲座中，我们的目的是围绕着 Ｐｏｉｎｃａｒｅ′ 猜测介绍一些近代数学的基本概念，例如，在该猜测的表述中，象“单连通”、“闭”、“３维”、“流形”、“同胚”、“３维球面”等这些名词都是什么意思呢？这些名词都是现代几何学和拓扑学中的概念，要真正弄懂它们是很不容易的，我们希望通过举一些例子来使大家对这些概念有一个初步的印象。
如果有同学能对此感兴趣，以后在大学里能进一步学习深造并对数学有所贡献那是再好不过的了。
二、 什么是拓扑 Ｐｏｉｎｃａｒｅ′ 猜测是拓扑学（Ｔｏｐｏｌｏｇｙ）范畴里的一个问题，什么是拓扑学，拓扑学研究哪些问题呢？要准确地回答这些疑问是困难的，我们先从一个例子开始。
１．哥尼斯堡七桥问题（Ｋｏｎｉｇｓｂｅｒｇ’ｓ Ｂｒｉｄｇｅ Ｐｒｏｂｌｅｍ） １８世纪在哥尼斯堡城（今俄罗斯加里宁格勒）的普莱格尔河上有７座桥，将河中的两个岛和河岸连结，城中的居民经常沿河过桥散步，于是提出了一个问题：能否一次走遍７座桥，而每座桥只许通过一次，最后仍回到起始地点，这就是七桥问题。
七桥问题这个问题人们试了很多办法都没能解决，直到引起了瑞士数学家Ｅｕｌｅｒ（欧拉）的注意。
探索探险实现梦想
探索探险实现梦想
佚名
【期刊名称】《科技中国》
【年(卷),期】2007(000)002
【摘要】俄罗斯数学家格里高里·佩雷尔曼于2002年11月首次发表了自己关于证明“庞加莱猜想”的文章，随后他又陆续发表了余下的两篇文章。
经过近4年的工作，数学家在2006年终于达成一致意见认定佩雷尔曼的研究解答了数学界最古老的问题之一。
对于他们来说，俄罗斯数学家格里高里·佩雷尔曼对“庞加莱猜想”的证明可以称为是10年来的突破。
这项研究在美国《科学》杂志2006年12月22日公布的2006年度十大科学进展中名列榜首。
【总页数】30页(P26-55)
【正文语种】中文
【中图分类】N8
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evolution
equation
d dt
gij
(t)
=
−2Rij
for
a
riemannian
metric
gij (t).
In
his
seminal paper, Hamilton proved that this equation has a unique solution for
a short time for an arbitrary (smooth) metric on a closed manifold. The
and of the curvature operator in dimension four are getting pinched point-
ห้องสมุดไป่ตู้
wisely as the curvature is getting large. This observation allowed him to
prove the convergence results: the evolving metrics (on a closed manifold) of
1
in dimension four converge, modulo scaling, to metrics of constant positive curvature.
Without assumptions on curvature the long time behavior of the metric evolving by Ricci flow may be more complicated. In particular, as t approaches some finite time T, the curvatures may become arbitrarily large in some region while staying bounded in its complement. In such a case, it is useful to look at the blow up of the solution for t close to T at a point where curvature is large (the time is scaled with the same factor as the metric tensor). Hamilton [H 9] proved a convergence theorem , which implies that a subsequence of such scalings smoothly converges (modulo diffeomorphisms) to a complete solution to the Ricci flow whenever the curvatures of the scaled metrics are uniformly bounded (on some time interval), and their injectivity radii at the origin are bounded away from zero; moreover, if the size of the scaled time interval goes to infinity, then the limit solution is ancient, that is defined on a time interval of the form (−∞, T ). In general it may be hard to analyze an arbitrary ancient solution. However, Ivey [I] and Hamilton [H 4] proved that in dimension three, at the points where scalar curvature is large, the negative part of the curvature tensor is small compared to the scalar curvature, and therefore the blow-up limits have necessarily nonnegative sectional curvature. On the other hand, Hamilton [H 3] discovered a remarkable property of solutions with nonnegative curvature operator in arbitrary dimension, called a differential Harnack inequality, which allows, in particular, to compare the curvatures of the solution at different points and different times. These results lead Hamilton to certain conjectures on the structure of the blow-up limits in dimension three, see [H 4,§26]; the present work confirms them.
of the Ricci tensor in dimension three and of the curvature operator in all
dimensions; moreover, the eigenvalues of the Ricci tensor in dimension three
positive Ricci curvature in dimension three, or positive curvature operator
∗St.Petersburg branch of Steklov Mathematical Institute, Fontanka 27, St.Petersburg 191011, Russia. Email: perelman@pdmi.ras.ru or perelman@ ; I was partially supported by personal savings accumulated during my visits to the Courant Institute in the Fall of 1992, to the SUNY at Stony Brook in the Spring of 1993, and to the UC at Berkeley as a Miller Fellow in 1993-95. I’d like to thank everyone who worked to make those opportunities available to me.
The most natural way of forming a singularity in finite time is by pinching an (almost) round cylindrical neck. In this case it is natural to make a surgery by cutting open the neck and gluing small caps to each of the boundaries, and then to continue running the Ricci flow. The exact procedure was described by Hamilton [H 5] in the case of four-manifolds, satisfying certain curvature assumptions. He also expressed the hope that a similar procedure would work in the three dimensional case, without any a priory assumptions, and that after finite number of surgeries, the Ricci flow would exist for all time t → ∞, and be nonsingular, in the sense that the normalized curvatures R˜m(x, t) = tRm(x, t) would stay bounded. The topology of such nonsingular solutions was described by Hamilton [H 6] to the extent sufficient to make sure that no counterexample to the Thurston geometrization conjecture can
R satisfies Rt = △R + 2|Ric|2, so by the maximum principle its minimum is non-decreasing along the flow. By developing a maximum principle for
tensors, Hamilton [H 1,H 2] proved that Ricci flow preserves the positivity
arXiv:math.DG/0211159 v1 11 Nov 2002
The entropy formula for the Ricci flow and its geometric applications
Grisha Perelman∗ May 21, 2006
Introduction
1. The Ricci flow equation, introduced by Richard Hamilton [H 1], is the