poincare conjectures-cao zhu

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1、ASIANJ.MATH.c2006InternationalPressVol.10,No.2,pp.165–492,June2006001ACOMPLETEPROOFOFTHEPOINCAREAND´GEOMETRIZATIONCONJECTURES–APPLICATIONOFTHEHAMILTON-PERELMANTHEORYOFTHERICCIFLOW∗HUAI-DONGCAO†ANDXI-PINGZHU‡Abstract.Inthispaper,wegiveacompleteproofofthePoincar´eandthegeometrizationconjec

2、tures.Thisworkdependsontheaccumulativeworksofmanygeometricanalystsinthepastthirtyyears.ThisproofshouldbeconsideredasthecrowningachievementoftheHamilton-PerelmantheoryofRicciflow.Keywords.Ricciflow,Ricciflowwithsurgery,Hamilton-Perelmantheory,Poincar´eConjec-ture,geometrizationof3-manifoldsA

3、MSsubjectclassifications.53C21,53C44CONTENTSIntroduction1671EvolutionEquations1721.1TheRicciFlow...............................1721.2Short-timeExistenceandUniqueness...................1771.3EvolutionofCurvatures..........................1831.4DerivativeEstimates...........................

4、.1901.5VariationalStructureandDynamicProperty..............1992MaximumPrincipleandLi-Yau-HamiltonInequalities2102.1PreservingPositiveCurvature.......................2102.2StrongMaximumPrinciple........................2132.3AdvancedMaximumPrincipleforTensors................2172.4Hamilton-

5、IveyCurvaturePinchingEstimate...............2232.5Li-Yau-HamiltonEstimates........................2262.6Perelman’sEstimateforConjugateHeatEquations...........2343Perelman’sReducedVolume2393.1RiemannianFormalisminPotentiallyInfiniteDimensions.......2393.2ComparisonTheoremsforPerelman’sRedu

6、cedVolume.........2433.3NoLocalCollapsingTheoremI......................2553.4NoLocalCollapsingTheoremII.....................2614FormationofSingularities2674.1CheegerTypeCompactness........................2674.2InjectivityRadiusEstimates........................2864.3LimitingSingularityMod

7、els........................2914.4RicciSolitons................................302∗ReceivedDecember12,2005;acceptedforpublicationApril16,2006.†DepartmentofMathematics,LehighUniversity,Bethlehem,PA18015,USA(huc2@lehigh.edu).‡DepartmentofMathematics,ZhongshanUniversity,Guang