morgan, j & tian, g ricci flow and the poincare conjecture (2007) ams外语英文电子书

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1、RicciFlowandthePoincar´eConjectureJohnW.MorganandGangTianarXiv:math/0607607v2[math.DG]21Mar2007AMSsubjectclassification:53C44,57M40.ContentsIntroduction51.OverviewofPerelman’sargument62.BackgroundmaterialfromRiemanniangeometry113.BackgroundmaterialfromRicciflow144.Perelman’sadvanc

2、es195.Thestandardsolutionandthesurgeryprocess246.ExtendingRicciflowswithsurgery277.Finite-timeextinction308.Acknowledgements339.Listofrelatedpapers34Chapter1.PreliminariesfromRiemanniangeometry351.RiemannianmetricsandtheLevi-Civitaconnection352.CurvatureofaRiemannianmanifold373.G

3、eodesicsandtheexponentialmap414.ComputationsinGaussiannormalcoordinates465.Basiccurvaturecomparisonresults486.Localvolumeandtheinjectivityradius50Chapter2.Manifoldsofnon-negativecurvature511.Busemannfunctions512.Comparisonresultsinnon-negativecurvature523.Thesoultheorem544.Endso

4、famanifold575.Thesplittingtheorem576.ǫ-necks597.Forwarddifferencequotients61Chapter3.BasicsofRicciflow631.ThedefinitionoftheRicciflow632.SomeexactsolutionstotheRicciflow643.Localexistenceanduniqueness664.Evolutionofcurvatures685.Curvatureevolutioninanevolvingorthonormalframe696.Varia

5、tionofdistanceunderRicciflow737.Shi’sderivativeestimates778.GeneralizedRicciflows851Chapter4.Themaximumprinciple891.Maximumprincipleforscalarcurvature892.Themaximumprinciplefortensors913.Applicationsofthemaximumprinciple924.Thestrongmaximumprincipleforcurvature955.Pinchingtowardpo

6、sitivecurvature100Chapter5.ConvergenceresultsforRicciflow1071.GeometricconvergenceofRiemannianmanifolds1072.GeometricconvergenceofRicciflows1133.Gromov-Hausdorffconvergence1154.Blow-uplimits1215.Splittinglimitsatinfinity122Chapter6.AcomparisongeometryapproachtotheRicciflow1251.L-leng

7、thandL-geodesics1252.TheL-exponentialmapanditsfirst-orderproperties1313.MinimizingL-geodesicsandtheinjectivitydomain1354.Second-orderdifferentialinequalitiesforLeτandLτx1375.Reducedlength1466.LocalLipschitzestimatesforlx1507.Reducedvolume156Chapter7.CompleteRicciflowsofboundedcurva

8、ture1631.ThefunctionsLxandlx1632.Aboundforminlx