A Brief Description Of The Proof Of Poincaré Conjecture

In this thesis, we mainly discuss how Huaidong Cao and Xiping Zhu use the method of Ricci flow to prove Poincare conjecture. We can get the short-time existence of the to Ricci Flow by computing the evovling equation of the metric and De Turck change.We can get the evovling equation of curvature by computing,and hence we can get the derivative estimate of curvature of Shi.The evovling equation of the defined Mαβwhich is relevant to curvature satisfies the conditon of Hamilton's Strong Maximum Principle,so Mαβsatisfies Hamilton's Strong Maximum Principle.Futhermore,the scalar curvature also satisfies the Li-Yau-Hamilton inequality.In order to prove the no local collapsing theorem,Perelman use the notions of L length and Perelman's Reduced Volume.The nondecreasing of Perelman's Reduced Volume can prove no local collapsing theorem. And we can get local injective radius estimate by no local collaps-ing theorem, the local injective radius estimate is called Little Loop Lemma. In order to get the limit of the sequence of evolving marked complete Manifolds, Hamilton proved Hamilton's compactness theorem,which requires uniform lower bound of injective radius of the sequence of marked Manifolds,no local collapsing theorem deal well with the problem. Hamilton's com-pactness theorem is very useful in proving the sphere theorems and canonical neighborhood theorem. Moreover,Hamilton establish the singularity models of Ricci flow.The maximal solu-tions have three types, corresponding to the three types of singularity models,Hamilton also get the properties of each singularity model.Hamilton proved the 3-D sphere theorem in which the compact three-manifolds with posi-tive Ricci curvature are diffeomorphic to S3 or its metric quotients.Besides,three-manifolds sat-isfy the pinching estimate, for which the limit solution of three-manifolds must have nonnegetive curvature operator.Now we consider the 3-D ancientκ-solutions.On the one hand, Perelman tell us the ancient K-solutions have good properties such as elliptic type estimate and canonical neighborhood theo-rem, canonical neighborhood theorem tells us every time-space point has one of the three types of open neighborhoods including spherical,necklike and caplike neighborhoods;On the other hand, Perelman show the singularity structure of Ricci flow,when the curvature is high enough ,it has similar structure of ancient K-solutions,thus the singular solution also have properties such as derivative estimate and canonical neighborhood theorem.Then we will talk about the main part of the thesis after the preparations above. Huaidong Cao and Xiping Zhu show how to perform surgeries to the Ricci flow of compact orientable three-manifolds,the procedures include cutoff surgery and removing some compact components. More-over,we need to choose proper sugeries and radius of canonical neighborhood assumption so that the solutions not only satisfy pinching assumption and canonical neighborhood assumption but also do finite sugeries in finite time interval. Finally,we will get the long-time existing theorem proposed by Perelman. The solution of Ricci flow with surgery becomes extinct at finite time by Colding-Minicozi for compact simply connected three-manifolds,thus by using the the long-time existing theorem they are diffeomorphic to S3.