ByZheZhangADISSERTATIONSubmittedtoMichiganStateUniversityinpartialfulfillmentoftherequirementsforthedegreeofMathematics—DoctorofPhilosophy2023THEUNIQUENESSANDSMOOTHNESSOFCONFORMALNORMALMETRICSABSTRACTItisknownthateveryRiemannianmetriconaclosedmanifoldisconformaltoametricwhoseexponentialmappreservestheEuclideanvolumenearapoint.Thisthesisconcernstheclassificationproblemofsuch“conformalnormalmetrics”onaconformalmanifold(X,[g])ofdimensionn≥3.Wefirstprovetheuniquenessofaconformalnormalmetricwithinafixed1-jetclassofmetrics.Fortheproof,wemainlyfollowCao’smethodin[Cao91]byanalyzinganon-linearsingularellipticequationintheframeworkofweightedHölderspaces.Oursecondresultconcernsthesmoothdependenceofconformalnormalmetricsonparameters.Asapplications,wefirstconstructasmoothRiemannianmetriconX×Xthatisconformalnormalnearthediagonaloneachfiber,andthenusethismetrictogiveasimplifiedproofoftheregularityofHabermann’scanonicalmetric.CopyrightbyZHEZHANG2023ACKNOWLEDGEMENTSIwouldliketoexpressmyheartfeltgratitudetotheindividualsandgroupswhoseunwaveringsupportandcontributionshavebeenpivotalinthesuccessfulcompletionofmythesis:Firstandforemost,Iamdeeplyindebtedtomyadvisor,TomParker.Yourguidance,expertise,andunwaveringdedicationtomyacademicjourneyhavebeeninstrumentalinshapingthisresearch.Yourmentorshiphasbeenasourceofinspiration.Iextendmysincereappreciationtotheesteemedmembersofmythesiscommittee:Dr.BenSchmidt,DemetreKazaras,RajeshKulkarniandXiaodongWang.Yourvaluableinsights,criticalfeedback,andscholarlyexpertisehavegreatlyenhancedthequalityanddepthofmywork.IwouldliketoacknowledgeChairpersonDr.JeffreySchenkerforhisinvaluableadviceandgenerousfundingsupport,whichhaveplayedacrucialroleinthesuccessfulexecutionofthisresearch.ToDr.HonghaoGao,Ioweaspecialdebtofgratitudeforyourunwaveringassistanceandsupportincountlessways.Yourmentorshipandfriendshiphavebeenindispensablethroughoutthisacademicjourney.Iextendmythankstomyfriendsandcolleagues:YuShen,WenchuanTian,KeshavSutrave,ZhixinWang,JosephMichaelMelby,DeanDemetriSpyropoulos,andChuang-tianGuan,foryourcamaraderie,encouragement,andintellectualexchangethatmadethisjourneyallthemoreenriching.AheartfeltthanksgotoDr.WenminGongforyourinvaluablesupportduringthe2019summerconferenceinBeijingandforextendingtheinvitationtopresentatalkatBeijingNormalUniversity.Yourguidanceandencouragementhavebeeninstrumentalinbroadeningmyacademichorizons.Lastly,Iwanttoexpressmydeepestgratitudetomywifeandbothfamiliesfortheirunwaveringsupport,love,andunderstanding.Yourpatienceandencouragementsustainedmeduringthechallengingtimesofthisendeavor,andIamforevergratefulforyourpresenceivinmylife.Toallofyou,yourcontributions,whetheracademicorpersonal,havebeeninvaluable.Thankyouforbeingapartofmyacademicjourneyandformakingthisthesispossible.vTABLEOFCONTENTSCHAPTER1INTRODUCTION...........................1CHAPTER2PRELIMINARIES..........................42.1PreliminariesonExponentialMaps.....................42.2PreliminariesonJetSpacesandJetBundles................72.3PreliminariesonConformalGeometry....................9CHAPTER3CONFORMALNORMALMETRICSATJETLEVELS.....133.1AJetLevelRelationofFlat,ConformallyFlatandConformalNormalMetrics.....................................153.2JetLevelExistenceofConformalNormalMetrics.............19CHAPTER4LOCALEXISTENCERESULTS..................264.1LinearPDEResults..............................264.2LocalExistenceofConformalNormalMetrics...............32CHAPTER5UNIQUENESSOFCONFORMALNORMALMETRICSATTHEGERMLEVEL............................365.1ConformalNormalMetricsonLocallyConformallyFlatManifolds....39CHAPTER6SMOOTHDEPENDENCEOFCONFORMALNORMALMETRICSONPARAMETERS.........................426.1SmoothDependenceofConformalNormalMetrics.............426.2AnApplicationofSmoothFamilyofConformalNormalMetrics.....48BIBLIOGRAPHY.....................................58APPENDIXAPROOFOFLEMMAA.3......................60APPENDIXBPROOFOFLEMMAB.1......................65viCHAPTER1INTRODUCTIONAgeneralconformalstructureondimensionn≥3manifoldsislocallynontrivial.Indeed,bytheWeyl-SchoutenTheorem,aconformalmetriconmanifoldsofdimensionn≥4islocallyconformallyflatifandonlyiftheWeyltensorvanishes(Cottontensorfor3-manifolds).Inconsequence,genericmetricsonmanifoldsofdimensionn≥3donotadmitisothermalcoordinates.Nonetheless,withinagivenconformalclass,onecanalwaysfindametricwiththeso-calledconformalnormalcoordinates,aspecialcoordinatesystemfirstintroducedbyLeeandParkerin[LP87].Explicitly,wehave:Theorem1.1(cf.Theorem5.1,[LP87]).LetXbeasmoothmanifoldtogetherwithacon-formalclassC.Atapointp∈X,thereisaconformalmetricg∈CsuchthatforeachN󰃍2detgij=1+O󰀃|x|N󰀄ing-normalcoordinates{xi}atp.LeeandParker’sresulthaslateronbeenimprovedbyCao[Cao91],andGünther[Gün93]independently.Theyprovedthelocalexistenceofaconformalnormalmetricinaneighbor-hoodUpofagivenpointp.Ononehand,underconformalnormalcoordinates,localanalysisonaconformalman-ifoldcanbesimplifiedtoagreatextent.Ontheotherhand,thesetofconformalnormalmetricsisofinterestonitsown.ByTheorem5.6in[LP87],onthejet-level,theconfor-maltransformationgroup󰁧CO(n)actsonthesetofconformalnormalcoordinatesfreeandtransitively,whichindicatesarelationbetweentheconformalnormalmetricsandtheglobalconformalstructureC(seeTheorem3.2.4).Ourfirstresultconcernsthegermleveluniquenessofconformalnormalmetrics.InChapter5,weprovethefollowingresult:Theorem1.2.Let(X,[g])beaconformalmanifold.Atp∈X,any1-jetclassj1p(g)ofmetricsin[g]containsaconformalnormalmetricthatisuniqueuptothegermlevel.1TheproofisbasedonCao’sapproachtotheexistencetheorem.In[Cao91],Caorephrasedthelocalexistenceofaconformalnormalmetricastheproblemoffindingsolutionsofanon-linearsingularellipticequationinaclassoffunctionsofweightedHöldernorms.Briefly,fixapointp∈Xandabackgroundmetricg0,andletr0=dist(p,·)betheg0-distancefromp.Thenaconformalmetricg=Φg0isconformalnormalatpifandonlyifitsdistancefunctionrfrompsatisfies∆gr=n−1r.(1.1)Letwbethefunctiondefinedbyr=r0ew(x).ThentheconformalfactorΦandwdetermineeachotherbytheformulaΦ=󰀂dr󰀂2g0=󰀃1+2xiwi+r20󰀂dw󰀂2g0󰀄e2w.TheEquation(1.1)canbeconvertedintoanequationforwing0normalcoordinatesoftheformV(x,∂w,∂2w)=L0(w)+G(x,∂w)+Q(x,w,∂w)=−∂r0ln󰁳det(g0(x))r0,(1.2)whereVisanon-linearellipticequationofwwhosesymbolissingularattheorigin,andL0isthescale-invariantlinearizationofV.SeeChapter4formoredetails.TounderstandEquation(1.2),wefirststudythelinearoperatorL0.WeshowthatkerL0={0}whenrestrictedtofunctionsthatvanishtotheinfiniteorderattheorigin(cf.Lemma4.1.5).WeusethisfacttogiveauniquenesstheoremforthesolutionofCao’sequation(1.2)(cf.Theorem5.2).ThislocalanalysisresultisthenusedtoprovethatthereisauniquegermofaconformalnormalmetricineachequivalenceclassinthesetJ1[g]=󰀋j1p(g)󰀏󰀏g∈[g],p∈X󰀌of1-jetsofmetricsintheconformalclass.Theorganizationofthethesisisasfollows.InChapter2,wefixsomenotationsandreviewbackgroundmaterialswewillusethroughoutthediscussion.Thisincludesreviewingsomebasicfactsabouttherelationbetweenmetricandexponentialmaps,jetspacesandjetbundles,andsomebasicfactsaboutlocalconformalgeometry.InChapter3,wedefineconformalnormalmetricsandreviewLeeandParker’swork2in[LP87]ontheexistenceofconformalnormalmetricsuptothejetlevel.AsanapplicationofLeeandParker’smethod,inSection3.1,wegiveajetlevelrelationbetweenconformalnormalmetricsandflatmetricsontheconformallyflatmanifoldswhichwillbegeneralizedtoarelationonthegermlevelbytheuniquenesstheorem.InChapter4,wereviewCao’sapproachtothelocalexistenceofconformalnormalmetricsandproveauniquecontinuationlemmafortheL0operator.InChapter5,westateandprovethemaintheoremontheuniquenessofconformalnormalmetrics.Finally,inChapter6,weshowthesmoothdependenceofconformalnormalmetricsonthemodulispace.Thisleadstotheproofofoursecondmainresult:theexistenceofametrich=g⊕Φ2gonX×Xsuchthat,inaneighborhoodofthediagonal,therestrictionofhtothesliceSp={(p,y)|y∈X}isaconformalnormalmetric.Asafurtherapplication,weusethemetrichtogiveasimplifiedproofoftheregularityofHabermann’scanonicalmetric.3CHAPTER2PRELIMINARIES2.1PreliminariesonExponentialMapsWewillworkatthelevelofgerms.Lemma2.1.1.Considertwoconformalmetricsgand˜g=Φ2gdefinedinaneighborhoodofpwithLevi-Civitaconnections∇and˜∇,distancefunctionsr=dist(p,·)and˜r,andexponentialmapsexpgp:TpM→Mandexp˜gp.Then,atthelevelofgermsatp,thefollowingareequivalent:(a)∇=˜∇(b)expgp=exp˜gp(c)˜r=crwherec=Φ(p)(d)Φisconstant.Inparticular,conformalmetricsgand˜ghavethesamedistancefunctionifandonlyifg=˜g.Proof:Theexponentialmapisdefinedbyexpgp(v)=γ(1)whereγ:[0,1]→Misthesolutiontothegeodesicequation∇˙γ˙γ=0withγ(0)=pand˙γ(0)=v.Thissolutionisunique,so(a)implies(b).If(b)holds,thenthepullbackfunctionsρ=(expgp)∗rand˜ρ=(exp˜gp)∗˜rareequal.Butρ(v)isthenorm|v|gfortheinnerproductgponTpM,andtherefore˜ρ2(v)=˜gp(v,v)=Φ(p)2gp(v,v)=Φ(p)2ρ2(v)forallv∈TpM.Theseexponentialmapsarelocaldiffeomor-phisms,so(c)holds.Thedistancefunctionforgsatisfies|dr|g=1,soif(c)holdsthen1=|d˜r|˜g=|cdr|Φg=cΦ−1|dr|g=cΦ−1,soΦisconstant.Finally,theimplication(d)=⇒(a)isclearfromthelocalcoordinateformulafortheChristoffelsymbols.□Foranalternativeproofthat(a)and(d)areequivalent,considerthedifferenceD(X,Y)=∇XY−˜∇XY(2.1.1)forvectorfieldsXandY.ThepropertiesoftheLevi-CivitaconnectionshowthatD(X,Y)istensorialinbothXandYandissymmetric.ThusDisatensorD∈Γ(Sym2(T∗M)⊗TM)thatvanishesifandonlyif∇=˜∇;bypolarizationthisisalsoequivalenttothevanishing4ofD(X,X).Forconformalmetricsgand˜g=e2fg,theformulafortheChristoffelsymbolsshowsthatD(X,Y)=Xf·Y+Yf·X−g(X,Y)·∇f.IfΦ=efisaconstant,thenDvanishes.Conversely,ifD=0then,usinginnerproductsforg,wehave0=〈X,D(X,X)〉=2Xf·|X|2−|X|2〈X,∇f〉=|X|2·XfforallvectorfieldsX,sofisconstant.Asanaside,werecordthefollowingvariationofLemma2.1.1inwhichthemetricsarenotassumedtobeconformal,buttheexponentialmapsareassumedtobeequalatall(ornearlyall)points.Proposition2.1.2.Supposethatgand˜garemetricsonanopensetU,thatUisgeodesicallyconvexforbothgand˜gandthatS⊂Uisanon-emptysubmanifoldofcodimension1.Thenthefollowingareequivalent:(a)∇=˜∇(b)expgp=exp˜gpforallp∈U(c)expgp=exp˜gpforallp∈S.Proof:TheproofofLemma2.1.1showsthat(a)implies(b).Obviously(b)implies(c).Nowassumethat(c)holds,andfixp∈S.ThenexpgpisadiffeomorphismfromaneighborhoodVof0inTpUtoU.BecauseTpSisacodimension1linearsubspaceofTp,soU∗=󰀋expp(v)∈U󰀏󰀏v∈TpU\TpS󰀌isanopendensesubsetofU.Foreachq=expp(v)∈U∗,thepathfromptoqdefinedbyγq(t)=expp(tv),0≤t≤1,isageodesicforthemetricg,andistransversetoSatthepointp.Byassumption,γqisalsoageodesicforthemetric˜g.Reversingperspective,wecanwritep=expq(w),wherew=−˙γq(1)∈TqU.Becauseγqisageodesicforbothmetrics,wehaveD(w,w)=0.Furthermore,foranyw′∈TqUsufficientlyclosetow,thereisaτcloseto1suchthattheg-geodesicexpq(τw′)intersectsStransversallyatapointp′=expq(τw′)closetop.Applyingthepreviousargumentwithpreplacedbyp′showsthatD(τw,τw)=τ2D(w′,w′)=0.Therefore|D(w′,w′)|4,whichisaquadricpolynomialonTqU,vanishesonaneighborhoodofwso,byanalyticity,iszero.5ThisistrueateachpointqinthedensesetU∗⊂U.ThereforethetensorDvanishesonU,so∇=˜∇.Thus(c)implies(a).