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..................120.2.

2、2Manifoldsofnon-negativecurvature..............130.2.3CanonicalNeighborhoods....................130.3BackgroundMaterialfromRicciFlow.................150.3.1Firstresults............................150.3.2GradientShrinkingSolitons...................160.3.3Controllinghigher

3、derivativesofcurvature...........160.3.4GeneralizedRicciflows.....................160.3.5Themaximumprinciple.....................180.3.6GeometricLimits.........................190.4Perelman’sadvances...........................200.4.1Thereducedlengthfunction...............

4、....200.4.2Applicationtonon-collapsingresults..............210.4.3Applicationtoancientκ-non-collapsedsolutions.......220.4.4BoundedCurvatureatBoundedDistance...........240.5Thestandardsolutionandthesurgeryprocess............250.5.1Thestandardsolution.....................

5、.250.5.2RicciFlowswithsurgery.....................260.5.3TheInductiveConditionsNecessaryfordoingSurgery....270.5.4Surgery..............................280.5.5Topologicaleffectofsurgery...................280.6SurgeryandcanonicalneighborhoodsforgeneralizedRicciflows...290.7F

6、initeExtinction.............................310.8Acknowledgements............................340.9Listofrelatedpapers...........................351PreliminariesfromRiemannianGeometry361.1RiemannianmetricsandtheLevi-Civit´aconnection.........361.2Curvatureofariemannianman

7、ifold..................381.2.1ConsequencesoftheBianchiidentities.............401.2.2FirstExamples..........................411.3Cones...................................421.4Geodesicsandtheexponentialmap...................431.4.1Geodesicsandtheenergyfunctional............

8、..431.4.2FamiliesofgeodesicsandJacobifields.............441.4.3Minimalgeodesics........................4