ricci flow and the poincare conjecture-morgan&gang tian

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1、RicciFlowandthePoincar´eConjectureJohnW.MorganandGangTianJuly25,2006arXiv:math.DG/0607607v125Jul2006Contents0.1OverviewofPerelman’sargument....................90.2Backgroundmaterialfromriemanniangeometry...........120.2.1Volumeandinjectivityradius........

2、..........120.2.2Manifoldsofnon-negativecurvature..............130.2.3CanonicalNeighborhoods....................130.3BackgroundMaterialfromRicciFlow.................150.3.1Firstresults............................150.3.2GradientShrinkingSolitons...........

3、........160.3.3Controllinghigherderivativesofcurvature...........160.3.4GeneralizedRicciflows.....................160.3.5Themaximumprinciple.....................180.3.6GeometricLimits.........................190.4Perelman’sadvances.........................

4、..200.4.1Thereducedlengthfunction...................200.4.2Applicationtonon-collapsingresults..............210.4.3Applicationtoancientκ-non-collapsedsolutions.......220.4.4BoundedCurvatureatBoundedDistance...........240.5Thestandardsolutionandthesurgerypr

5、ocess............250.5.1Thestandardsolution......................250.5.2RicciFlowswithsurgery.....................260.5.3TheInductiveConditionsNecessaryfordoingSurgery....270.5.4Surgery..............................280.5.5Topologicaleffectofsurgery........

6、...........280.6SurgeryandcanonicalneighborhoodsforgeneralizedRicciflows...290.7FiniteExtinction.............................310.8Acknowledgements............................340.9Listofrelatedpapers...........................351PreliminariesfromRiemannianG

7、eometry361.1RiemannianmetricsandtheLevi-Civit´aconnection.........361.2Curvatureofariemannianmanifold..................381.2.1ConsequencesoftheBianchiidentities.............401.2.2FirstExamples..........................411.3Cones..........................

8、.........421.4Geodesicsandtheexponentialmap...................431.4.1Geodesicsandtheenergyfunctional..............431.4.2FamiliesofgeodesicsandJacobifields.............441.4.3Minimalgeodesics........................4