□TheconditionsofProposition2.1.2arealsoequivalentto˜g=cgifweimposeoneadditionalassumption.RecallthataRiemannianmanifold(U,g)isirreducibleif,forsomep∈U,theactionoftheholonomygroupHp(U,g)atponTpUisirreducible.Thispropertyisindependentofp(cf.[KN63]).Corollary2.1.3.Supposethatgand˜garemetricsonanopensetUandthat(U,g)isirreducible.ThentheLevi-Civitaconnectionsofgand˜gareequalifandonlyif˜g=cgforsomeconstantc.Proof:AsintheproofofLemma2.1.1,if˜g=cgthen∇=˜∇.Fortheconverse,assumethat∇=˜∇.ThisimmediatelymeansthattheholonomygroupHp(U,g),whichisdefinedbyparalleltransportwithrespectto∇,isequaltotheholonomygroupHp(U,˜g)definedby˜∇.DefineavectorbundlemapA:TU→TUbytheconditiong(AX,Y)=˜g(X,Y)(2.1.2)forallvectorfieldsX,Y.Because˜gissymmetric,thisimpliesthatAisself-adjointforthemetricg,andhenceisdiagonalizable.Differentiatingandnotingthat∇g=0and∇˜g=˜∇˜g=0,oneseesthat∇A=0.Nowsupposethatγ(t)isapathinUstartingandendingatpwithvelocityvectorT.IfX(t)isavectorfieldalongγthatisparallel,i.e.∇TX=0,then∇TA(X)=(∇TA)X+A(∇TX)=0,soAXisalsoparallel.ItfollowsthatAcommuteswiththeactionofHp(U,g)onTpU.BySchur’sLemma,Aisamultipleoftheidentity.Thisistrueforeveryp∈U,so(2.1.2)showsthat˜g=Φ2gforsomesmoothfunctionΦ.ApplyingLemma2.1.1,weconcludethat˜g=cgforaconstantc>0.□62.2PreliminariesonJetSpacesandJetBundlesJetSpacesDefinition2.2.1.ThesetC∞p(Rn,Rm)ofgermsofsmoothfunctionsf:Rn→Rmatapointp∈RnisamoduleoverC∞(Rn).Letmp⊆C∞(Rn)betheidealoffunctionsthatvanishatp.Definethek-jetspaceatpby:Jkp(Rn,Rm)=C∞p(Rn,Rm)mk+1p·C∞p(Rn,Rm)Thejetspaceoffunctionsfsuchthatf(p)=qisJkp(Rn,Rm)q=󰀋jkp(f)∈Jkp(Rn,Rm)|f(p)=q󰀌Inparticular,wedenoteLkn,m=Jk0(Rn,Rm)0,elementsinLkn,mcanbeidentifiedasthekthorderTaylorpolynomialsofthegeneratingfunctions.Explicitly,let{xi}and{yj}becoordinatesonRnandRmrespectively,andαbeamulti-index,theneveryelementjkp(f)inLkn,mhasapolynomialrepresentative:f(x)=(fj(x))=󰀳󰁃󰁛1≤|α|≤kcjαxα󰀴󰁄DefinethefollowingtwonaturaloperationsonLkn,m:(a)Forl≤k,thejetprojectionmapπkl:Lkn,m→Lln,misdefinedbythenaturalprojectionofmodules:C∞0(Rn,Rm)0mk+10·C∞0(Rn,Rm)0→C∞0(Rn,Rm)0ml+10·C∞0(Rn,Rm)0󰀡ml+10·C∞0(Rn,Rm)0mk+10·C∞0(Rn,Rm)0,(b)ThejetcompositionmapLkn,m×Lkm,d→Lkn,disdefinedbycomposingthepolynomialrepresentativesandtruncatingtodegreek.LetDiff0(Rn,Rn)0bethegroupofgermsofdiffeomorphismsofRnfixingtheorigin.Thek-jetspaceofDiff0(Rn,Rn)0,denotedasGLkn,isaLiegroupconcerningthejetcompositionoperation.Inparticular,fork=1,GL1n=GL(n),thegenerallineargroup.ForaLiesubgroupG⊆Diff0(Rn,Rn)0,thek-jetspaceofGisaLiesubgroupofGLkn,denotedasGk.Denoteg1theLiealgebraofG1,thenthemanifoldstructureonGkisinheritedfromthefollowingidentity:Gk=󰀋(A,τ1,...,τk−1)|A∈G1,τi∈Symi+1(Rn),τi(−,v1,...,vi)∈g1󰀌,7whereSymi+1(Rn)isthespacethesymmetric(i+1)-linearmapsRn×(i+1)→Rn.Thejetprojectionπk+1k:Gk+1→Gkisgivenbyπk+1k(A,τ1,...,τk)=(A,τ1,...,τk−1).Thekernelofπk+1kiscalledthek-jetprolongationofg1andisdenotedasgk+1.DefinetheorderofGtobethesmallestintegerksuchthatπk+1kisanisomorphism,namelythesmallestksuchthatgk+1=0.ForagroupGoforderk,theLiealgebragofGdecomposesas:g=g1⊕···⊕gk,whichisagradedLiealgebra.Forτp∈gpandτq∈ggtheLiebracket[τp,τq]∈gp+qandisgivenasfollows(cf.[Kob12]):[τp,τq](v0,v1,...,vp+q)=1p!(q+1)!󰁛τp(τq(vj0,...,vjq),vjq+1,...,vjp+q)−1(p+1)!q!󰁛τq(τp(vk0,...,vkp),vkp+1,...,vkp+q)Definition2.2.2.LetXbeasmoothmanifoldofdimensionsnandU⊂Rnbeanopenneighborhoodof0.Wesaytwolocaldiffeomorphismsf,g:U→Xdefinethesamek-frameatapointp∈Xiff(0)=g(0)=pandjk0(ϕ−1◦f)=jk0(ϕ−1◦g)∈GLkn,whereϕisanarbitrarylocalchartatp.Itisclearthatak-frameatp∈Xiswell-definedindependentofthechoiceofthelocalchartϕ.ThegroupGLknactsonthesetofk-framesatpfreeandtransitivelybythejetcomposition.Definition2.2.3.ForaLiesubgroupG⊆GLkn,aprincipalGbundleπ:P→XdefinesaG-structureoforderkonXiffor∀p∈XthefiberFp=π−1(p)consistsofk-framesatpandtheprincipalbundleactionofGonFpisbythejetcomposition.ClassicalexamplesofG-structuresonamanifoldXinclude:GL+nstructuredefinesanorientationonX;SLnstructuredefinesavolumeelementonX;O(n)structuredefinesaRiemannianmetriconXandsoon.WewillbefocusingontheG-structurecharacterizationofconformalstructures,seeSection2.3and[Kob12]formoredetails.8JetBundlesDefinition2.2.4([Par]).Letπ:E→Mbeasmoothvectorbundle,thesetΓ(E)ofsmoothsectionsisamoduleoverC∞(M).Denotemp={f∈C∞(M)|f(p)=0}theidealofsmoothfunctionsvanishesatp∈M.Wedefinethek-jetsofsectionsofEatPby:Jk(E)p=Γ(E)/mk+1p·Γ(E)Jk(E)pisavectorspacewithvectorsummationas[ξ]k+[η]k=[ξ+η]kandscalarmultipli-cationasλ·[ξ]k=[λ·ξ]k.Jk(E)=∪p∈MJk(E)pisavectorbundleoverMcalledthekthjetbundleofthevectorbundleE.Inparticular,J0(E)≃E.Incoordinates{xi}nearpandabasis{σα}ofEp,thek-jetofasectionξisuniquelyrepresentedbyitsdegreekTaylorpolynomial[ξ]k=󰁛α󰀣aα0+󰁛iaαi(x−p)i+···+󰁛aαi1i2···ik(x−p)i1i2···ik󰀤σαWehavethefollowingexactsequenceofjetbundles:0→Sk(T∗M)⊗E→Jk(E)→Jk−1(E)→0,(2.2.1)wherethemapπk:Jk(E)→Jk−1(E)isthenaturaljetprojection,andSk(T∗M)isthek-foldsymmetrictensorproductofT∗M,themapSk(T∗M)⊗E→Jk(E)isdefinedbyidentifyingthekernelofπkwithSk(T∗M)⊗E.Inparticular,whenE=M×Rnbeingatrivialvectorbundle,wehaveacanonicalisomor-phismJ1(E)≃T∗M⊕n⊕Rn(2.2.2)definedasfollows:Lets∈Γ(E)bearepresentativeofanelementj1p(s)∈J1(E).Takes2=π2◦s∈C∞(M,Rn),whereπ2:M×Rn→RnistheprojectiontotheRncompo-nent.Defineη:J1(E)→T∗M⊕nbyη(j1p(s))=d(s2)(p).Itisclearthatηiswelldefinedindependentofthechoiceoftherepresentativesandgivesasplittingoftheshortexactsequence(2.2.1),andhencewehavetheisomorphism(2.2.2).2.3PreliminariesonConformalGeometryWeuse[Kob12]asamainreferenceforthissection.9LetˆRn=Rn∪{∞}betheextendedEuclideanspace.Let󰁧CO(n)bethegroupofconformalautomorphismofˆRnhavingtheoriginfixed.Ageneralelementϕ∈󰁧CO(n)isoftheform:ϕ(x)=λAx−x2η1−2η·Ax+x2η2,(2.3.1)whereλ>0,η∈Rn,A∈O(n).Denote󰁧COk(n)thegroupofk-jetsof󰁧CO(n)asaLiesubgroupofDiff0(Rn,Rn)0.Fork=1,󰁧CO1(n)=CO(n),thelinearconformalgrouponRn.Explicitly,wehave:CO(n)={A∈GL(n);AtA=λI,λ∈R+}∼=O(n)×R+.Fork=2,wehavethefollowingshortexactsequence:1→co1τ−→󰁧CO2(n)−→󰁧CO1(n)→1,(2.3.2)whereco1(n)isthefirstprolongationoftheLiealgebraco(n)ofCO(n).Bydefinitionco1(n)={f∈Sym2(Rn)|∀v∈Rn,f(−,v)∈co(n)}.co1(n)isisomorphictoRnbythefollowingmap:t:Rn→co1(n)v󰀁→tv(a,b)=12(〈v,a〉b+〈v,b〉a−〈a,b〉v).Themapτin(2.3.2)canbegivenexplicitlyasτ(t)(x)=x+t(x,x).Letϕ∈󰁧CO(n)beageneralelementasgivenin(2.3.1).Bytakingthe2-jetofϕattheorigin,wehave:j20(ϕ)=ϕkixi+ϕkijxixj=λ(Ax+tη(Ax,Ax)).(2.3.3)ThecoefficientmatrixC=ϕkiofthelineartermsofj20(ϕ)equalsthematrixλA,henceλ=(det(C))1nandA=C(det(C))1n.Let{ei}bethestandardbasisofRn,thenthequadratictermofj20(ϕ)atA−1(ei)equalsη/2.Hencethedata{λ,A,η}isuniquelydeterminedbyj20(ϕ).Thereforeelementsin󰁧CO(n)areuniquelydeterminedbyj20(ϕ),namely󰁧CO(n)isasubgroupofGL2n.Bythefactthat󰁧CO(n)isasubgroupofGL2n,wedefinethe󰁧CO(n)structureoforder2onasmoothmanifoldXasindefinition2.2.3,tobeaprincipal󰁧CO(n)bundleπ:P→Xof2-frameswith󰁧CO(n)actsoneachfiberbythejetcomposition.10Explicitly,inlocalcoordinates{x}atp∈X,letf(x)=clixi+clijxixjbethepolynomialrepresentativeofa2-frameatp.Theactionofϕ∈󰁧CO(n)onfisgivenasf·ϕ=j20(f◦ϕ)=(clkϕki)xi+(clkϕkij+clksϕkiϕsj)xixj,(2.3.4)whereϕkiandϕkijarethecoefficientsofj20(ϕ)givenin(2.3.3).Infact,a󰁧CO(n)structurePisuniquelydeterminedbyaconformalstructure[g]andviceversa.Weargueasfollows(cf.[Kob12]):Ononehand,supposePisa󰁧CO(n)structureonX.ProjectingPtothecorresponding1-jetbundleP1bythejetprojectionmap,weobtainP1asaprincipalCO(n)bundle.SinceCO(n)=O(n)×R+,asectionsoftheorbitbundleP1/O(n)definesaprincipalbundlereductionofP1toaprincipalO(n)bundleH=H(s).Ateachpointp∈Xchooseaframeθ∈Hp,defineametriconTpXasg(p)(v,w)=󰁓viwi,whereviandwiarecomponentsofvandwwithrespecttotheframeθ.Letθ′∈Hpbeadifferentframesuchthatθ′=AθwithA∈O(n).Withrespecttoθ′,themetricg′(p)(v,w)=󰁓(Av)i(Atw)i=󰁓viwi=g(p).Hencethemetricgiswelldefinedindependentofthechoiceofθ∈Hp.Differentchoicesofsectionswillgivemetricsconformaltog,hencedefiningaconformalstructure[g].Ontheotherhand,suppose[g]isaconformalstructureonX.DenoteOg(n)theor-thonormalframebundlewithrespecttog∈[g],defineP1=󰁞p∈X,g∈[g]Ogp(n).P1isaprincipalCO(n)bundle.Thecorresponding󰁧CO(n)structurePonXisdefinedasthefirstprolongationofP1asfollows:Wefirstembedco1(n)asasubgroupofEnd(Rn⊕co(n))bythefollowingmap:ι:co1(n)→End(Rn⊕co(n))t󰀁→󰀻󰀿󰀽¯t(v)=v+t(−,v),forv∈Rn¯t(A)=A,forA∈co(n).LetΛ2Rn∗⊗Rnbethespaceofskew-symmetricbilinearmappings,definealinearmap∂:co(n)⊗Rn∗→Λ2Rn∗⊗Rnby(∂f)(v1,v2)=f(v1)v2−f(v2)v1,11wheref∈co(n)⊗Rn∗,v1,v2∈Rn∗.Wechooseonceandforalladirectsumcomplementof∂(co(n)⊗Rn∗)inΛ2Rn∗⊗Rn,denotedasC.Letθ∈Ω1(P1,Rn)bethecanonical1formonP1.Atapointu∈P1,an-dimensionalsubspaceHofTuP1iscalledhorizontalifθ:H→Rnisanisomorphism.AhorizontalspaceHiscalledCadmissibleifdθ(TH⊕TH)∈C.EveryCadmissiblespaceHdeterminesalinearframeofTuP1asfollows:Letf:co(n)→TuP1bethemapthatsendsA∈co(n)toA∗u,whereA∗isthefundamentalvectorfieldonP1withrespecttoA.Thedirectsumoffwiththemapθ−1:Rn→H⊆TuP1definesalinearframeθ−1⊕f:Rn⊕co(n)→TuP1.TheunionofalllinearframesinducedfromCadmissiblehorizontalspacesisaprincipalι(co1(n))bundleof1-framesPoverP1.AsbundleoverX,Pisaprincipal󰁧CO(n)bundleof2-framesdefininga󰁧CO(n)structureonX.12CHAPTER3CONFORMALNORMALMETRICSATJETLEVELSLet(X,g)beaRiemannianmanifold(allmanifoldsareassumedtobesmoothofdimensionn≥3).Wefirstgiveacoordinate-freedefinitionofthedeterminantofthemetricgnearapointp∈X.LetU⊆TpXbeanopenneighborhoodoftheoriginonwhichtheRiemannianexpo-nentialmapisawell-defineddiffeomorphismontoanopenneighborhoodVofp.Letdvgbetheg-Riemannianvolumeformwhichcandefinedinacoordinate-freeway.Thepullbackexp∗g,pdvg|VbytheexponentialmapatpisavolumeformonU.Ontheotherhand,theinnerproductspace(TpX,g(p))hasacanonicalvolumeformdvp.BothvolumeformsarenonvanishingsectionsofΛtop(TpU)whichisareallinebundle,andhencethedivisionexp∗dvgdvpisawell-definedsmoothfunctiononU.Definedet(exp∗g):=(exp∗dvgdvp)2.Definition3.1.Ametricgiscalledconformalnormalatapointp,ifthereexistsanopenneighborhoodVofpsuchthatdet(exp∗g)=1onU=exp−1(V).Also,wesaygiskthorderconformalnormalatpifthek-jetjk0(det(exp∗g))=1.AnarbitraryRiemannianmetricisnotnecessarilyconformalnormal,indeedtheword“conformal”indicatesthefollowingfact(cf.[Cao91,Corollary0.1]):Theorem3.2.ForanyRiemannianmetricgonX,andapointp∈X,thereexistsaconformalmetric˜g=Φgwhichisconformalnormalinasmallneighborhoodofp.Giventwoconformalmetricsgand˜g=e2fg,assumethattheyarebothconformalnormalanddefinethesameexponentialmapatapointp,wethenhave:det(exp∗(˜g))=det(exp∗(e2fg))=e2nfdet(exp∗g)=e2nf=1.(3.1)Thusg=˜gwithintheinjectiveradius,andconformalnormalmetricsatpareuniquelydeterminedbytheexponentialmapsonTpX.Infact,aswewillshowinTheorem5.1,aconformalnormalmetricislocallyuniquelydeterminedbyits1-jetclass.Inthischapter,weworkonjetlevelsandfixthefollowingnotations:Onaconformalmanifold(X,[g])atapointp∈X,fork=1,2,...,∞,definethesetof13k-jetsofRiemannianmetricsatpwithkthorderconformalnormaldeterminantasCNk(p)={jkp(g)|g∈[g],jk0(det(exp∗pg))=1}.