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1、Ricciflowwithsurgeryonthree-manifoldsGrishaPerelman∗February1,2008Thisisatechnicalpaper,whichisacontinuationof[I].Hereweverifymostoftheassertions,madein[I,§13];theexceptionsare(1)thestatementthata3-manifoldwhichcollapseswithlocallowerboundforsectionalc
2、urvatureisagraphmanifold-thisisdeferredtoaseparatepaper,astheproofhasnothingtodowiththeRicciflow,and(2)theclaimaboutthelowerboundforthevolumesofthemaximalhornsandthesmoothnessofthesolutionfromsometimeon,whichturnedouttobeunjustified,and,ontheotherhand,i
3、rrelevantfortheotherconclusions.TheRicciflowwithsurgerywasconsideredbyHamilton[H5,§4,5];unfortu-nately,hisargument,aswritten,containsanunjustifiedstatement(RMAX=Γ,onpage62,lines7-10fromthebottom),whichIwasunabletofix.Ourapproachissomewhatdifferent,andisai
4、medateventuallyconstructingacanonicalRicciflow,definedonalargestpossiblesubsetofspace-time,-agoal,thathasnotbeenachievedyetinthepresentwork.Forthisreason,weconsidertwoscalebounds:thecutoffradiush,whichistheradiusofthenecks,wherethesurg-eriesareperformed,
5、andthemuchlargerradiusr,suchthatthesolutiononthescaleslessthanrhasstandardgeometry.Thepointistomakeharbitrarilysmallwhilekeepingrboundedawayfromzero.NotationandterminologyB(x,t,r)denotestheopenmetricballofradiusr,withrespecttothemetricattimet,centered
6、atx.P(x,t,r,△t)denotesaparabolicneighborhood,thatisthesetofallpointsarXiv:math/0303109v1[math.DG]10Mar2003(x′,t′)withx′∈B(x,t,r)andt′∈[t,t+△t]ort′∈[t+△t,t],dependingonthesignof△t.AballB(x,t,ǫ−1r)iscalledanǫ-neck,if,afterscalingthemetricwithfactorr−2,i
7、tisǫ-closetothestandardneckS2×I,withtheproductmetric,whereS2hasconstantscalarcurvatureone,andIhaslength2ǫ−1;hereǫ-closereferstoCNtopology,withN>ǫ−1.AparabolicneighborhoodP(x,t,ǫ−1r,r2)iscalledastrongǫ-neck,if,afterscalingwithfactorr−2,itisǫ-closetothe
8、evolvingstandardneck,whichateach∗St.PetersburgbranchofSteklovMathematicalInstitute,Fontanka27,St.Petersburg191011,Russia.Email:perelman@pdmi.ras.ruorperelman@math.sunysb.edu1timet′∈[−1,0]haslength2ǫ−1andscalarcurvature(1−t′)−1.AmetriconS2×I,su
