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1、Ricci?owwithsurgeryonthree-manifoldsGrishaPerelman?August21,2006Thisisatechnicalpaper,whichisacontinuationof[I].Hereweverifymostoftheassertions,madein[I,§13];theexceptionsare(1)thestatementthata3-manifoldwhichcollapseswithlocallowerboundforsectionalcurvatureisagraphmanifold-thisisdeferre
2、dtoaseparatepaper,astheproofhasnothingtodowiththeRicci?ow,and(2)theclaimaboutthelowerboundforthevolumesofthemaximalhornsandthesmoothnessofthesolutionfromsometimeon,whichturnedouttobeunjusti?ed,and,ontheotherhand,irrelevantfortheotherconclusions.TheRicci?owwithsurgerywasconsideredbyHamilt
3、on[H5,§4,5];unfortu-nately,hisargument,aswritten,containsanunjusti?edstatement(RMAX=Γ,onpage62,lines7-10fromthebottom),whichIwasunableto?x.Ourapproachissomewhatdi?erent,andisaimedateventuallyconstructingacanonicalRicci?ow,de?nedonalargestpossiblesubsetofspace-time,-agoal,thathasnotbeenac
4、hievedyetinthepresentwork.Forthisreason,weconsidertwoscalebounds:thecuto?radiush,whichistheradiusofthenecks,wherethesurg-eriesareperformed,andthemuchlargerradiusr,suchthatthesolutiononthescaleslessthanrhasstandardgeometry.Thepointistomakeharbitrarilysmallwhilekeepingrboundedawayfromzero.
5、NotationandterminologyarXiv:math.DG/0303109v110Mar2003B(x,t,r)denotestheopenmetricballofradiusr,withrespecttothemetricattimet,centeredatx.P(x,t,r,△t)denotesaparabolicneighborhood,thatisthesetofallpoints(x′,t′)withx′∈B(x,t,r)andt′∈[t,t+△t]ort′∈[t+△t,t],dependingonthesignof△t.AballB(x,t,??
6、1r)iscalledan?-neck,if,afterscalingthemetricwithfactorr?2,itis?-closetothestandardneckS2×I,withtheproductmetric,whereS2hasconstantscalarcurvatureone,andIhaslength2??1;here?-closereferstoCNtopology,withN>??1.AparabolicneighborhoodP(x,t,??1r,r2)iscalledastrong?-neck,if,afterscalingwithfact
7、orr?2,itis?-closetotheevolvingstandardneck,whichateach?St.PetersburgbranchofSteklovMathematicalInstitute,Fontanka27,St.Petersburg191011,Russia.Email:perelman@pdmi.ras.ruorperelman@math.sunysb.edu1timet′∈[?1,0]haslength2??1andscalarcurvature(1?t′)?1.AmetriconS2×I,suchthate