(3.2)Andcorrespondingly,definetheconformalnormalkframesas:CNCk(p)={jk0(ϕ)|ϕg,β=expg,p◦β,β∈O(TpX,gp),g∈[g]s.t.jk0(det(exp∗pg))=1}.Fork=1,j10(det(exp∗pg))=1holdsforanymetricg.ThuswehaveCN1(X)=J1[g]=󰀋j1p(g)|g∈[g],p∈X󰀌.DenoteC∞(Sym2T∗X)thespaceofsmoothsymmetric(0,2)tensors,whichisaFréchetspacewithC∞topology.FixabackgroundRiemannianmetricg0,byTheorem1.5in[FM77],theconformalclass[g0]astheorbitoftheC∞(X)actionong0isasmoothsub-manifoldofC∞(Sym2T∗X),i.e,themapΦ0:C∞(X)→C∞(Sym2T∗X)byΦ0(f)=efg0isasmoothembedding.DefinethedescendingofΦ0to1-jetsbythefollowingdiagram:C∞(X)C∞(Sym2T∗X)J1(R)J1(Sym2T∗X)j1p(f)j1p(efg0)Φ0π1π1J1Φ0J1Φ0isabundleisomorphismontoJ1[g].Indeed,theimagej1p(efg0)=ef(p)(j1p(g0)+j1p(f−f(p))g0(p))isuniquelydeterminedbyj1p(f).Insummary,wehavethefollowinglemma:Lemma3.3.Ametricg0∈[g]inducesasmoothbundleisomorphism:η:=J1Φ0◦ι−1:R⊕T∗X−→J1[g]=CN1,whereιisthebundleisomorphismdefinedin(2.2.2):ι:J1(R)−→R⊕T∗Xj1p(f)󰀁−→(f(p),df(p)).143.1AJetLevelRelationofFlat,ConformallyFlatandConformalNormalMetricsFirst,recallthefollowingtheoremonflatnessandconformalflatnessofRiemannianmetrics(cf.[Lee18,Theorem7.10&Theorem7.37]).Asusual,aRiemannianmanifold(X,g)isflatifandonlyifitsRiemanncurvaturetensorvanishesidentically.Weyl-SchoutenTheorem.Amanifold(X,g)ofdimensionn≥4islocallyconformallyflatifandonlyifitsWeyltensorisidenticallyzero.A3-manifoldislocallyconformallyflatifandonlyifitsCottontensorisidenticallyzero.TheCottonandWeyltensorsaredefinedinequations(3.1.13)and(3.1.12)below.InspiredbytheWeyl-SchoutenTheorem,wegivethefollowingdefinitionofjet-levelflatnessandconformalflatness:Definition3.1.1.WesayaRiemannianmetricgisflatuptoorderratapointp∈XifitsRiemanncurvaturetensorvanishestoorderratp.Similarly,wesaythataconformalstructure[g]onamanifoldofdimensionn≥4isconformallyflatuptoorderriftheWeyltensorWvanishestoorderratp.A3-manifoldisconformallyflatuptoorderratpifitsCottontensorvanishestoorderr−1atp.Tobebrief,wefixedsomenotations.Weworkunderag-normalcoordinatechartandusetheEinsteinsummationconvention.Letµ=(µ1...µk)beakmulti-index,with1≤µi≤nand|µ|=k≥2.ThepermutationgroupSkactsonµbypermutingµiandtheorbitsofthisactiondefineanequivalentrelationµ∼µ′.Eachequivalenceclasshasauniquerepresentative¯µ=(µ1...µk)inascendingorder1≤µ1···≤µk≤n.Whenwehavealistofmulti-indices,weuseupperindices:µ1,...,µl.LetRiemµdenotethematrixwhoseijthentryisthecovariantderivativeRiµ1µ2j;µ3...µk.ThendefineRiemµν:=RiemµRiemtν+RiemνRiemtµ,(3.1.1)wheretheuppertdenotesthetransposematrix.DefineRiemµ1...µlbythisformulaandinduction,i.e.takeµtobeµ1andνtobeµ2···µlin(3.1.1).15Denote∂Riemµ=∂µ3...µkRiµ1µ2jforthepartialderivatives.SincewehaveRj1j2j3j4;i=∂iRj1j2j3j4−Γkj1iRkj2j3j4−···−Γkj4iRj1j2j3k,(3.1.2)Riemµand∂Riemµdifferbyderivativesoforderlessthan|µ|−2attheorigininnormalcoordinates.DenotethecovariantderivativesandpartialderivativesoftheRiccicurvaturetensorRicsimilarlyasabove.Withthisnotation,onecangiveclosedformulasfortheTaylorexpansionsofganddet(g)underthenormalcoordinates.ThiswasneatlydonebySchubertandvandeVenin[MSvdV99].Specifically,g=I+∞󰁛l=1∞󰁛|µi|=2cµ1...µlRiemµ1...µl(0)xµ1+···+µl,(3.1.3)wherethecoefficientscµ1...µlareconstantsdependonlyonabsolutevalues|µi|ofµi.Inparticular,forl=1,and|µ|=k,wehaveck=2k−2(k+1)!.Remark3.1.2.Theexpansion(3.1.3)isobtainedin[MSvdV99]bywritingg(x)=et(x)e(x),Then,asshownin[MSvdV99],thecoefficientsintheTaylorseriese(x)=I+󰁛|µ|≥2eµ(0)xµaregiven,foreachµwith|µ|=k,by(k+1)eµ(0)=(k−1)Riemµ(0)+k−2󰁛n=2(n+1)󰀕k−1n+1󰀖Riem(µ1...µk−n)(0)e(µk−n+1...µk)(0)Combiningtheabovethreeformulasyields(3.1.3)andalsodefinesthecoefficientscµ1...µlrecursively.Tocalculatedet(g),writeg(x)=exp(A(x))withA(x)=A¯µx¯µ.ThenwehaveTr(A(x))=Tr(A¯µ)x¯µ,det(g)=det(exp(A(x)))=exp(Tr(A(x))).Henceexp(Tr(A(x)))=1+Tr(A¯µ)x¯µ+···+1l!l󰁜i=1Tr(A¯µi)x¯µi+···(3.1.4)Sinceg=I+O(r2)innormalcoordinates,wehaveA¯µ=0for|¯µ|󰃑1;thusthe16summationinA¯µx¯µistakenover¯µwith|¯µ|≥2.DenotetheTaylorexpansionofexp(A(x))as:exp(A(x))=I+A¯µx¯µ+12!A¯µ1A¯µ2x¯µ1+¯µ2+···+1l!l󰁜i=1A¯µix¯µi+···(3.1.5)Foreach¯µ,comparing(3.1.3)and(3.1.5),wehave󰁛µ1+···+µl∼¯µcµ1...µlRiemµ1...µl(0)=A¯µ+12!󰁛¯µ1+¯µ2∼¯µA¯µ1A¯µ2+···(3.1.6)Byinductionon|¯µ|fortheequalities(3.1.6),wecancalculateA¯µ(0)recursivelyasfollows:A¯µ(0)=c2󰁛µ∼¯µRiemµ(0),for|¯µ|=2;A¯µ(0)=c3󰁛µ∼¯µRiemµ(0),for|¯µ|=3;A¯µ(0)=󰁛µ1+···+µl∼¯µcµ1...µlRiemµ1...µl(0)+−[|µ|2]󰁛k=21k!󰁛¯µ1+···+¯µk∼¯µk󰁜i=1A¯µi,for|¯µ|≥4.(3.1.7)For|¯µ|=k,by(3.1.2)and(3.1.7),wehave:A¯µ(0)=ck󰁛µ∼¯µ∂Riemµ(0)+P(Riemµ′(0)),(3.1.8)whereP(Riemµ′(0))isapolynomialofRiemµ′with2≤|µ′|≤k−2.Taketraceoftheidentities(3.1.8)andsubstitutebackinto(3.1.4),weobtainaclosedformulafortheTaylorexpansionofdet(g):det(g)=exp(Tr(A(x)))=1+∞󰁛|¯µ|=2󰀳󰁃Tr󰀳󰁃󰁛µ1+···+µl∼¯µcµ1...µlRiemµ1...µl󰀴󰁄−T¯µ󰀴󰁄x¯µ,(3.1.9)whereT¯µisasymmetrictensordefinedbyapolynomialofTr(A¯ν)with2≤|ν|≤|µ|−2,inparticularT¯µ=0for|µ|=2,3.ThegeneralT¯µcanbecalculatedrecursivelybytheformula(3.1.7).Basedontheabovediscussions,wecannowgiveajet-levelconditionformetricsbeingflat,conformallyflat,andconformalnormalatapointpbythetheorembelow:Theorem3.1.3.Supposethataconformalstructure[g]onXiskthorderconformallyflatatapointp.Theng∈[g]is(k+2)thorderconformalnormalatpifandonlyifgiskth17orderflatatp.Proof:Supposethatgiskthorderflatatp.Then(3.1.8)showsthatA¯µ(p)vanishesfor2󰃑|¯µ|󰃑k+2.HenceTr(A¯µ(p))vanishestothesameorder,andhencedetg=1+O(rk+3)by(3.1.4).Thusgis(k+2)thorderconformalnormalatp.Ontheotherhand,assumethatg∈CNk+2(p),i.e.jk+2pdet(g)=jk+2p(exp(Tr(A)))=1.Hence,byinductionon|¯µ|in(3.1.4),wehave:Tr(A¯µ(p))=0,for|¯µ|≤k+2(3.1.10)Taketraceof(3.1.8)onbothsidesupto|¯µ|=k+2,alongwith(3.1.10),wehave:Ricij(p)=0,for¯µ=(ij),(3.1.11)0Ricij,k(p)+Ricik,j(p)+Ricjk,i(p)=0,for¯µ=(ijk),(3.1.11)1...ck+2󰁛µ∼¯µRicµ(p)+P(Riemµ′(p))=0,(3.1.11)kwhere|¯µ|=k+2andP(Riemµ′)isapolynomialofRiemµ′with|µ′|≤k.LetCbetheCottontensordefinedforthemetricg,inlocalcoordinateswehaveCijk=Ricij,k−Ricik,j+12(n−1)(Sjgik−Skgij)=Pij,k−Pik,j,(3.1.12)whereP=Ric−12(n−1)Sgistheso-calledSchoutentensor.WeclaimthefollowinglemmawiththedetailedproofgiveninAppendixA.LemmaA.3.Equations(3.1.11)0to(3.1.11)k,togetherwiththeassumptionjk−1p(C)=0,implythatjkp(Ric)=0.Sinceconformallyflatnessisdefinedbytwocases:Namely,dimensionn=3andn≥4,wediscussaccordinglybelow:First,recalltheRiccidecompositionoftheRiemanniancurvaturetensor:Rijkl=Wijkl+Sn(n−1)󰀃gilgjk−gikgjl󰀄+1n−2󰀃Zilgjk−Zjlgik−Zikgjl+Zjkgil󰀄,(3.1.13)whereZij=Ricij−1nSgij.18Indimension3,theWeyltensorWvanishesidentically,andhenceby(3.1.13),theRie-manniancurvaturetensorisdeterminedbytheRiccicurvature.Indimension3,conformallyflatnesstoorderkmeansjk−1p(C)=0.AndhencebyLemmaA.3,wehavejkp(Ric)=0.Andjkp(Riem)=0inturnby(3.1.13).Indimensionn≥4,conformallyflatnesstoorderkmeansjkp(W)=0.TherelationbetweentheWeyltensorandtheCottontensorisgivenbelow:∇aWabcd=(n−3)Cbcd.Andthusjkp(W)=0⇒jk−1p(C)=0.AgainbyLemmaA.3,wehavejkp(Ric)=0.Thus,bytheRiccidecomposition(3.1.13),jkp(W)=0togetherwithjkp(Ric)=0implyjkp(Riem)=0.□3.2JetLevelExistenceofConformalNormalMetricsThejetlevelexistenceofconformalnormalmetricsisprovedbyLeeandParkerin[LP87]usingaGrahamnormalizationprocess.Inthissection,wereviewtheirproofwithafocusontheconformalfactors,theconstructionofwhichwillbeappliedlater.Lemma3.2.1.Let(X,g)beann-dimensionalRiemannianmanifold,andx:Rn→Xbetheg-normalcoordinatechartatapointp∈X.Forasmoothfunctionf∈C∞(Rn)suchthatf=O(|x|k),taketheconformalmetricgf:=e2fx∗gandlet˜x:Rn→Rnbethecorrespondingcoordinatetransformationtothegf-normalcoordinatesystem,thenwehave:˜x(x)−x=O(|x|k).Proof:Intheg-normalcoordinatesx,letΓabcand˜ΓabcbetheChristoffelsymbolsforgandgfrespectively.Foreachvectorx,theradialrayγ=x·tisag-geodesicfromthepointp,hencesatisfiestheg-geodesicequation:Γabc(x·t)xbxc=0.(3.2.1)Wealsohavetheconformaltransformationformulafor˜Γabcbelow:˜Γabc=Γabc+fbδac+fcδab−fdgdagbc.(3.2.2)Hencef=O(|x|k)⇒Tabc=˜Γabc−Γabc=O(|x|k−1).19Letγx(t)bethegf-geodesicsatisfyingtheinitialconditionsγx(0)=0,γ′x(0)=x.Thenbydefinitionwehave˜x(x)=γx(1)andγx(t)satisfiesthegf-geodesicequation:d2dt2γax+˜Γabc(γ)ddtγbxddtγcx=0(3.2.3)TaketheTaylorexpansionofγx(t)attheoriginandevaluateatt=1,wehave:˜x(x)=γx(1)=x+···+γ(k)x(0)k!+···,whereγ(k)x(0)k!isthedegreektermintheTaylorexpansionof˜x(x).Hencetoprove˜x(x)−x=O(|x|k)istoproveγ(i)x(0)=0foralli=2,...,k−1,whichweprovebelowbyinductiononk.Sinceγx(0)=0,thestatementistruefork=1.Assumethestatementistruefork≤m+1.Fork=m+2,takethe(m+1)thorderderivativeofγ(x),byEquation(3.2.3)attheorigin,wehave:−γa(m+1)x(0)=d(m−1)dt(m−1)(˜Γabc(γ)ddtγbxddtγcx)|t=0=d(m−1)dt(m−1)(Γabc(γx)˙γbx˙γcx)|t=0+d(m−1)dt(m−1)(Tabc(γx)˙γbx˙γcx)|t=0,wherethesecondequalityfollowsfrom(3.2.2).BythefactthatTabc=O(|x|m+1),thesecondtermvanishesattheorigin.Bytheinductionassumption,wehaveγ(i)x(0)=0for2≤i≤m,,andhencethefirsttermequals:0=󰁛α,b,cΓa(m−1)bc(0)n󰁜i=1(˙γix(0))αi˙γbx(0)˙γcx(0)=d(m−1)dt(m−1)(Γabc(x·t)xbxc)|t=0,whereαaremulti-indicesofabsolutevaluem−1andthelastequalityfollowsfromtheeuqation(3.2.1).□Theorem3.2.2(Lee-Parker).Let(X,[g])beaconformalmanifoldofdimensionn≥3.Atapointp∈X,for∀g∈[g]andl=1,...,∞,thereisauniqueformalpolynomialhlinnvariablesandofdegree≤lsuchthatforanysmoothfunctionfsatisfyingjlp(f)=hlwithrespecttotheg-normalcoordinates,theconformalmetric˜g=e2fghasthefollowingproperties:(a)j1p(˜g)=j1p(g),20(b)det(exp∗˜g)(p)=1+O(rl+1)inthe˜gnormalcoordinates,namely˜giskthordercon-formalnormalatp.Proof:Theproofisbyinductiononthejetlevell.Forl=1,itisclearthatasmoothfunctionfpreservesthe1-jetofgatpifandonlyifj1p(f)=0,andhenceh1(x)=0andisunique.Forthesamereason,weseethattheproperty(a)istrueifandonlyifj1p(hl)=0foranyl.Wehenceassumehlhasquadraticleadingtermsandprovetheuniqueexistenceofsuchhlsatisfyingtheproperty(b)byinduction.Assumethestatementistrueforl=k,namely,there’sauniqueformalpolynomialhkofdegree≤ksuchthatforanysmoothfunctionfwithjkp(f)=hkundertheg-normalcoordinates,theconformalmetricgk:=e2fgisconformalnormaltothekthorderatp.Explicitly,letgk=e2hk·g,with{xk=(xik)}beingthegknormalcoordinatesatp.Withrespectto{xk}wehave:det(gk)=1+󰁛|µ|=k+1{ck+1󰁛µ∼¯µ∂Ricµ(0)}xµk+S¯µx¯µk+O(rk+2),whereS¯µisasymmetrictensordefinedbyapolynomialofRiemνwith2≤|ν|≤k−1.Forl=k+1,lete2fk+1beaconformalfactorsuchthatgk+1:=e2fk+1gkisconformalnormalto(k+1)thorderatp.Ononehand,ifjkp(fk+1)∕=0,thenbyassumptiongk+1=e2(fk+1+hk)·gsatisfies(b)forl=k,howeverjkp(fk+1+hk)=jkp(f)+hk∕=hkcontradictswiththeuniquenessofhkintheinductionassumption.Hencejkp(fk+1)=0andtheS¯µtermisinvariantundertheconformalchangebyefk+1.Letxk+1bethegk+1normalcoordinates,byLemma3.2.1,wehavexk+1−xk=O(|xk|k+1).Hencefor|µ|=k+1,∂(Ricgk+1)µ(0)havethesamevalueunderbothcoordinatesxk+1andxk.Ontheotherhand,foranarbitraryconformalmetricgf=e2fgk,theconformaltrans-formationformulaoftheRiccicurvatureis:Ricgf=Ricgk−(n−2)(d2f−df⊗2)+(∆f−(n−2)|df|2)gk(3.2.4)21By(3.2.4),weseethatifjkp(f)=0,thenconformalchange∂(Ricgf)µ(0)with|µ|=k+1dependsonlyonjk+1p(f).Insummary,∂(Ricgk+1)µ(0)dependsonlyonthehomogeneousdegreek+1termsoffk+1andhasthesamevaluewithrespecttoboth{xk+1}and{xk}coordinates.TheTaylorexpansionofgk+1inthe{xk+1}coordinatesisdet(gk+1)=1+󰁛|µ|=k+1ck+1(∂Ricgk+1µ(0))x¯µk+1+S¯µx¯µk+1+O(rk+2).Thusgk+1isconformalnormaltotheorder(k+1)ifandonlyifforany¯µwith|¯µ|=k+1,wehave:󰁛|µ|=k+1ck+1(∂Ricgk+1µ(0))+S¯µ)x¯µk+1=0.Since∂(Ricgk+1)µ(0)havethesamevaluewithrespecttobothxk+1andxkcoordinates,byacoordinatetransformationtothe{xk}coordinatesystem,wehave:󰁛|µ|=k+1ck+1(∂Ricgk+1µ(0))xµk+˜S¯µ)x¯µk=0.