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1、Ricciflowwithsurgeryonthree-manifoldsGrishaPerelman∗August21,2006Thisisatechnicalpaper,whichisacontinuationof[I].Hereweverifymostoftheassertions,madein[I,§13];theexceptionsare(1)thestatementthata3-manifoldwhichcollapseswithlocallowerboundforsectionalcurvatureisagraphmanifold-thisisdeferre
2、dtoaseparatepaper,astheproofhasnothingtodowiththeRicciflow,and(2)theclaimaboutthelowerboundforthevolumesofthemaximalhornsandthesmoothnessofthesolutionfromsometimeon,whichturnedouttobeunjustified,and,ontheotherhand,irrelevantfortheotherconclusions.TheRicciflowwithsurgerywasconsideredbyHamilt
3、on[H5,§4,5];unfortu-nately,hisargument,aswritten,containsanunjustifiedstatement(RMAX=Γ,onpage62,lines7-10fromthebottom),whichIwasunabletofix.Ourapproachissomewhatdifferent,andisaimedateventuallyconstructingacanonicalRicciflow,definedonalargestpossiblesubsetofspace-time,-agoal,thathasnotbeenac
4、hievedyetinthepresentwork.Forthisreason,weconsidertwoscalebounds:thecutoffradiush,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,ǫ−
6、1r)iscalledanǫ-neck,if,afterscalingthemetricwithfactorr−2,itisǫ-closetothestandardneckS2×I,withtheproductmetric,whereS2hasconstantscalarcurvatureone,andIhaslength2ǫ−1;hereǫ-closereferstoCNtopology,withN>ǫ−1.AparabolicneighborhoodP(x,t,ǫ−1r,r2)iscalledastrongǫ-neck,if,afterscalingwithfact
7、orr−2,itisǫ-closetotheevolvingstandardneck,whichateach∗St.PetersburgbranchofSteklovMathematicalInstitute,Fontanka27,St.Petersburg191011,Russia.Email:perelman@pdmi.ras.ruorperelman@math.sunysb.edu1timet′∈[−1,0]haslength2ǫ−1andscalarcurvature(1−t′)−1.AmetriconS2×I,suchthate
