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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´eandthe
2、geometrizationconjectures.Thisworkdependsontheaccumulativeworksofmanygeometricanalystsinthepastthirtyyears.ThisproofshouldbeconsideredasthecrowningachievementoftheHamilton-PerelmantheoryofRicciflow.Keywords.Ricciflow,Ricciflowwithsurgery,Hamilton-Perelmantheory,Poincar´e
3、Conjec-ture,geometrizationof3-manifoldsAMSsubjectclassifications.53C21,53C44CONTENTSIntroduction1671EvolutionEquations1721.1TheRicciFlow...............................1721.2Short-timeExistenceandUniqueness...................1771.3EvolutionofCurvatures..................
4、........1831.4DerivativeEstimates............................1901.5VariationalStructureandDynamicProperty..............1992MaximumPrincipleandLi-Yau-HamiltonInequalities2102.1PreservingPositiveCurvature.......................2102.2StrongMaximumPrinciple...............
5、.........2132.3AdvancedMaximumPrincipleforTensors................2172.4Hamilton-IveyCurvaturePinchingEstimate...............2232.5Li-Yau-HamiltonEstimates........................2262.6Perelman’sEstimateforConjugateHeatEquations...........2343Perelman’sReducedVolume239
6、3.1RiemannianFormalisminPotentiallyInfiniteDimensions.......2393.2ComparisonTheoremsforPerelman’sReducedVolume.........2433.3NoLocalCollapsingTheoremI......................2553.4NoLocalCollapsingTheoremII.....................2614FormationofSingularities2674.1CheegerTyp
7、eCompactness........................2674.2InjectivityRadiusEstimates........................2864.3LimitingSingularityModels........................2914.4RicciSolitons................................302∗ReceivedDecember12,2005;acceptedforpublicationApril16,2006.†Depart
8、mentofMathematics,LehighUniversity,Bethlehem,PA18015,USA(huc2@lehigh.edu).‡DepartmentofMathematics,ZhongshanUniversity,Guang