(3.2.5)Weworkwith{xk}coordinatesandletfk+1=󰁛ci1···ik+1xi1k···xik+1k.(3.2.6)Takethe(k+1)thorderderivativeof(3.2.4)inthe{xk}coordinates,wehave:󰁛|µ|=k+1ck+1(∂Ricgk+1µ(0))xµk+˜S¯µx¯µk=󰁛|µ|=k+1ck+1∂(Ricgk)µ(0)xµk−(n−2)d2fk+1(xk,xk)+r2∆0fk+1+˜S¯µx¯µk.ByEuler’sformula,d2fk+1(xk,xk)=(xkd)2fk+1−(xkd)f=k(k+1)fk+1.ThusEquation(3.2.5)isequivalenttothefollowingequationoffk+1:(r2∆−(n−2)k(k+1))fk+1=−ck+1(󰁛|µ|=k+1∂Ricµ(0)+˜S¯µ)xµk.(3.2.7)Comparingthecoefficientsofxi1···ik+1konbothsidesof(3.2.7)andbyLemma5.3in[LP87],weobtainanon-degeneratelinearsystemofequationsfortheunknowns,i.e.thecoefficientsci1···ik+1offk+1.Thusgivesauniquesolutionforci1···ik+1.Lethk+1(y)=hk(y)+󰁓ci1···ik+1yi1···yik+1withybeingtheformalvariables.De-note{x}forthegnormalcoordinates,thenanyfunction˜fk+1satisfyingjk+1p(˜fk+1)=22󰁓ci1···ik+1xi1···xik+1inthe{x}coordinates,bychangingtothe{xk}coordinates,wehave:˜fk+1=󰁛ci1···ik+1xi1···xik+1+O(rk+2)=󰁛ci1···ik+1xi1k···xik+1k+O(rk+2)=fk+1+O(rk+2)(3.2.8)Hencejk+1p(˜fk+1)=jk+1p(fk+1)andthemetric˜gk+1:=e2˜fk+1·gk=e2˜fk+1+2hk(x)·gwithjk+1p(˜fk+1+hk)=hk+1(x)isconformalnormaltothe(k+1)thorder.Wetherebyprovedthetheoremforallk<∞byinduction.Fork=∞,wefirstrecallBorel’sLemmabelow(cf.[Hör15]):Lemma3.2.3(Borel’sLemma).ThecanonicalmapfromtheringofgermsofC∞functionat0∈RntotheringofformalpowerseriesobtainedbytakingtheTaylorseriesat0issurjective.Explicitly,letf=󰁓∞|α|=2cαxαbetheuniqueformalpowerseriesobtainedbytheabovealgorithm,whereαisamulti-indexwithabsolutevalue|α|.FixψasmoothbumpfunctiononRsuchthatψ=1onB1,andsupp(ψ)⊆B2.For|α|=m,letHα=max0≤l≤k<m󰀕22m−lm!(k+1)!|cα|󰀂ψ(k−l)󰀂∞(m−k)!(k−l)!l!󰀖1m−k.Thenf:=∞󰁛|α|=2cαψ(Hα|x|)xα(3.2.9)isasmoothfunctioninC∞(Rn)ofwhichtheTaylorseriesattheoriginequalsf.Foranyfiniteintegerk≥2,letfkbethetruncationoffuptodegreek.Thene2fg=e2f−2fk(e2fkg),wherebydefinitionthefunction2f−2fk=O(rk+1)andtheconformalmetric(e2fkg)isofkthorderconformalnormalatp.Letxfandxfkbethenormalcoordinatesofe2fgande2fkgrespectively,thenbyLemma3.2.1,wehavexf−xfk=O(|xfk|k+1).Hencethecoordinatetransformationtothexfcoordinatespreservesthekthorderconformalnormalproperty.Hencee2fgisconformalnormalatptoanyfiniteorderkandthusthestatementistruefork=∞.□AswehaveseeninSection2.3,givenaconformalstructure[g],onecandefinethecorresponding󰁧CO(n)structurePwhichisaprincipal2-framebundleoverX.Byapplying23theconformalnormalcoordinates,wecangiveageometricconstructionofPastheprincipal󰁧CO(n)bundleofconformalnormal2-frames:Theorem3.2.4.LetCNC2(X)=⊔p∈XCNC2(p)bethebundleof2-jetsofthesecondorderconformalnormalcoordinates,thenCNC2(X)isaprincipal󰁧CO(n)bundleonXwhichcharacterizestheconformalstructure[g]asthe󰁧CO(n)structure.Proof:CNC2(X)isasub-bundleofthegenerallinear2-framebundleP2,andhencehasalocaltrivializationinducedfromP2.ToseeCNC2(X)isaprincipal󰁧CO(n)bundle,wedefinethe󰁧CO(n)groupactiononCNC2(X)asfollows:Foranelementj20(ϕ)∈CNC2(p)definedbyametricg∈[g]suchthatj2p(g)∈CN2(p),andβ∈O(TpX,gp),theactionof󰁧CO(n)onj20(ϕ)isdefinedasfollows:By(2.3.1),ageneralelementof󰁧CO(n)isoftheform:h(x)=λAx−x2η1−2η·Ax+x2η2,whereλ>0,η∈Rn,A∈O(n).WenowfocusonRnanddenoteϕ∗gbrieflyasg,byadirectcalculationwehave:j10(h∗g)=λ2(1+4ηAx)j1(g).Hence󰁧CO(n)actsfreeandtransitivelyonCN1(p).BytheLemma(3.2.2),forany˜g∈j10(h∗g),thereisauniquehomogeneousdegree2polynomialfsuchthatdet(e2f˜g)(p)=1+o(r3)andnoticethath∗βisanorthonormalframeofh∗g(p)=e2f˜g(p).Hencethenormalcoordinates˜ϕdefinebye2f˜gandh∗βgivesanelementj20(˜ϕ)∈CNC2.Wedefinetheactionof󰁧CO(n)onCNC2(p)ash·j20(ϕ)=j20(˜ϕ).(3.2.10)Sincetheactionof󰁧CO(n)onCN1=J1([g])isfreeandtransitiveandO(n)⊆󰁧CO(n)actsontheorthonormalframesfreeandtransitive,the󰁧CO(n)actiononCNC2(X)isfreeandtransitive.ToshowCNC2(X)definesa󰁧CO(n)structure,weneedtoshowthe󰁧CO(n)actiondefinedaboveisthejetcompositionaction(2.3.4).Letι:CNC2(X)→P2bethenaturalinclusionmap,itissufficienttoshowιis󰁧CO(n)equivariant.24Recallnearapointp∈P2,wehavenaturallocalcoordinatesu=(ui;uij;uijk)andanaturallocalcoordinatesfor󰁧CO(n):Forageneralelementh=hλ,A,η∈󰁧CO(n),thecorrespondingcoordinatesish=(hij,hijk)=(λaij,12(ηjδik+ηkδij−ηiδpq)λ2apjaqk)Letu=(0,δij,0),thentheGL2nactiononuinlocalcoordinatesis:h·u=(hij,hijk)·(0,δij,0)=(0,λaij,12(ηjδik+ηkδij−ηiδpq)λ2apjaqk).(3.2.11)Ontheotherhand,theactionof󰁧CO(n)onCNC2isdefinedbyexponentialmaps,andthustoobtainalocalexpressionoftheaction(3.2.10),considertheinitialvalueproblemofthegeodesicequation:󰀻󰁁󰁁󰁁󰀿󰁁󰁁󰁁󰀽¨γi+󰁨Γijk˙γj˙γk=0,˙γ(0)=dh(X),γ(0)=p,(3.2.12)where󰁨ΓijkistheChristoffelsymboloftheconformalnormalmetric˜g=e2fg,withj10(e2f)=λ2(1+4η·Ax).TheconformaltransformationformulaofChristoffelsymbolsisgivenasbelow:󰁨Γijk=Γijk+∂f∂xjδik+∂f∂xjδik−∂f∂xlgligjk.Thus󰁨Γijk=−(bjδik+bkδij−biδjk)+o(r).TakingtheTaylorseriesiterationof(3.2.12),atstep1,wehave(˙γi)1=λaijxj+(bjδik+bkδij−biδjk)λ2aipxpajqxq(t).Bytakingtheintegralofton[0,1],wehavej20(h·˜ϕ)=λaijxj+12(bjδik+bkδij−biδjk)λ2aipxpajqxq(3.2.13)Theinclusionmapιis󰁧CO(n)equivariantbycomparing(3.2.11)and(3.2.13).□25CHAPTER4LOCALEXISTENCERESULTSInthischapter,wediscusslinearPDEresultswhichwillbeappliedintheproofofTheorem5.1andreviewCao’sproofofthelocalexistenceofconformalnormalmetricswithcarefultrack-ingofthebackgroundmetricsandconstantsinestimations.4.1LinearPDEResultsLetRnbethestandardEuclideann-spacewithcoordinates{x=(xi)}.DenoteR=|x|=(󰁓x2i)12,θ=x|x|,Bρ={|x|≤ρ|x∈Rn}.Take∆0=󰁓∂2∂x2ithestandardLaplacianand∆∗thesphericalLaplacian.TwoLaplacianoperatorsarerelatedby:∆0(v)=∂2v∂R2+n−1R∂v∂R+1R2∆∗v.(4.1.1)Considerthefollowinglineardifferentialoperator:L0(v)=∆0(v)+(n−2)∂2∂R2(4.1.2)DefinethefollowingweightedHöldernormsandspacesonwhichtheoperatorL0isapplied:For0<α<1,3≤k<N,defineCk,α;N,ρtobethespaceoffunctionsintheHölderspaceCk,α(Bρ)forwhichthefollowingnormisfinite:󰀂|v|󰀂k,α;N,ρ=sup0<r≤ρ{r−N|v|Ck,α(Br−Br/2)}=sup0<r≤ρ{r−N(k󰁛|β|=0r|β|supr2󰃑|x|󰃑r{|∂βv(x)|}+rk+αsup{|∂βv(x)−∂βv(y)||x−y|α|x∕=y,r2󰃑x,y󰃑r,|β|󰃑k})}(4.1.3)Remark4.1.1.In[Cao91],CaogeneralizedtheoperatorL0toacollectionofsingularellipticdifferentialoperators,seealso[PR00]forasemi-lineargeneralization.In[PR00],theweightedHöldernorm󰀂󰀂k,α;N,ρisgeneralizedtodefineafamilyofBanachspacesCk,αν,σ(¯Ω\Σ),see[PR00]page23Lemma2.1fordetails.Inparticular,forσ=ρ,¯Ω=Bρ,Σ={0},ν=N󰃍2k,wehaveaBanachspaceCk,αN,ρ(Bρ\{0})={v|v∈Ck,αloc(Bρ\{0}),󰀂|v|󰀂k,α;N,ρ<∞}.Ontheotherhand,foranelementv∈Ck,αN,ρ(Bρ\{0}),bytakingthetermwith|β|=0inthe(k,α;N,ρ)norm,wehave|x|−Nsup0<|x|󰃑ρ|v(x)|<∞.Thereforev(x)=O(|x|N)near26theorigin,andthusv(x)∈Ck,αloc(Bρ)=Ck,α(Bρ)andCk,α;N,ρisaBanachspace.Lemma4.1.2.Let0<ρ≤1,andv∈Ck,α(B1),then󰀂|v(ρx)|󰀂k,α;N,1≤ρN−k󰀂|v(x)|󰀂k,α;N,ρ.(4.1.4)Proof:Bydefinition󰀂|v(ρx)|󰀂k,α;N,1=sup0<r≤1{r−N(k󰁛|β|=0r|β|supr2󰃑|x|󰃑r{|∂βv(ρx)|}+rk+αsup{|∂βv(ρx)−∂βv(ρx′)||x−x′|α|x∕=x′,r2󰃑x,x′󰃑r,|β|󰃑k})}Substitutingλ=ρrandy=ρxtotheaboveequation,wehave󰀂|v(ρx)|󰀂k,α;N,1=sup0<λ≤ρ󰀻󰀿󰀽ρNλN(k󰁛|β|=0λ|β|supλ2󰃑|x|󰃑λ{|∂βv(y)|}+λk+αsup{ρ(|β|−k)|∂βv(y)−∂βv(y′)||y−y′|α|y∕=y′,λ2󰃑x,x′󰃑λ,|β|󰃑k})󰀞≤ρN−ksup0<λ≤ρ󰀻󰀿󰀽λ−N(k󰁛|β|=0λ|β|supλ2󰃑|x|󰃑λ{|∂βv(y)|}+λk+αsup{|∂βv(y)−∂βv(y′)||y−y′|α|y∕=y′,λ2󰃑x,x′󰃑λ,|β|󰃑k})󰀞=ρN−k󰀂|v(x)|󰀂k,α;N,ρ.□DefineZρ:={f∈C∞(Bρ)|j∞0(f)=0},wehavethefollowinglemma:Lemma4.1.3.Zρ=∞󰁟k,NCk,α;N,ρProof:Withoutlossofgenerality,weproveforρ=1andomittheindexρ.Forasmoothfunctionf∈C∞(B1),weclearlyhavethefollowingequivalence:sup0<r≤1supr2≤|x|≤r|f(x)|rN<∞⇔f(x)=O(|x|N).(4.1.5)Tosee󰁗∞k,NCk,α;N⊆Z1,foreachv∈Ck,α;N,takethe|β|=0terminitsweightednorm,wehavesup0<r<1supr2≤|x|≤r|v(x)|rN<∞,thusby(4.1.5)wehavev∈O(|x|N).LetN→∞,wehavev∈Z1.27Conversely,forv(x)∈Z1,foreach∂βterminthenorm󰀂󰀂k,α;N,wehave:O(|x|N−|β|)=sup0<r<1r−N+|β|supr2≤|x|≤r|∂βv(x)|<∞⇔∂βv(x).Fortheαcontinuousterms,wehavesup0<r<1rk+α−Nsup{|∂βv(x)−∂βv(y)||x−y|α|x∕=y,r2≤|x|,|y|<r,|β|≤k}<2sup0<r<1rk+α−Nsup{|∂βv(x)−∂βv(y)||x−y||x∕=y,r2≤|x|,|y|<r,|β|≤k}<2sup0<r<1rk+α−Nsupr2≤|x|≤r{∂β′v(x)||β′|≤k+1}<∞⇒∂β′v(x)=O(|x|N−k−α)Thusv∈󰁗∞k,NCk,α;N.□L0mapsCk,α;N,ρtoCk−2,α;N−2,ρ,ZρtoZρandhasaboundedlinearrightinverseoperatorSonZρdefinedasfollows:Theeigenvaluesof−∆∗areλl=l(l+n−2)withl=0,1,2,....Thecorrespondingeigenspaceofλlisofdimensionnl=󰀃n+l−1n−1󰀄−󰀃n+l−3n−1󰀄.Denote{ϕm}theorthonormalbasisofL2(Sn−1)suchthateachϕmisaneigenvectorofλl(m),withl(m)=lfor1+󰁓l−1i=0ni≤m≤󰁓li=0ni.Forafunctionf∈Zρ,considertheinhomogeneousequationL0v=f.(4.1.6)Insphericalcoordinates(R,θ),taketheFourierseriesf(R,θ)=∞󰁛m=0fm(R)ϕm(θ),wherefm(R)=󰁕Sn−1f(R,θ)ϕm(θ)dθ.Bytheseparationofvariablesmethod,aformalFourierseriesv(R,θ)=󰁓∞m=0βm(R)ϕm(θ)solves(4.1.6)iffβmsatisfiesthefollowinginhomogeneousCauchy-Eulerequationforeachm:(n−1)β′′m+n−1Rβ′m−λl(m)R2βm=fm.(4.1.7)mByvariationofparameters,weobtainthefollowingparticularsolutionsof(4.1.7)m:β0(f)=1n−1󰁝R0r·lnRrf0(r)dr,βm(f)=Re󰁛i=1,2Rγim󰁝R0fm(r)r1−γim2(n−1)γimdrform≥1,whereγim=(−1)i󰁴λl(m)n−1.28Takeasmoothcut-offfunctionζ∈C∞(R)suchthat:ζ(R)=󰀻󰀿󰀽1,R󰃑34ρ;0,R󰃍45ρ.WecannowdefinethepromisedoperatorSas:S(f)=ζ(R)∞󰁛m=0βm(f)·ϕm.Theorem4.1.4(cf.Corollary2.14,[Cao91]).SisalinearoperatoronZρsuchthat:a)L0(Sf)(y)=f(y)forall|y|≤34ρ.b)Ifft(y)=f(ty),0<t≤1and|y|≤34ρ,then(Sft)(y)=t−2(Sf)(ty).c)Foranyk,α,N,ρ,thereisaconstantnumberK1suchthat󰀂|S◦f|󰀂k,α;N,ρ≤K1{󰀂|f|󰀂0,α;N,ρ+󰀂|f|󰀂k−2,α;N−2,ρ}.(4.1.8)Afact:Given(X,µ),(Y,ν)twoσ-finitemeasurespaces.Let{fn},{gk}betwocountableorthonormalbasisfortheHilbertspacesL2(X)andL2(Y)respectively.Then{fngk}isacompleteorthonormalbasisforL2(X×Y).Proofofthefact:<fngk,fmgl>=<fn,fm><gk,gl>=δnmδkl,hence{fngk}areor-thonormal.Leth∈L2(X×Y)suchthat<h,fngk>=0foranyn,k.Namely,wehave󰁝X󰀕󰁝Yhgkdν󰀖fndµ=0.(4.1.9)Denoteuk(x):=󰁕Yh(x,y)gkdν,bytheHölderinequality,wehave󰀂uk󰀂2L2=󰁝X󰀕󰁝Yhgkdν󰀖2dµ≤󰁝X󰀕󰁝Yh2dν󰀖󰀕󰁝Yg2kdν󰀖dµ=󰁝X󰁝Yh2dνdµ=󰀂h󰀂2L2<∞.Andhenceuk∈L2(X),andby(3.1)uk=0almosteverywhereonX.DenoteEk={x∈X|uk(x)∕=0},µ(Ek)=0,thusthecountableunionE=󰁖∞0Ekhasµ(E)=0.Thusuk=0onX\Eforallk,namely󰁕Yh(x,y)gk(y)dν=0.Thush(x,y)=0almosteverywhereonY,forallx∈X\E.󰀂h󰀂2L2=󰁝X󰁝Yh2dνdµ=󰁝X\E󰁝Yh2dνdµ=0.29Thush=0,and{fngk}isacompleteorthonormalbasisforL2(X×Y).