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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´ea
2、ndthegeometrizationconjectures.Thisworkdependsontheaccumulativeworksofmanygeometricanalystsinthepastthirtyyears.ThisproofshouldbeconsideredasthecrowningachievementoftheHamilton-PerelmantheoryofRicciflow.Keywords.Ricciflow,Ricciflowwithsurgery,Hamilton-Perelmantheory
3、,Poincar´eConjec-ture,geometrizationof3-manifoldsAMSsubjectclassifications.53C21,53C44CONTENTSIntroduction1671EvolutionEquations1721.1TheRicciFlow...............................1721.2Short-timeExistenceandUniqueness...................1771.3EvolutionofCurvatures...
4、.......................1831.4DerivativeEstimates............................1901.5VariationalStructureandDynamicProperty..............1992MaximumPrincipleandLi-Yau-HamiltonInequalities2102.1PreservingPositiveCurvature.......................2102.2StrongMaximumPrin
5、ciple........................2132.3AdvancedMaximumPrincipleforTensors................2172.4Hamilton-IveyCurvaturePinchingEstimate...............2232.5Li-Yau-HamiltonEstimates........................2262.6Perelman’sEstimateforConjugateHeatEquations...........2343P
6、erelman’sReducedVolume2393.1RiemannianFormalisminPotentiallyInfiniteDimensions.......2393.2ComparisonTheoremsforPerelman’sReducedVolume.........2433.3NoLocalCollapsingTheoremI......................2553.4NoLocalCollapsingTheoremII.....................2614Formationo
7、fSingularities2674.1CheegerTypeCompactness........................2674.2InjectivityRadiusEstimates........................2864.3LimitingSingularityModels........................2914.4RicciSolitons................................302∗ReceivedDecember12,2005;accepte
8、dforpublicationApril16,2006.†DepartmentofMathematics,LehighUniversity,Bethlehem,PA18015,USA(huc2@lehigh.edu).‡DepartmentofMathematics,ZhongshanUniversity,Guang