□Lemma4.1.5.Fork,Nbigenough,thekernelofL0inCk,α;N,ρinpolarcoordinatesconsistsofu=∞󰁛m=mNcmRγl(m)ϕm(θ),(4.1.10)whereγl(m)=󰁴λl(m)n−1,λl(m)andϕmaregivenasabove.mNisthesmallestintegersuchthatγl(m)≥N.Inparticular,KerL0∩Zρ=0.Proof.BySturm-LiouvilleTheorem,thecountableset󰀝ψ0=󰁵1ρ,ψl=󰁵2ρsin󰀕nπrρ󰀖󰀞∞l=1isacompleteorthonormalbasisofC0[0,ρ],thusofL2[0,ρ].Bythefactprovedabove,{ψl·ϕm}isacompleteorthonormalbasisofL2([0,ρ]×Sn−1).Fork>3,takeu∈Ck,α;N,ρsuchthatL0u=0.Undersphericalcoordinateswehaveu(r,θ)∈C0([0,ρ]×Sn−1)⊂L2([0,ρ]×Sn−1).Thusu(r,θ)=∞󰁛m=0(∞󰁛l=0cmlψl(r))ϕm(θ)=M󰁛m=0(L󰁛l=0cmlψl(r))ϕm(θ)+ξML,wheretheremainderξMLsatisfies:limM,L→∞󰀂ξML󰀂L2=0.(4.1.11)Foreachindexm,denoteβm=󰁓∞l=0cmlψl(r),thenu(r,θ)=󰁓∞m=0βm(r)ϕm(θ).Since{ϕm}∞0isacompleteorthonormalbasisofL2(Sn−1),wehaveβm(r)=󰁝Sn−1ru(r,θ)ϕm(θ)dvolSn−11.Thusβm(r)∈Ck[0,ρ],with󰁓∞l=0cmlψl(r)itsFourierexpansion.Letg(r)∈C∞0([0,ρ])beanarbitrarytestfunction.30Foreachindexm0,andM>m0,consider0=󰁝BρL0u·g(r)ϕm0=󰁝BρL0(M󰁛m=0(L󰁛l=0cmlψl(r))ϕm(θ)+ξML)g(r)ϕm0=󰁝BρL0(M󰁛m=0(L󰁛l=0cmlψl(r))ϕm(θ))g(r)ϕm0+󰁝BρL0(ξML)g(r)ϕm0Ononehand,integrationbypartsandusingHölderinequality,wehave:󰀏󰀏󰀏󰀏󰀏󰁝BρL0(ξML)g(r)ϕm0󰀏󰀏󰀏󰀏󰀏=󰀏󰀏󰀏󰀏󰀏󰁝BρξMLL0(g(r)ϕm0)󰀏󰀏󰀏󰀏󰀏≤󰁝Bρ|ξML|·|L0(g(r)ϕm0|≤󰀂ξML󰀂L2󰀂L0(g(r)ϕm0󰀂L2≤C󰀂ξML󰀂L2Thusby(4.1.11),wehavelimM→∞,L→∞󰁝BρL0(ξML)g(r)ϕm0=0.Ontheotherhand,󰁝BρL0󰀣M󰁛m=0(L󰁛l=0cmlψl(r))ϕm(θ)󰀤g(r)ϕm0=󰁝Bρ󰀣M󰁛m=0L0(βLm)ϕm(θ)󰀤g(r)ϕm0=󰁝Bρ((n−1)β′′Lm0+n−1rβ′Lm0−λm0r2)g(r)ϕ2m0=󰁝ρ0((n−1)β′′Lm0+n−1rβ′Lm0−λm0r2)rn−1gdr.Thusinsummary󰁝ρ0limL→∞((n−1)β′′Lm0+n−1rβ′Lm0−λm0r2)g(r)rn−1dr=0Sinceg(r)isarbitrary,wehavelimL→∞((n−1)β′′Lm0+n−1rβ′Lm0−λm0r2)=0(4.1.12)Sinceβm0∈Ckwithk>3,bythestandardFourierseriesfact,β′′Lm0convergesuniformlytoβ′′m0,thus(4.1.12)gives(n−1)β′′m0+n−1rβ′m0−λm0r2=0,whichistheCauchy-Eulerequationwithcharacteristicpolynomialγ2=λm0n−1.31Thus±γm0=±󰁴λm0n−1.andthegeneralsolutionisoftheform:ugen=∞󰁛0(cmrγm+dmr−γm)ϕm.Sinceu∈Ck,α;N,ρ,bydefinition,wehavesupx∈Sn−1R|u(x)|RN≤supx∈BR\BR2|u(x)|RN≤󰀂|u|󰀂k,α;N,ρ<∞,hence|u(x)|≤cN|x|N,andu=∞󰁛mNcmRγmϕm(θ),wheremNthesmallestintegersuchthatγm≥N,andtheequalityholdsinL2sense.Ifu∈Zρ,wethenhaveU(r)=󰁝Sr|u|2dvolSr≤CNr2NVol(Sn−1r)=Cr2N+n−1(4.1.13)holdsforanyN.Assume,bycontradiction,thatthereexistsasmallestm0suchthatthecoefficientcm0∕=0,thenU(r)=󰁝Sr|u|2dvolSr=󰁝Sr󰁛(c2mr2γmϕ2m)dvolSr≥󰁝Src2m0r2γm0ϕ2m0dvolSr=Cr2γm0+n−1,contradictswith(4.1.13)whenN>γm0.Thusu=0.Corollary4.1.6.Onthefunctionspace,Zρ,SisaninverseoperatorofL0onbothsides.Proof:Forv∈Zρ,sinceSisarightinverseofL0,wehave:L0(S(L0(v))−v)=(L0◦S)(L0(v))−L0(v)=0.ThusS(L0(v))−v∈ker(L0)∩Zρ,byLemma4.1.5,wehaveS(L0(v))=v,henceSisaleftinverseofL0onZρ.□4.2LocalExistenceofConformalNormalMetricsTheorem4.2.1.Let(X,g0)beaC∞RiemannianmanifoldandletpbeapointonX.Thenthereexistsaconformalmetricg=Φg0suchthatdetgij(y)=1forallsufficientlysmall󰀂y󰀂,i.e.,theexponentialmapofgatp,expp,isalocalvolumepreservingmapinasmallneighborhoodofp.TheabovetheoremisprovedasCorollary0.1in[Cao91],wegiveabriefreviewofCao’s32workfocusingonconformalnormalmetrics.Wefirstrephrasetheexistenceproblemofaconformalnormalmetricintosolvingasingularellipticequationasfollows:Let(X,g0),p∈Xbegivenasinthetheoremabove.Withinag0normalcoordinatechartatp,multiplyingg0withaconformalfactorefdefinedin(3.2.9),weobtainaconformalmetric˜g=efg0suchthatj∞p(det(exp∗˜g))=1.Wecanthereforeassumeinthebeginningthatj∞p(det(exp∗g0))=1andalsouptoaconstantrescaling,weassumethattheinjectiveradiusofg0atpisgreaterthan1.Letg=Φg0beametricconformaltog0.Thefactthatgisconformalnormalinaneighborhoodofpisequivalenttothefactthatdet(g)=|g|isasolutionoftheinitialvalueproblemofthefollowingordinarydifferentialequation:󰀻󰀿󰀽∂rln|g|=0,ln|g(p)|=0Namely|g|=1⇔∂rln󰁳|g|=0.Undertheg0-normalcoordinates{xi},wetaker(x)=distg(p,x)theg-distancefunctionfromapointxtop,forwhichwehave:∆gr=1󰁳|g|∂j(󰁳|g|gij∂ir)=∂j(gij∂ir)+1󰁳|g|∂j(󰁳|g|)gij∂ir=∂j(dr)+∂rln(󰁳|g|)=n−1r+∂rln(󰁳|g|)Insummary,gislocallyconformalnormalinaneighborhoodUofpifandonlyifgsatisfiesthefollowingequationonU:∆gr=n−1r.(4.2.1)Denoter0(x)theg0distancefunctionfromapointxtop,wehave1=󰀂dr󰀂2g=󰀂dr󰀂2Φg0=1Φ󰀂dr󰀂2g0.HenceΦ=󰀂dr󰀂2g0.(4.2.2)33Definethefunctionw(x)byr(x)=r0(x)ew(x)andsubstitutewinto(4.2.2),wehaveΦ=󰀂dr󰀂2g=(1+2xiwi+r20󰀂dw󰀂2g0)e2w.(4.2.3)ItisclearbydefinitionthatΦ=1⇔r=r0⇔w=0.Furthermore,bycomparingpartialderivatives,wehavejk0(w)=0⇔jk0(Φ)=1.Thusgandg0havethesamek-jetatpiffjk0Φ=1iffjk0w=0.By(4.2.3)wecanrewrite(4.2.1)asanequationofw(x):V(x,∂w,∂2w)=f,(4.2.4)wheref=−∂r0ln󰁳det(g0(x))r0∈Z1V(x,∂w,∂2w)=L0(w)+G(x,∂w)+Q(x,w,∂w),(4.2.5)whereL0istheoperator(4.1.2)definedabove,andGandQaresmoothfunctionssatisfying:G=󰁓xixj|x|2Gij(x,∂w)andGij(x,0)=0;(4.2.6)Q=󰁓ijQij(x,∂w)wijandQij=󰁓xk∂kQij(4.2.7)Theorem4.2.1aboveisthenacorollaryofthefollowingresult:Theorem4.2.2([Cao91],CorollaryB).Givenf∈Z1,thenforasmallenoughconstant0<ρ≤1,thereexistsafunctionw∈ZρsolvingEquation(4.2.4).Proof:Fix∗=(2n,12;4n,1),wewillprovetheexistenceofasolutionw∈C2n,12;4n,ρof(4.2.4).Fortheregularityofw∈Zρ,seeCorollaryBin[Cao91].TakethecompletemetricspaceDK0={v∈C∗|󰀂|v|󰀂∗≤K0},whereK0=8󰀂|S(f)|󰀂∗+1.Forv∈DK0andρ>0,defineFρ(v)=S[L0(v)−V(ρx,ρ∂v,∂2v)+f(ρx)].(4.2.8)Bytheestimate(4.1.8)inTheorem4.1.4,weseethatfortwofunctionsv1andv2inDK0,thereexistsaconstantK1dependsonnandK0suchthat:󰀂|Fρ(v1)−Fρ(v2)|󰀂∗≤K1ρ󰀂|v1−v2|󰀂∗(4.2.9)34Takeρ=ρf=14(K1+K0),thenK1ρ<14and󰀂|Fρ(v)|󰀂∗≤󰀂|Fρ(v)−Fρ(0)|󰀂∗+󰀂|Fρ(0)|󰀂∗≤K04+󰀂|S(f(ρx))|󰀂∗=K04+󰀂|(Sf)(ρx)|󰀂∗ρ2≤K04+󰀂|(Sf)(x)|󰀂∗≤K0,wheretheequalityfollowsfromTheorem4.1.4.(b)andthethirdinequalityfollowsfromLemma4.1.4.Thus,FρfisacontractionmappingonDK0.HencebytheBanachfixedpointtheorem,FρfhasafixedpointinDK0,denotedasv.ApplyL0onbothsidesofFρf(v)=vandbyTheorem4.1.4.(a),weseethatvisasolutionofV(ρx,ρ∂v,∂2v)=f(ρx)inC∗,andhencew(x)=ρ2v(xρ)isasolutionof(4.2.4)inC2n,12;4n,ρ.□35CHAPTER5UNIQUENESSOFCONFORMALNORMALMETRICSATTHEGERMLEVELLet(X,[g])beaconformalmanifold.Atapointp∈X,thegermsatpofmetricsgintheconformalclass[g]isasetGp[g]={germp(g)|g∈[g]}.ByTheorem4.2.1,thissetcontainsatleastoneconformalnormalmetric.LetCN(p)=󰀋germp(g)∈Gp[g]|det(exp∗pg)=1󰀌denotethesubsetofa(germsof)conformalnormalmetricsintheconformalclass.LetJ1p[g]=󰀋j1p(g)|g∈[g]󰀌andπ1:Gp[g]→J1p[g]withπ1(germp(g))=j1p(g)betheprojectionmapto1-jets.Themaintheoremisstatedasfollows:Fixp∈X.Foreachmetricg,theconformalclass[g],the1-jetclassj1p(g)containsauniqueconformalnormalmetric.Theorem5.1(maintheorem).Atp∈X,fixanarbitrary1-jetclassoftheconformalmetrics,thereisaconformalnormalmetricgatpwithinthe1-jetclassandthemetricgisuniqueuptothegermlevel.Namely,thejetprojectionmapπ1restrictedtoCN(p):π1|CN(p):CN(p)→J1p[g]isabijectionontoJ1p[g].Wefirstproveauniquenesstheoremforsolutionsofequationsoftype(4.2.4)whichisacorollaryofLemma4.1.5.Theorem5.2.GiventheinhomogeneousequationV(x,∂w,∂2w)=L0(w)+G(w)+Q(w)=f,(5.1)whereL0isdefinedas(4.1.2),GandQsatisfies(4.2.6)and(4.2.7)respectively,andf∈Zλforsomeλ>0.Thenthereexistsapositiveconstant0<τ≤λsmallenoughsuchthatthereexistsauniquesolutionof(5.1)inZτ.Proof:Forρ>0,definetheoperatorFρasin(4.2.8).LetK0=8󰀂|S(f)|󰀂2n,12;4n,λ+1,thenbythesameargumentasTheorem4.2.2,thereexistsaconstantK>0dependsonK036suchthatforρ≤14(K+K0),FρisacontractionmappingonDK0={v∈C2n,12;4n,λ|󰀂|v|󰀂2n,12;4n,λ≤K0}Forsuchaρ>0,ononehand,ifv∈ZλisafixedpointofFρ,thenbyapplyingL0onbothsidesofFρ(v)=v,weseethatvisasolutionoftheequation:V(ρy,ρ∂v,∂2v)=f(ρy).(5.2)Ontheotherhand,ifv∈Zλsolves(5.2),thenFρ(v)=S(L0(v))=v,wherethelastequalityfollowsfromCorollary4.1.6.Hence,bytheuniquenesspropertyofBanach’sfixedpointtheorem,Equation(5.2)hasauniquesolutioninZλ.WeclearlyhaveabijectionbetweensolutionsofEquation(5.2)inZλandsolutionsofEquation(5.1)inZρλbylettingw(x)=ρ2v(xρ).Setτ=ρλ,andtheconclusionfollows.□Proofofthemaintheorem::Weprovethisbyconstructingamapsp:J1p[g]→CNp,andproveitisawell-definedinversemapofπ1|CN(p)onbothsides.Fixabackgroundmetricg0∈[g]togetherwithanothonormalframeθ0of(TpX,g0(p)).ByLemma3.3,g0inducesanisomorphismη:R⊕T∗pX→J1p[g].Let(ϕ0,x=(xi))betheg0normalcoordinatechartatpspecifiedbyθ0.Withrespecttowhich,wecanwriteηexplicitly:Foreachα=(λ,v)∈R⊕T∗pX,takeα(x)=λ+󰁓ni=1cixi,whereci=v(θ0i).Letgαij(x)=eα(x)g0ij(x),thenη(α)=j1p((ϕ−1)∗(gαijdxidxj)).Letθλ=e−λ2θ0,thenθλisanorthonormalframeof(TpX,gα(p)).Let(ϕα,xα=(xαi))bethegαnormalcoordinatechartatpspecifiedbyθλ.Wenowworkwith{xα}:ByLemma3.2.2,weseethatbychoosingasmoothbumpfunctionψonR,onecanconstructasmoothfunctionh=h(α)inthexαchartdefinedas(3.2.9)suchthath(xα)=O(r2)andj∞0(det(exp∗(ehgα)))=1.Denotegh=ehgα,itisclearthatθλisanorthonormalframeof(TpX,gh(p)).Againlet(ϕh,xh=(xhi))bethegh37normalcoordinatechartatpspecifiedbyθλ.Wenowworkwith{xh}:Letρ=ρ(h)betheinjectiveradiusofghatp,thenthefunctionf=−∂rln󰁴det(ghij(xh))r∈Zρ.Letw∈ZτbetheuniquesolutionofEquation(4.2.4)withfbeingtheinhomogeneousterm.TheexistenceanduniquenessofwisensuredbyTheorem4.2.2andTheorem5.2.Letgw=Φ(w)gh,withΦ(w)=(1+2󰁓xhi∂iw+|xh|2󰀂dw󰀂2gh)e2w,thengwisaconformalnormalmetriconBτ.Definethemapspasfollows:sp(j1p(g))=germp(gw)=germp((Φ(w)◦ϕ−1h)·(eh◦ϕ−1α+α◦ϕ−10)·g0)(5.3)Themapspiswell-definedindependentofthechoiceoftheframeθpandthesmoothbumpfunctionψ.Forθpindependency,wecheckthefollowing:1.Forα=(λ,ν)∈R⊕T∗pX,α◦ϕ−10(v)=λ+ν(v),henceαiswell-definedindependentofθ.2.Toshowthefunctionhisindependentofthechoiceofθp,itissufficienttoshowitsTaylorseriesattheoriginisindependentofthechoiceofθ.Bychoosingadifferentorthonormalframe,wehavealinearcoordinatetransformationy=Ax,withA∈O(n).Thereforeattheorigin,wehave∂nh∂xi1···∂xin(0)=∂nh∂yj1···∂yjn∂yj1∂xi1···∂yjn∂xin(0).HencetheTaylorseriesofhattheoriginisindependentofθ.3.ThefunctionΦ(w)isasolutionoftheequation:∆ghr+n−22〈d(lnΦ),dr〉gh=−∂rln󰁳det(exp∗gh)r.By1and2above,themetricghisindependentofthechoiceofθ,andhencetheequationofΦisindependentofθandsoisthesolutionΦ.Forψindependency:Let˜hbeadifferentBorelextensionoftheformalpowerseriesinLemma3.2.2andg˜hthecorrespondingconformalmetric.BysolvingCao’sequation(4.2.4),38weobtainaconformalnormalmetric˜g.Let˜g=Φ(˜w)gw,withΦtheconformalfactorand˜wdefinedin(4.2.3).Thenunderthegwnormalcoordinates,˜w∈Zρforsomeρ>0smallenoughandsatisfiestheequationV(x,∂˜w,∂2˜w)=0.(5.4)ByTheorem5.2,thesolutionof(5.4)isuniqueinasmallneighborhoodofp,hence˜w=0andgerm(gw)=germ(˜g).Toseespisarightinverseofπ1|CNp,observethattheconformalfactorsΦ(w)andhbothvanishtothesecondorderandthusdonotaffectthe1-jetofgw.Hencewehaveπ1◦sp(j1p(g))=j1p(gw)=j1p(eα◦ϕ−10g0)=η−1(η(j1p(g)))=j1p(g).Toseespisaleftinverseofπ1|CNp,takeg∈[g]suchthatgermp(g)∈CNp,andletgwbethecorrespondingconformalmetricobtainedasabovethatisalsoconformalnormalatp.Hencewecanwriteg=Φ(x)gwwithrespecttogw-normalcoordinatesxatp,andapplythesameargumentasψindependencyabove,weseethatΦ(x)=1,hencegermp(g)=germp(gw).Thus,theleftinversefollows:sp◦π1(germp(g))=sp(j1p(g))=germp(gw)=germp(g).□Insummary,wehavethefollowingdiagramofmaps:CNpGp[g]R⊕T∗pXJ1p[g]ιηπ1spη−1Andbythesectionmaps,weseethatR⊕T∗Xservesasaroughmoduliofthesetofgermsofconformalnormalmetricinclass[g].5.1ConformalNormalMetricsonLocallyConformallyFlatManifoldsInsection3.1,weprovedTheorem3.1.3,ajetlevelrelationamongconformalflatness,flatness,andconformalnormal.ByapplyingTheorem5.1,wecanobtainasimilarresultinthegermlevelasfollows:39Corollary5.1.1.Let(X,[g])beasmoothmanifoldwithaconformalstructure[g].If[g]islocallyconformallyflatonanopenneighborhoodUofp∈X,thenforaconformalmetricg∈[g],germp(g)isflatifandonlyifgermp(g)isconformalnormal.Proof.Itisclearbydefinitionthataflatmetricisconformalnormal.Conversely,letg∈[g]beaconformalnormalmetriconU.Since[g]isconformallyflatonU,thereexistsaconformalfactore2fsuchthatgf=e2f·gisaflatmetriconU.Withrespecttothegf-normalcoordinates{x},gf(x)=δijisthestandardEuclideanmetric.Pullbackδijbytheconformalmappingϕ=ϕλ,A,ηdefinedin(2.3.1),wehaveϕ∗δij=λ2(1−2Ax·η+x2η2)2δij.ϕ∗δijhasapoleatx=Atηη2.Sinceϕ∗Riem=0,ϕ∗δijisaflatmetriconU∩B(0,|η|−1).Uptopullingbackbyaconformalmappingϕ,wecanassumej1p(ϕ∗gf)=j1p(g),withbothmetricsbeingconformalnormalatp.HencebyTheorem5.1,wehavegermp(gf)=germp(g).Remark5.1.2.ByCorollary5.1.1,weseethatthegermofaflatmetricisuniquelydeter-minedbyits1-jet.ThisfactcanbeproveddirectlywithoutreferringtoTheorem5.1:Proof:Withoutlossofgenerality,leth∈[g]beaflatmetriconU.Letϕbeasmoothfunctionsuchthat˜h=e2ϕhremainsflatonsomeopenneighborhoodV⊆U.BytheconformaltransformationformulafortheRiemanncurvaturetensor,wehave0=󰁨Riem=e2ϕRiem−e2ϕh∧©(Hessϕ−dϕ⊗dϕ+12|dϕ|2h)=−e2ϕh∧©(Hessϕ−dϕ⊗dϕ+12|dϕ|2h).Hencewehaveh∧©(Hessϕ−dϕ⊗dϕ+12|dϕ|2h)=0,(5.1.1)where∧©istheKulkarni-Nozumiproduct.Write(5.1.1)inthehnormalcoordinates,wehaveδik(ϕjl−ϕjϕl+12|dϕ|2δjl)+δjl(ϕik−ϕ1ϕk+12|dϕ|2δik)−δjk(ϕil−ϕiϕl+12|dϕ|2δil)−δil(ϕjk−ϕjϕk+12|dϕ|2δjk)=0.40Thisisanoverdeterminedsystemofequationsofϕ,amongwhichwehavethefollowingnon-trivialrelationsfori∕=j:󰀻󰀿󰀽ϕii+ϕjj=ϕ2i+ϕ2j−|dϕ|2,ϕij=ϕiϕj,(5.1.2)wherethefirstequationisforthecyclicpairs(i,j)=(1,2),(2,3),...,(n,1).DenoteAthecoefficientmatrixofϕii.ItisclearbyelementaryrowoperationthatAisnonsingular.Viewthefirstordertermsϕiasconstantsandsolveforϕii,wegetϕii=ϕ2i−12|dϕ|2.SinceAisnonsingular,theabovesolutionisunique.Defineu:Rn→Rnasu=(ϕi)ni=1,andα:Rn→Mn×n(R)z󰀁→(αji)=󰀻󰀿󰀽(zi)2−12z2,fori=jzizj,fori∕=jAsonecancheck,wehave󰁓nl=1αlk∂αij∂zl=󰁓nl=1αlj∂αik∂zl.Hencethefunctionsuandαsatisfythecompatibleconditionsinthefollowinglemmaforanover-determinedsystemwhichistheproposition19.29in[Lee12].Lemma5.1.3.SupposeWisanopensubsetofRn×Rm,andα=(αlj):W→M(m×n,R)isasmoothmatrix-valuedfunctionsatisfying∂αij∂xk+αlk∂αij∂zl=∂αik∂xj+αlj∂αik∂zlforalli,j,k,wherewedenoteapointinRn×Rmby(x,z)=(x1,...,xn,z1,...,zm).Forany(x0,z0)∈W,thereisaneighborhoodUofx0inRnandauniquesmoothfunctionu:U→Rmsuchthatu(x0)=z0andtheJacobianofusatisfies∂ui∂xj󰀃x1,...,xn󰀄=αij󰀃x1,...,xn,u1(x),...,um(x)󰀄Henceforanyfixedinitialconditionϕ(0)=canddϕ(0)=v,wehaveaunqiueconformalfactorϕsuchthate2ϕgisflatinsomesmallopenneighborhoodVofp.󰃈41CHAPTER6SMOOTHDEPENDENCEOFCONFORMALNORMALMETRICSONPARAMETERSInChapter5,weseethatforasmoothbackgroundmetricg,thereexistsauniqueconformalnormalmetric˜gatanypointp∈X.Inthischapter,weprovethesmoothdependenceofconformalnormalmetricsonafamilyofbackgroundmetricsandgiveanapplicationtotheregularityofthecanonicalmetricgCinaYamabe-positiveconformalclassCintroducedbyHabermannandJostin[HJ99].6.1SmoothDependenceofConformalNormalMetricsWefirstworkoverthebackgroundEuclideanspaceRnandgivealocalsmoothdepen-denceresult(Lemma6.1.7).SupposeB1⊆RnistheunitballcenteredattheoriginanddenotethespaceofsmoothmetricsonB1asMet∞(B1)whichisanopenconeintheFréchetspaceC∞(B1,Sym2Rn).Consideral-parameterfamilyofsmoothmetricsonMet∞(B1)asfollows:γ:Rl→Met∞(B1)t󰀁→γ(t)=gt(x),(6.1.1)withgt(x)=(gij(t,x))satisfiesgij(t,x)∈C∞(Rl×B1).Recallthefollowingwell-knownresultonthesmoothdependenceofsolutionsofasystemofordinarydifferentialequations,seeSec1.6in[Tay96]fordetails:Theorem6.1.1.SupposeD={(s,x,t)}⊆R⊕Rn⊕Rmisopenandf:D→Rnisasmoothvectorvaluedfunction.Considertheinitialvalueproblem:󰀻󰁁󰀿󰁁󰀽dxds=f(s,x(s),t)x(s0)=v,(6.1.2)with(s0,v,t)∈D.Thenthereexistsaconstantδ>0suchthaton[−δ,δ]×Dtheinitialvalueproblem(6.1.2)hasauniquesolutionx=x(s;s0,v,t)∈C∞([−δ,δ]×D).Lemma6.1.2.Supposeγisanarbitraryl-parameterfamilyofmetricsonB1⊆Rngivenas(6.1.1).ForanyK⊆Rlcompact,thereexistsδ>0smallenoughsuchthat˜gt=exp∗gt(gt)42satisfies˜gij(t,x)∈C∞(K×Bδ).Inparticular,thecoordinates{xi}onBδarenormalcoordinatesfor˜gij(t,x)foreacht.Proof:ApplyTheorem6.1.1tothegeodesicequationswithmetricsinγ(K)⊆Met∞(B1).Namely,considerthefollowingsystemofequations:󰀻󰁁󰀿󰁁󰀽duads=−Γabc(x(s),t)ybycu(0)=(p,v),whereu(s)=(x(s),y(s)),y(s)=dx(s)dsandΓabc(x(s),t)istheChristoffelsymbolofgt(x).Suppose(p,v,t)∈B1×B1×K.Weseethat,uptorestrictingtoasmallerballBδ⊆B1,theRiemannianexponentialmapisuniformlydefinedonBδandissmoothwithrespectto(p,v,t).□Wegiveaparameter-dependentversionofBorel’sLemmaofasymptoticexpansionsbe-low:Lemma6.1.3.Suppose{ak(t)∈C∞(K)|K⊆Rcompact,k∈N}isanarbitrarycollectionofsmoothfunctions.Thereexistsasmoothfunctionf(t,x)∈C∞(K×R)suchthat∂kxf(t,0)=ak(t)foranyk∈N.Proof:Letρ(x)beasmoothbumpfunctionsuchthatρ=1on|x|󰃑34andsupp(ρ)=B1.Foreachk∈N,takeAk=󰀂ak(t)󰀂Ck(K),Bk=󰀂ρ(x)󰀂Ck(R),Mk=supi≤k󰀫i󰁛j=0󰀕ij󰀖1(k−j)!󰀬Definef(t,x)=∞󰁛k=0ak(t)xkk!ρ(hkx),withhk=(2kAkBkMk)2k.Weclaimthatthefunctionf(t,x)definedabovesatisfiestherequirements.Formallywehave∂kxf(t,0)=ak(t).Henceitissufficienttoprovethatf(t,x)∈C∞(K×R).Weprovebycheckthefollowingfact:Foreachn,m∈N,theseries∞󰁛k≥2(n+m)󰀏󰀏󰀏󰀏∂nx∂mt󰀕ak(t)xkk!ρ(hkx)󰀖󰀏󰀏󰀏󰀏convergesonK×R.Indeed,wehave:43∞󰁛k≥2(n+m)󰀏󰀏󰀏󰀏∂nx∂mt󰀕ak(t)xkk!ρ(hkx)󰀖󰀏󰀏󰀏󰀏=∞󰁛k≥2(n+m)󰀏󰀏󰀏󰀏󰀏a(m)k(t)n󰁛j=0󰀕nj󰀖xk−j(k−j)!·hn−jk·ρ(n−j)(hkx)󰀏󰀏󰀏󰀏󰀏≤∞󰁛k≥2(n+m)AkBkMkhj−kk·hn−jk≤∞󰁛k≥2(n+m)AkMkBkh2/kk=󰁛k≥2(n+m)12k≤1,wherethefirstinequalityfollowsfromthedefinitionofAk,Bk,Mk,andthefactthatρ(hkx)=0for|x|>1hk,thesecondinequalityisby−k2󰃍n−k.□Remark6.1.4.Byacompletelysimilarargument,onecanshowthatLemma6.1.3holdsforthemultivariablecases.Theorem6.1.5.For{˜gt∈Met∞(Bδ)|t∈K},thesmoothl-parameterfamilyofmetricsobtainedinLemma6.1.2,thereexistsafunctionΨ(t,x)overK×Bδsuchthat:a)Ψ(t,x)∈C∞(K×Bδ),b)Ψt(x)=1+O(|x|2),c)Foreacht∈K,themetricg∞t(x):=Ψ(t,x)˜gt(x)is∞-orderconformalnormalattheoriginin{t}×Bδ.Proof:ApplyLemma3.2.1(cf.[LP87,Theorem5.])toeach˜gt(x),weobtainauniqueformalpowerseries:h∞t(x)=∞󰁛|α|=2cα(t)xα,withcα(t)apolynomialof∂αRicgt(0)and∂βRiemgt(0)with|β|<|α|−2,suchthatforht(x),anarbitraryBorel’sextensionofh∞t(x),thefunctionΨt(x)=eht(x)satisfiesrequirementsb)andc).Bydefinitioncα(t)isasmoothfunctionoftoverRl.On∀K⊆Rlcompact,byLemma6.1.3,wecanchooseh(t,x)∈C∞(K×Bδ),henceΨ(t,x)=eh(t,x)∈C∞(K×Bδ).□Insummary,weobtainasmoothl-parameterfamilyofmetrics:{g∞t∈Met∞(Bδ)|t∈K}(6.1.3)suchthatforeacht∈K,g∞tis∞-orderconformalnormalattheoriginandj10(g∞t)=j10(gt).44ByapplyingLemma6.1.2tothefamily{g∞t},wecanassumethecoordinates{xi}onBδarenormalcoordinatesforg∞tforeacht∈K.Foreacht∈K,wethencorrectthemetricg∞ttoaconformalnormalmetricbyCao’sPDEapproach.OnBδ,writedownCao’sequation(4.2.4)withrespecttog∞t:L0(w)+󰁛i,jxixjx2·Gij(x,∂w,t)+󰁛ijkxk·wij·Qijk(x,∂w,t)=f(x,t),(6.1.4)twheretheexplicitformulaforf(x,t),Gij(x,ζ,t),andQijk(x,ζ,t)aregivenasfollows:f(x,t)=−∂rdetg∞t(x)2rdetg∞t(x)(6.1.5)DenoteΦ1(x,∂w)=1+2󰁓xiwi+x2󰁓(g∞t)ijwiwj,wehave(usingEinstein’sconventionbelow):Gij(x,ζ,t)=(g∞t)ab·Γcab(x,t)ζc·δij+n−2Φ1(2ζiζj+(g∞t)ab·ζaζbζixj)+n−2Φ1(2(g∞t)abζaζb(1+xcζc)+12∂c(g∞t)abζaζb(xc+x2(g∞t)cdζd))δij(6.1.6)Qijk(x,ζ,t)=󰁝10∂k((g∞t)ij−δij)(xs)ds+n−2Φ1(2xk(g∞t)ia(g∞t)jbζaζb+(g∞t)iaζaδjk)(6.1.7)UptochoosingasmallercompactsetKandδ>0ifnecessary,wecanassumethat12<Φ1(x,ζ,t)<2onBδ×Bδ×Kand12<det(g∞(x,t))<2onBδ×K.Hencebytheconstruction,f(x,t)isasmoothfunctioninx,tandft(x)∈Zδwithrespecttox,Gij(x,ζ,t)andQijk(x,ζ,t)aresmoothfunctionswithrespecttox,ζ,t.LetK0=8󰀂|S(f)|󰀂2n,12;4n,λ+1,thenbythesameargumentasTheorem4.2.2,thereexistsaconstantK3>0dependsonK0suchthatforρ≤14(K3+K0),FρisacontractionmappingonDK0={v∈C2n,12;4n,λ|󰀂|v|󰀂2n,12;4n,λ≤K0}.AsintheproofofTheorem5.2,fix∗=(2n,12;4n,δ).Withrespecttoeachg∞t,takeK0(t)=8󰀂|S(ft)|󰀂∗+1,byconstructionK0(t)iscontinuousint,hencefort∈Kcompact,wehaveamaximum¯K0.DenotetheconvexsetD¯K0={v∈C∗|󰀂|v|󰀂∗≤¯K0}.DenoteK3(t)thecontractionconstantin(4.2.9)forthemapFρ,t(v).Namely,wehave:󰀂|Fρ,t(v)−Fρ,t(˜v)|󰀂∗≤K3(t)ρ󰀂|v−˜v|󰀂∗.ThefollowinglemmaisprovedinAppendixB:45LemmaB.1.ForK0>0,andfunctionsinDK0={v|󰀂|v|󰀂k,α;N,δ≤K0},defineTρ(v)=Gρ(v)+Qρ(v)=󰁛i,jxixjx2·Gij(ρx,ρ∂v)+󰁛ijkρxk·vij·Qijk(ρx,ρ∂v),whereQijk(x,ζ)andGij(x,ζ)aresmoothfunctionswithrespecttox,ζ.ThereexistsaconstantK2suchthatfor0<ρ<1andanypairoffunctionsv1,v2∈DK0,wehave󰀂Tρ(v2)−Tρ(v1)󰀂k−2,α;N−2,δ+󰀂Tρ(v2)−Tρ(v1)󰀂0,α;N−1,δ≤K2ρ󰀂˜v−v󰀂k,α;N,δ.Infact,K2=C(α)P(K0)M,whereC(α)isaconstantdependsonα,P(K0)isapoly-nomialofK0andM=maxijk󰀋󰀂Gij(x,ζ)󰀂Ck(D(δ,K0)),󰀂xkQijk(x,ζ)󰀂Ck(D(δ,K0))󰀌,withD=D(δ,K0)=Bδ×BK0⊆Rn⊕Rn.LetK1betheconstantin(4.1.8)for∗=(2n,12;4n,δ):󰀂|S◦f|󰀂∗≤K1{󰀂|f|󰀂0,1/2;4n,δ+󰀂|f|󰀂2n−2,1/2;4n−2,δ}.ApplyLemmaB.1tothefunctionalTρ,tdefinedbythemetricg∞t,forapairoffunctionsv1,v2∈DK0,wehave:󰀂|Fρ,t(v1)−Fρ,t(v2)|󰀂∗=󰀂|S◦(Tρ,t(v1)−Tρ,t(v2))|󰀂∗≤K1·K2(t)ρ󰀂|v−˜v|󰀂∗.WeseethatK3(t)=K1·K2(t).SinceGijandQijkdependsmoothlyonthemetricsg∞t,hencesodotheirCknorms.WeobtainanupperboundofK3(t)bytakethemaximumvaluesforC2n(D)normsofGijandQijk.Ononehand,letτ(t)=18(K3(t)+K0),thenbytheargumentaboveτ(t)isuniformlyboundedbelowbyτ=τ(K)forallt∈K,thesolutionw(x,t)ofCao’sequation(6.1.4)texistsonBτ;ontheotherhand,themapFρ,tisasmoothfamilyofcontractionmapsoverD¯K0×Kwithauniformcontractionconstant.Recallthefollowingclassicalresultofuniformcontractions(cf.[CH12,Theorem2.2]):Lemma6.1.6(UniformContractionPrinciple).LetU,VbeopensetsinBanachspacesX,Yrespectively.Let¯UbetheclosureofU,T:¯U×V→¯Uauniformcontractionon¯Uandletg(y)betheuniquefixedpointofT(·,y)in¯U.IfTissmoothover¯U×V,theng(·)∈C∞(V,X).46ApplyLemma6.1.6toFρ,t,weseethatthesolutionw(x,t)ofCao’sequation(6.1.4)twithrespecttothemetricg∞t(x)dependssmoothlyont.Insummary,wehavethefollowingresult:Lemma6.1.7.Giveasmoothfamilyofmetricsγ:Rl→Met∞(B󰂃),foranyK⊂Rlcompact,thereexist0<δ<εsmallenoughsuchthatthereisanuniquefunctionΦ∈C∞(K×Bδ,R+)suchthatΦt(x)=1+O(r2)andΦ2tgisconformalnormalonBδforanyt∈K.ApplyLemma6.1.7,weconstructasmoothRiemannianmetriconX×Xwhichisconformalnormalnearthediagonalinthefollowingsense:Corollary6.1.8.Given(X,g),thereexistsasmoothRiemannianmetricg⊕Φ2gonX×X,whereΦisafunctionsatisfiesthefollowingconditions:(a)Φ=Φ(p,q)∈C∞(X×X,R+).(b)Nearthediagonal∆⊂X×X,Φ=1+O(r2),whereristhedistancefromthediagonal.(c)ThereexistsanopenneighborhoodUofthediagonal∆suchthatfor∀p∈X,Φ2pgisconformalnormalonUp=({p}×X)∩U.Proof:Letϕ0:U0→Xbeachartcenteredatapointp0∈X.Fixθ0∈Op0(X),whereO(X)istheg-orthonormalframebundle.Forδ>0smallenough,andtheδballBδ⊂Rnattheorigin,wedefineasmoothfamilyofmetricsγ:U0→Met∞(Bδ)asfollows:Foranyx∈U0,letθx∈Oϕ(x)(X)betheorthonormalframeobtainedbytheparalleltransportationfromθ0withrespecttothemetricg.WeidentifyTϕ(x)XwithRnbyθx:Rn→Tϕ(x)Xasθx(v)=󰁓vi(θx)i.Letδ>0smallenoughsuchthatθx(Bδ)isintheinjectivitydomainoftheexponentialmaponTϕ(x)Xforanyx∈U0.Defineγ(x)=(expϕ(x)◦θx)∗g∈Met∞(Bδ).Itisclearbydefinitionthatγisasmoothfamilyofmetrics.47ApplyLemma6.1.7toγ,weobtainasmoothfamilyofconformalnormalmetric˜g=Φ2g,withΦ∈C∞(U0×Bδ).Let˜ϕ:U0×Bδ→exp(TX|ϕ(U0))⊂X×Xbethemaptoanopenneighborhoodofthediagonal∆obtainedbyθxandtheexponentialmap.Thenbyconstruction,themetricg⊕˜ϕ∗˜gisasmoothmetricon˜ϕ(U0×Bδ).Foranotherpoint,p1∈XtogetherwithanopenneighborhoodU1,bythesameargu-ment,weobtainasmoothfamilyofconformalnormalmetricsparametrizedoverU1andbytheuniquenessofconformalnormalmetrics,thetwofamiliesofmetricscoincideontheoverlapU0∩U1andhenceweobtainasmoothfamilyofmetricsg⊕Φ2goverXinanopenneighborhoodU∆ofthediagonal.TakeK⊂U∆compactandV⊃Kopen.Respectively,letµbeabumpfunctiononX×Xsuchthatµ=1onKandsupp(µ)⊆V.Thenthemetricg⊕(Φ2µ+1−µ)gsatisfiestherequirements.□6.2AnApplicationofSmoothFamilyofConformalNormalMetricsInthissection,weapplythesmoothfamilyofconformalnormalmetricsinCorollary6.1.8togiveashorterproofoftheregularityofthecanonicalmetricgCinaYamabe-positiveconformalclassCintroducedbyHabermannandJostin[HJ99].WebeginbyreviewingsomebackgroundknowledgeonGreen’sfunctionsforRiemannianmanifoldsandthemassofasymptoticallyflatmanifolds.ConformalLaplacianandGreen’sfunction.OnaRiemannianmanifold(X,g)ofdimensionn≥3,byaddingamultipleofthescalarcurvatureSgtotheLaplace-Beltramioperator∆g,weobtaintheso-calledconformalLaplacianoperator:Lg=4(n−1)(n−2)∆g+Sg.Lgisconformallycovariantinthefollowingsense.Suppose˜g=u4n−2g,thentheconformalLaplacianchangescorrespondinglyas:un+2n−2◦L˜g=Lg◦u.(6.2.1)Remark6.2.1.Lgisthemostfamousexampleofageneralhierarchyofconformallycovari-48antoperators.Leta,b∈R,alineardifferentialoperatorPgofordermon(X,g)iscalledconformallycovariantofbi-degree(a,b)ifwithrespecttoaconformalchange˜g=ϕ2g,wehave:ϕb◦P˜g=Pg◦ϕa.LgcanbegeneralizedtotheconformalpowersP2N,gofLaplacian(alsoknownasGJMSoperators):P2N,g=∆Ng+LOT,where“LOT”indicatestermsoforderlowerthan2N.P2N,gisofbi-degree(n−2N2,n+2N2).InparticularP2,g=Lg,P4,gistheso-calledPaneitzoperatorwhichwasdiscoveredindepen-dentlybyPaneitz[Pan08],Eastwood-Singer[ES85]andRiegert[Rie84].Explicitly:P4,g=∆2g+δ󰀕n−22(n−1)Sg·g−4P󰀖d+n−42Q,whereδistheformaladjointofd,PistheSchoutentensor,whichisdefinedby(n−2)P=Ricg−12(n−1)Sg·g,andQ=n4(n−1)S2g−2|P|2−12(n−1)∆gSgistheso-calledQ-curvaturetensor.See[Juh09]formoredetails.TheGreen’sfunctionGoftheoperatorLgisasmoothfunctiononX×X−∆.Following[LP87],wewillnormalizeitbyrequiringthat󰁝XG(p,q)Lg(ϕ(q))dvolg(q)=(n−2)ωn−1ϕ(p)(6.2.2)forallϕ∈C∞0(X).ApplyingEquation(6.2.1)to(6.2.2),weseethatundertheconformalchange˜g=u4n−2g,theGreen’sfunctiontransformsas󰁨G(p,q)=1u(p)u(q)G(p,q).(6.2.3)Next,recallthattheYamabeconstantY(C)ofaconformalclassCisdefinedbyY(C)=infg∈C󰁕XSgdvolg󰀃󰁕Xdvolg󰀄n−2n.TheproofofLemma6.1in[LP87](seealsoProp2.2.9in[Hab00])showsthatY(C)>0ifandonlyifthereexistsametricg∈Cwithpositivescalarcurvature.Furthermore,foreachg∈C,thesmallesteigenvalueλ1ofLghasthesamesignasY(C),sointhecasethatY(C)>0,Lgisinvertibleandwehavethefollowingresult.Theorem6.2.2.IfY(C)>0,thenforeachg∈C,thereisauniqueGreen’sfunctionG49forLg.Moreover,Gissymmetricandpositive.WeworkwithY(C)>0throughoutthissection.Forconvenience,weadoptthefollowingnotationsasin[LP87].Notation.Wewritef=O′󰀃rk󰀄tomeanf=O󰀃rk󰀄and∇f=O󰀃rk−1󰀄.O′′isdefinedsimilarly.NearthediagonalofX×X,Ghasthefollowingasymptoticexpansionunderthecon-formalnormalcoordinates.Lemma6.2.3([LP87],Lemma6.4).LetXbeasmoothmanifoldofdimensionnwithaconformalstructureCsuchthatY(C)>0andeithern=3,4,5orCisconformallyflat.Fixp∈X.Then,inconformalnormalcoordinates{xi}atp,theGreen’sfunctionGp=G(p,·)hasanasymptoticexpansionoftheformGp(x)=|x|2−n+α(p)+O′′(|x|)forsomeconstantα(p).Infact,theregularpartα(p)+O′′(|x|)ofGp(x)canbeexpressedintermsoftheheatkernelkofLg.RecallthattheheatkernelkofLgisasmoothfunctionk:X×X×R+→R+suchthat:(a)kp(q,t)=k(p,q,t)asafunctionofqandt>0solvestheLgheatequation:(∂t+Lg)kp(q,t)=0.(b)For∀p∈Xand∀ϕ∈C∞(X),limt→0󰁝Mk(p,q,t)ϕ(q)dvolg(q)=ϕ(p).Itisawell-knownfactthattheheatkernelkforLgisuniquelydeterminedbythemetricg.Furthermore,kdependssmoothlyonthemetricinthefollowingsense(cf.[PR87,Lemma1.1]or[BGV03,Theorem2.48]).Lemma6.2.4.Supposeγ:Rl→Met∞(X)withγ(s)=gsisasmoothfamilyofsmoothmetrics,thenthecorrespondingfamilyofheatkernelsk(s,p,q,t)isasmoothfunctiononRl×X×X×R+.50WewillusethefollowingrelationbetweenGreen’sfunctionGandtheheatkernelk(cf.[BGV03,Theorem2.38]).Lemma6.2.5.Foreach(p,q)∈X×X,wehaveG(p,q)=(n−2)ωn−1󰁝∞0k(p,q,t)dt,(6.2.4)whereωn−1isthevolumeoftheunitsphereSn−1.Inparticular,thestandardEuclideanheatkernelcenteredattheoriginonRnisk0=(4πt)−n2exp(−|x|24t),anddirectintegrationusingtheGammafunctionshowsthat(n−2)ω󰁝∞0k0dt=|x|2−n.(6.2.5)Proposition6.2.6.IfthemanifoldXisofdimensionn=3,4,5orCisconformallyflat,theninconformalnormalcoordinates{xi}onaneighborhoodUpofapointp,Gp(x)=|x|2−n+Φ0(p,x)(6.2.6)whereΦ0(p,x)=(n−2)ωn−1󰁝∞0(k−k0)dt(6.2.7)isaboundedfunctiononUp.Proof.By(6.2.4),(6.2.5)and(6.2.7),wehaveGp(x)=(n−2)ωn−1󰀕󰁝∞0k0(p,x,t)dt+󰁝∞0(kp(x,t)−k0(p,x,t))dt󰀖=|x|2−n+Φ0(p,x).ToseethatΦ0isboundedonUp,weintroducethefunctionk1=k0(1+a1(x)t),wherea1(x)=󰁝10S(xt)dt(6.2.8)istheintegralofthescalarcurvatureS.ByTheorem2.2in[PR87],k1isaparametrixforkindimension≤5,andthereareboundedfunctionsϕjsuchthatΦ0(p,x)=󰀻󰁁󰁁󰁁󰁁󰁁󰁁󰀿󰁁󰁁󰁁󰁁󰁁󰁁󰀽ϕ1+ϕ3−ϕ4n=3orClocallyconformallyflatϕ′1+ϕ3−ϕ4+a1(x)󰀃ϕ2−ln(|x|)󰀄,n=4ϕ′1+ϕ3−ϕ′4−a1(x)|x|n=5.(6.2.9)51Specifically,withthesamelabelingasin[PR87],ϕ1=(n−2)ωn−1󰁝140(k−k0)dt,ϕ′1=(n−2)ωn−1󰁝140(k−k1)dt,ϕ2=󰁝∞1λ−1e−λdλ+󰁝1|x|2λ−1(e−λ−1)dλ,ϕ3=(n−2)ωn−1󰁝∞14kdt,ϕ4=(n−2)ωn−1󰁝∞14k0dt,ϕ′4=(n−2)ωn−1󰁝∞14k1dtThePropositionfollowsbecause,inconformalnormalcoordinatesatp,S(x)=O(|x|2)(cf.[LP87,Theorem5.1]),andhencea1(x)=O(|x|2).Remark6.2.7.TherestrictionofΦ0tothediagonalisasmoothfunctionΦ(p,p)onX.Thiscanbeseenfrom(6.2.9).By(6.2.8),a1(x)vanishesonthediagonal,andϕ3,ϕ4andϕ′4areclearlysmooth.Asforϕ1andϕ′1,theargumentsinSection2.5of[BGV03]showthatthefunctionsκ(p,t)=k(p,p,t)−k0(p,p,t)andκ′(p,t)=k(p,p,t)−k1(p,p,t)satisfy|∂lpκ(p,t)|≤cl√t,fordimn=3,|∂lpκ′(p,t)|≤c′l√t,fordimn=4,5.foranyl∈N+.Becauseκandκ′aresmoothfort>0andthefunction1/√tisintegrable,thestandardtheoremondifferentiatingundertheintegralshowsthatϕ1andϕ′1aresmooth.MassofAnAsymptoticallyFlatManifoldDefinition6.2.8.Ann-dimensionalRiemannianmanifold(X,h)iscalledasymptoticallyflatoforderτ>0ifthereexistacompactsubsetK⊂XandadiffeomorphismΨ:X\K→{z∈Rn:|z|>1}suchthat,inthecoordinatesz1,...,zninducedonX\K,hij(z)−δij=O′′󰀃ρ−τ󰀄asρ:=|z|→∞.Thecoordinates{zi}arecalledasymptoticcoordinates.Givenanasymptoticallyflatmanifold(X,h),letSrdenotethesphereofradiusr≫0intheasymptoticcoordinatesystem{xi}.Wecanthendefinethefollowingfundamentalquantity.52Definition6.2.9.Themassofanasymptoticallyflatmanifold(X,h)isthenumbermass(h)=1ωn−1limr→∞󰁝Srn󰁛i,j=1(hij,i−hii,j)(−1)j+1dz1∧···∧󰁦dzj∧···∧dzn,(6.2.10)whereωn−1isthevolumeoftheunitsphereSn−1.Bythisdefinition,themassisameasureofhowquicklythemetricapproachestheEuclideanmetricnearinfinity.ThefollowingtheoremofR.Bartnik(cf.[Bar86])showsthatthemassdependsonlyonthemetrich.Theorem6.2.10(Bartnik).If(X,h)isasymptoticallyflatoforderτ>n−22>0,thenmass(h)isindependentofthechoicesofasymptoticcoordinates,soisaninvariantoftheRiemannianmetrich.Weshalllaterapplythefollowingversionofthen-dimensionalPositiveMassTheoremofSchoenandYau(cf.[SY79],[SY81],[LP87]).Theorem6.2.11.Let(X,h)beaRiemannianmanifoldofdimensionn≥3thatisasymp-toticallyflatoforderτ>n−22.If(X,h)hasnon-negativescalarcurvature,thenmass(h)≥0,withequalityifandonlyif(X,h)isisometrictotheEuclideanRn.ConformalblowupUnlessspecificallystatedotherwise,weworkwiththecasesn=3,4,5orCisconformallyflat,andweassumethatY(C)>0.In[Sch84],Schoenintroducedtheideaofconformallyblowingupametricg∈CbytheGreen’sfunctionofLgtoturnXintoanasymptoticallyflatmanifold(cf.[LP87]).Explicitly,theconformalblowupof(X,g)atapointp∈XofgisthemanifoldX\{p}withtheRiemannianmetrichpdefinedbyhp=h(p,q):=󰀃G(p,g)󰀄4n−2g(q).(6.2.11)NotethathpisasmoothmetricbyTheorem6.2.2.Supposethat{xi}areconformalnormalcoordinatescenteredatpdefinedonaneighbor-hoodUofpasinLemma6.2.3.OnU\{p}define“invertedconformalnormalcoordinates”byzi=xi|x|2.BytheasymptoticexpansionofGinLemma6.2.3(cf.[LP87,Theorem6.5]),53wehave:hij(z)=γ4n−2(z)󰀃δij+O′′󰀃|z|−2󰀄󰀄,(6.2.12)whereγ(z)=1+α(p)|z|2−n+O′′󰀃|z|1−n󰀄.Thisshowsthathpisanasymptoticallyflatmetricoforder1ifn=3,order2ifn=4,5,andordern−2ifgisconformallyflatnearp.Hence,ineachofthesecases,Bartnik’sTheoremimpliesthatthemassm(hp)iswell-definedanddependsonlyonthemetricg∈C.Lemma6.2.12.(cf.[LP87,Lemma10.5])mass(hp)=limq→p(G(p,q)−rp(q)2−n)=4(n−1)α(p).(6.2.13)Proof.Takethesphericalcoordinates,ρ=|z|andξ=z|z|.Withrespectto(ρ,ξ),thedefinition(6.2.10)ofmassgives:mass(hp)=1ωn−1limλ→∞λn−2󰁝Sλn󰁛i,j=1(hij,i−hii,j)zjdξ.(6.2.14)Bytheasymptoticformula(6.2.12),wehavehij(z)=(1+4α(p)n−2ρ2−n)δij+O′′(ρ1−n).Hencen󰁛i,j=1(hij,i−hii,j)zj=(1−n)n󰁛j=1zj∂j(1+4α(p)n−2ρ2−n+O′′(ρ1−n))=(1−n)ρ∂ρ(1+4α(p)n−2ρ2−n+O′′(ρ1−n))=4(n−1)α(p)ρ2−n+O(ρ1−n).Theresultfollowsbysubstitutingtheaboveresultinto(6.2.14).RegularityofHabermann’sCanonicalMetricDefinition6.2.13.ForaRiemannianmetricgonX,definethecorrespondingmassfunctionasmg:X→Rmg(p)=mass(hp)4(n−1),(6.2.15)wherehpistheasymptoticallyflatmetricatp∈Xwithrespecttogbyformula(6.2.11).Theorem6.2.14.Themassfunction(6.2.15)satisfiesthefollowingproperties:(a)m=0if(X,g)conformaltothestandardn-sphere,andm>0inallothercases.54(b)mf∗g=f∗mgforf:X→Xadiffeomorphism.(c)u2·mu4n−2g=mgforu∈C∞(X).(d)m∈C∞(X).Proof:(a)Foreachpointp,thePositiveMassTheorem6.2.11showsthatm(p)≥0,withequalityifandonlyif(X\{p},hp)isisometrictoeuclideanRn,inwhichcase(X,C)isconformallyequivalenttothesphereSnwithitsstandardmetric.Henceforalltheothercases,misstrictlypositiveonX.(b)Anisometryfpreservesthedistancefunction:rg(f(p),f(q))=rf∗g(p,q).ItalsopreservestheGreen’sfunction,asfollows:󰁝XG(f(p),f(q);g)Lf∗g(ϕ(q))dvolf∗g(q)=󰁝XG(f(p),f(q);g)Lg(ϕ(f(q)))dvolg(f(q))=(n−2)ωn−1(ϕ◦f)(p)HencebytheuniquenessofGreen’sfunction,wehaveG(f(p),f(q);g)=G(p,q;f∗g).By(6.2.13),wehavef∗mg=αg(f(p))=limq→p󰀏󰀏G(f(p),f(q);g)−r2−ng(f(p),f(q))󰀏󰀏=limq→p󰀏󰀏G(p,q;f∗g)−r2−nf∗g(p,q)󰀏󰀏=αf∗g(p)=mf∗g.(c)By(6.2.3)󰁨G(p,q)=1u(p)u(q)G(p,q).Temporarilysettingγ=4n−2,thedefinition(6.2.11)showsthat˜hp(q)=󰀓󰁨G(p,q)󰀔γ˜g=󰀕G(p,q)u(p)u(q)󰀖γu(q)γg(q)=u(p)−γhp(q).Itisclearthatif{zi}areasymptoticcoordinatesofh,then{λzi}areasymptoticcoordinatesforaconstantrescalingλ2h,andbyacoordinatechangingin(6.2.10),wehavem(λ2h)=λn−2m(h).Theconclusionfollowsbytakingλ=u(p)−γ/2=u(p)−2n−2.(d)LetΦ∈C∞(X×X,R+)betheconformalfactordefinedinLemma6.1.8.Noticethatproposition(c)ispointwisetrue,henceatapointp,letu4n−2(q)=Φ2(p,q),wehaveΦ1/(n−2)mΦ2g(p)=mg(p).(6.2.16)ByLemma6.1.8,gp=Φ2pgisthesmoothfamilyofmetricsthatareconformalnormalnearp.55Letk:X×X×R+×X→R+bethefamilyofheatkernelsparametrizedbythesmoothfamilyofmetrics{gp|p∈X},i.e.k(q1,q2,t;gp)=kp(q1,q2,t)istheheatkernelofLgp.ByLemma6.2.4,k(q1,q2,t;gp)issmoothinallthevariables.LetGp=G(q1,q2;gp)denotetheGreen’sfunctionrelatedtotheheatkernelofgpby(6.2.4).Sincegpareconformalnormalforeachp,by(6.2.13)and(6.2.6),wehavemass(hp(gp))=limx→0Gp(p,x)−|x|2−n=limx→0Φ0(p,x;gp)+O(|x|)=Φ0(p,p;gp).Byequation(6.2.7),Φ0=(n−2)󰁝∞0(k(p,p,t;gp)−k0(p,p,t))dt,wherek(p,p,t;gp)byLemma6.2.4isasmoothfunctiononX×R+.ByLemma6.2.6andRemark6.2.7,Φ0isasmoothfunctiononX.Henceby(6.2.16)mgisasmoothfunctiononX.□In[HJ99],HabermannandJostobservedthateachconformalclassConXhasacanon-icallyassociatedmetric,definedasfollows.Definition6.2.15.ForaconformalclassConX,supposeψ:X→Cisasmoothmap,thecanonicalmetricκCisthe(0,2)tensorbelow:κC(p)=mg(p)2n−2g(p).ByTheorem6.2.14,κCiswelldefinedindependentofthechoiceofg∈Candispre-servedbythepullbackmapbyisometry,andκCvanishesidenticallyifandonlyif(X,C)isconformallyequivalenttothesphereSnwithitsstandardmetric.Otherwise,κC∈CisasmoothRiemannianmetriconX.Remark6.2.16.OnaYamabepositivemanifold(X,C)ofdimensionn,considerP2NtheconformalNth-powerofLaplacianwith2N+1≤n≤2N+3.LetGbetheGreen’sfunctionofP2N.Byacompletelysimilarargumentasabove,onecanseethatthe(0,2)tensorgCdefinedbygC(p)=mass󰀕G4n−2Npg󰀖2n−2Ng(p)issmoothanddependsonlyontheconformalclassofthemetricg.Thiswasprovedby56B.Michelin[Mic10].Again,theproofofregularity([Mic10,Prop.3.3])canbesimplifiedusingLemma6.1.8.57BIBLIOGRAPHY[Bar86]R.Bartnik.Themassofanasymptoticallyflatmanifold.CommunicationsonPureandAppliedMathematics,39(5):661–693,1986.[BGV03]N.Berline,E.Getzler,andM.Vergne.HeatkernelsandDiracoperators.Springer,2003.[Cao91]J.Cao.Theexistenceofgeneralizedisothermalcoordinatesforhigher-dimensionalRiemannianmanifolds.TransactionsoftheAmericanMathematicalSociety,324(2):901–920,1991.[CH12]S-N.ChowandJ.K.Hale.Methodsofbifurcationtheory,volume251.Springer,2012.[ES85]M.EastwoodandM.Singer.AconformallyinvariantMaxwellgauge.PhysicsLettersA,107(2):73–74,1985.[FM77]A.E.FischerandJ.E.Marsden.Themanifoldofconformallyequivalentmetrics.CanadianJournalofMathematics,29(1):193–209,1977.[Gün93]M.Günther.Conformalnormalcoordinates.AnnalsofGlobalAnalysisandGeometry,11:173–184,1993.[Hab00]L.Habermann.Riemannianmetricsofconstantmassandmodulispacesofconformalstructures,volume1743.Springer,2000.[HJ99]L.HabermannandJ.Jost.Greenfunctionsandconformalgeometry.JournalofDifferentialGeometry,53(3):405–442,1999.[Hör15]L.Hörmander.TheanalysisoflinearpartialdifferentialoperatorsI:DistributiontheoryandFourieranalysis.Springer,2015.[Juh09]A.Juhl.OnconformallycovariantpowersoftheLaplacian.PreprintarXiv:0905.3992,2009.[KN63]S.KobayashiandK.Nomizu.Foundationsofdifferentialgeometry,volume1.Wiley,1963.[Kob12]S.Kobayashi.Transformationgroupsindifferentialgeometry.Springer,2012.[Lee12]J.M.Lee.Smoothmanifolds.Springer,2012.[Lee18]J.M.Lee.IntroductiontoRiemannianmanifolds,volume2.Springer,2018.[LP87]J.M.LeeandT.H.Parker.TheYamabeproblem.BulletinoftheAmericanMathematicalSociety,17(1):37–91,1987.[Mic10]B.Michel.MassedesopérateursGJMS.PreprintarXiv:1012.4414,2010.58[MSvdV99]U.Muller,C.Schubert,andAE.M.vandeVen.AclosedformulafortheRiemannnormalcoordinateexpansion.GeneralRelativityandGravitation,31(11):1759–1768,1999.[Pan08]S.Paneitz.Aquarticconformallycovariantdifferentialoperatorforarbi-trarypseudo-Riemannianmanifolds(summary).SIGMA.SymmetryIntegra-bilityGeom.MethodsAppl.,4:036,2008.[Par]T.H.Parker.Geometryprimer.[PR87]T.H.ParkerandS.Rosenberg.InvariantsofconformalLaplacians.JournalofDifferentialGeometry,25(2):199–222,1987.[PR00]F.PacardandT.Riviere.Linearandnonlinearaspectsofvortices:TheGinzburg-Landaumodel,volume39.Springer,2000.[Rie84]R.J.Riegert.Anon-localactionforthetraceanomaly.PhysicsLettersB,134(1-2):56–60,1984.[Sch84]R.Schoen.ConformaldeformationofaRiemannianmetrictoconstantscalarcurvature.JournalofDifferentialGeometry,20(2):479–495,1984.[SY79]R.SchoenandS.T.Yau.Ontheproofofthepositivemassconjectureingeneralrelativity.CommunicationsinMathematicalPhysics,65:45–76,1979.[SY81]R.SchoenandS.T.Yau.Proofofthepositivemasstheorem.II.Communica-tionsinMathematicalPhysics,79:231–260,1981.[Tay96]M.E.Taylor.Partialdifferentialequations:Basictheory,volume1.Springer,1996.59APPENDIXAPROOFOFLEMMAA.3ThisappendixsuppliestheproofofLemmaA.3.Theproofiscompletelyalgebraicandbeginswiththefollowingpreliminarylemma.Let(Rn,δ)bethestandardn-dimensionalEuclideanspacewithdimensionn≥3andletV=Sym2(Rn)⊗Syml(Rn).ConsiderthefollowinglinearoperatorsonV:Sym:V→Symk(Rn)αµ󰀁→1k!󰁛σ∈Skασ·µ(A.1)wherek=2+l.Tr:V→Syml(Rn)aij⊗bν󰀁→󰁛i,jδijaijbν(A.2)ThesymmetrizationoperatorSymgivesadirectsumdecomposition:V=Symk(Rn)⊕ker(Sym).Forε>0smallenough,definethefollowingoperator:Pε(α):V→Vα󰀁→α−ε·δ⊗Tr(α)(A.3)Wesayα∈Visε-symmetricifPε(α)∈Symk(Rn),anddenotetheε-symmetricsubspaceofVas:Symkε(Rn):={α∈V|Pε(α)∈Symk(Rn)}(A.4)LemmaA.1.Forε≤12(n−1),wehavefollowingthedirectsumdecomposition:V=Symkε(Rn)⊕ker(Sym).Proof.IfPε(α)=0,thenwehaveTr(Pε(α))=(1−nε)Tr(α)=0.Byassumption,ε≤12(n−1)<1n,henceTr(α)=0and0=Pε(α)=α−ε·δij⊗Tr(α)=α.ThusPεisanisomorphismonVanddim(Symkε(Rn))=dim(Symk(Rn)).WearelefttoshowthatSymkε(Rn)∩ker(Sym)={0}.Assumeα∈VsuchthatSym(α)=0andPε(α)∈Symk(Rn).BytakingTr◦SymonPε(α),wecanobtainthefollowingequation60onSyml(Rn):(1−ε·n)Tr(α)=Tr(Pε(α))=Tr(Sym(Pε(α)))=−εTr(Sym(δ⊗Tr(α))).(A.5)Tobebrief,denoteTr(α)=Sνwithν=(i1···il).SinceSν∈Syml(Rn),wemayassumetheindicesarenondecreasing:i1≤···≤il.Foreach1≤i≤n,letνibethenumberofi’sinν,thenitisclearthattheindexνisdeterminedbythenumbersν1,...,νn,andhenceinotherwordsbytheYoungdiagram(ν1,...,νn)ofnrowswithνimanyboxesontheithrowsuchthat󰁓ni=1νi=l.LetN=ε−1,multiplyEquation(A.5)with(l+2)(l+1)Nandwritewithindices,wehave:[(l+2)(l+1)(N−n)+n󰁛i=1(vi+1)(vi+2)]Sν+󰁛j∕=ivj>1,vi<l−1vj(vj−1)Sv′ij=0,(A.6)νwheretheindexv′ijisdeterminedbytheYoungdiagram(ν1,...,νi+2,...,νj−2,...,νn).WeseethattwoelementsSνandS′νarecorrelatedbyEquation((A.6)ν)ifthereexist1≤i∕=j≤nsuchthatν′i=νi+2,ν′j=νj−2andν′k=νkfork/∈{i,j}.Wethusdefineanequivalencerelationbetweentheindicesνandν′asfollows:wesayνandν′areelementarycorrelatedifthereexist1≤i∕=j≤nsuchthatν′i=νi+2,ν′j=νj−2andν′k=νkfork/∈{i,j}.Andν,ν′areequivalentifν′canbeobtainedfromνbyfinitelymanystepsofelementarycorrelations.Denote¯νforanequivalentclassofν,let|¯ν|beitscardinality.Foreachν∈¯ν,wehavethecorrespondingEquation(A.6)ν,andhencewehave|¯ν|manyhomogenouslinearequationsfor|¯ν|manyunknowns.Inmatrixform,wehaveA·(Sν)ν∈¯µ=0,(A.7)whereAisa|¯ν|×|¯ν|squarematrixdefinedasAνν′=󰀻󰁁󰁁󰁁󰁁󰁁󰁁󰀿󰁁󰁁󰁁󰁁󰁁󰁁󰀽(l+2)(l+1)(N−n)+󰁓ni=1(νi+1)(νi+2),ν′=ννj(νj−1),i∕=j,ν′=ν′ij0,otherwise.(A.8)WeclaimthatAisadiagonallydominatedmatrix.Indeed,fortheνthrow,thedifference61betweentheabsolutevalueofthediagonalandthesumofabsolutevaluesofelementsofthediagonalis:(l+2)(l+1)(N−n)+n󰁛i=1(νi+1)(νi+2)−󰁛i∕=jνj(νj−1)=(N−n)l2+l2−󰁛i∕=j(νi+νj)νj+(3(N−n)+n+2)l+2(N+1−n)≥(N−n)l2+l2−l󰁛i∕=jνj+(3(N−n)+n+2)l+2(N+1−n)=(N−2n+2)l2+(3(N−n)+n+2)l+2(N+1−n)≥(3(N−n)+n+2)l+2(N+1−n)>0,(A.9)wherethefirstequalityisbythefact:n󰁛i=1ν2i=(n󰁛i=1νi)2−󰁛i∕=jνiνj=l2−󰁛i∕=jνiνj,thefirstinequalityisbyνi+νj≤landthesecondinequalityisbytheassumptionthatN=ε−1≥2n−2.HencethecoefficientmatrixAisdiagonallydominated.Itisaclassicalfactthatdiagonallydominantmatricesareinvertible,andhence(Sν)ν∈¯ν=A−1(A(Sν))=0.Sincetheclass¯νisarbitrarilytaken,wehaveSν=0foranyindexν,andhenceTr(α)=0.HencePε(α)=α∈Symk(Rn)andSym(α)=0,whichmeansα=0.RemarkA.2.DiagonaldominanceisasufficientbutnotnecessaryconditionforthematrixAbeingnonsingular,andhence12n−2isnotasharpboundfortransversality.Indeed,ifweformallywritetheunknownsSνintheequations(A.6)νascoefficientsofahomogeneousdegreelpolynomialf,wecanthenwritethesystem(A.6)νinacompactwayas:((l+2)(l+1)(N−n)+4l+2n)f+r2∆(f)=0.(A.10)SeeLemma5.3in[LP87]forthefollowingfact:Theeigenvaluesofr2∆onthespaceofhomogeneousdegreelpolymonialsare{λj=−2j(n−2+2l−2j):j=0,...,[l/2]}.Hencetoensure(Sν)=0,itissufficienttorequirethefollowingsetdoesnotcontain0:Λ={((l+2)(l+1)(N−n)+4l+2n)−2j(n−2+2l−2j)|j=0,...,[l/2]}.62HencetransversalityresultholdsforageneralεawayfromacountablesubsetinR.Inparticular,forN=n=3,wehave0/∈Λ.Forthiscase,ifwetakeα=Ric,thenP1/3(Ric)isthetracelessRicci.LemmaA.3.Fork≥0,equations(3.1.11)0,...,(3.1.11)ktogetherwithjk−1p(C)=0deducejkp(Ric)=0.Proof.Weprovebyinductiononk:Fork=0,theclaimistrueby(3.1.11)0.Assumetheclaimistruefork≤m.Fork=m+1,bytheinductionassumption,wehavejmp(Ric)=0.InEquation(3.1.11)kwithk=m+1,wehave:2m+4(m+4)!󰁛µ∼¯µRicµ(p)+P(Rµ′(p))=0,where|µ|=m+3and|µ′|≤m+1andtheP(Rµ′(p))termconsistsofderivativesoftheRiemanniancurvatureRoforderlessthanmandthereforevanishesbytheinductionassumption.Wethushave:󰁛µ∼¯µRicµ(p)=0.Ontheotherhand,bytheconditionjmp(C)=0,foranyindexνwith|ν|=m,wehave:0=Cν(p)=Pij,kν−Pik,jν,(A.11)WhereP=Ric−12(n−1)SgistheSchoutentensor.SinceS=gijRicij,andjmp(Ric)=0,wehavejmp(S)=0.Hence(Sgij)kν(p)=δij(p)Skν(p).Forl=m+1,andµ=(ijν)with|ν|=m+1,inlocalcoordinateswehaveRicµ(p)∈Sym2(Rn)⊗Syml(Rn)=V.Letε=12n−2inLemmaA.1,wehaveP12n−2(Ricµ(p))=Ricµ(p)−12n−2δ(p)⊗Sν(p)=Pν,whichissymmetricby(A.11).63HenceRicµ(p)satisfiestheconditionsofLemmaA.1,bywhichwehaveRicµ(p)=0,for|µ|=m+1,namelyjm+1p(Ric)=0.64APPENDIXBPROOFOFLEMMAB.1ThisappendixsuppliestheproofofLemmaB.1.For0<α,δ<1,k≤N,anddenoteArtheannulusBr−Br/2,recallthedefinitionofthe󰀂·󰀂k,α;N,δnorm:󰀂f󰀂k,α;N,δ=sup0<r≤δr−N󰀕k󰁛|β|=0r|β|supAr{|∂βf(x)|}+rk+αsupx∕=y∈Ar|β|≤k|∂βf(x)−∂βf(y)||x−y|α󰀖.ForK0>0,letDK0={v|󰀂v󰀂k,α;N,δ≤K0},onwhichdefinethefunctional:Tρ(x,v)=Gρ(x,v)+Qρ(x,v)=󰁛i,jxixjx2·Gij(ρx,ρ∂v)+󰁛ijkρxkQijk(ρx,ρ∂v)∂ijv,whereQijk(x,ζ)andGij(x,ζ)aresmoothfunctionswithrespecttox,ζ.LemmaB.1.ThereexistsaconstantK2suchthatfor0<ρ<1andanypairoffunctionsv1,v2∈DK0,wehave󰀂Tρ(v2)−Tρ(v1)󰀂k−2,α;N−2,δ+󰀂Tρ(v2)−Tρ(v1)󰀂0,α;N−1,δ≤K2ρ󰀂v2−v1󰀂k,α;N,δ.(B.1)Infact,K2=C(α)P(K0)M,whereC(α)isaconstantdependsonα,P(K0)isapolyno-mialofK0andM=maxijk󰀋󰀂Gij󰀂Ck(D),󰀂xkQijk󰀂Ck(D)󰀌,withD=D(δ,K0)=Bδ×BK0⊆Rn⊕Rn.Proof:WriteTρ󰀏󰀏󰀏21=T(ρx,ρ∂v2)−T(ρx,ρ∂v1)andsimilarlyforGρandQρterms.First,considerthecasek=2.Inthiscase,thesecondtermin(B.1)dominatesthefirstterm,soitsufficestobound󰀂Tρ󰀏󰀏󰀏21󰀂0,α;N−1,δ.Since󰀂Tρ(v)󰀏󰀏󰀏21󰀂0,α;N−1,δ=󰀂Gρ(v)󰀏󰀏󰀏21+Qρ(v)󰀏󰀏󰀏21󰀂0,α;N−1,δ≤󰀂Gρ(v)󰀏󰀏󰀏21󰀂0,α;N−1,δ+󰀂Qρ(v)󰀏󰀏󰀏21󰀂0,α;N−1,δ.WewillboundtheGandQnormsseparately.FortheGpart,wehave:|Gρ󰀏󰀏󰀏21|=|󰁛ijxixjx2(Gij(ρx,ρ∂v2)−Gij(ρx,ρ∂v1))|≤󰁛ij|Gij(ρx,ρ∂v2)−Gij(ρx,ρ∂v1)|≤󰀂Gij󰀂C1(D)ρ|∂v󰀏󰀏󰀏21|,wherethelastinequalityisbytheMeanValueTheorem.65Byinductiononm,wehave∂mxixjx2≤C1rm,whereCdependsmandthedimensionn.Hence|∂mGρ󰀏󰀏󰀏21|≤C󰁛ijm󰁛l=0rl−m|∂lGij(ρx,ρ∂v))󰀏󰀏󰀏21|.(B.2)Applythecompositionruleto∂lGij(ρx,ρ∂v),weobtainalinearcombinationofthefollowingterms:∂m1+···+mlGij(ρx,ρ∂v)l󰁜j=1(∂j+1v)mj,where󰁓lj=1jmj=l.Evaluateeachtermatv1andv2andbytheMeanValueTheoremsimilartoabove,wehave:|∂lGij(ρx,ρ∂v))󰀏󰀏󰀏21|≤ρ󰀂Gij󰀂Cl+1(D)P1(K0)(l+1󰁛l′=1|∂l′v󰀏󰀏󰀏21|),(B.3)Combining(B.2)and(B.3),wehave|∂mGρ󰀏󰀏󰀏21|≤ρP(K0)maxij󰀋󰀂Gij󰀂Cm+1(D)󰀌m󰁛l=0rl−m|∂l+1v󰀏󰀏󰀏21|(B.4)HencefortheCαtermwehave:|∂mGρ(x,v)󰀏󰀏󰀏21−∂mGρ(y,v)󰀏󰀏󰀏21||x−y|α=|∂mGρ(x,v)󰀏󰀏󰀏21−∂mGρ(y,v)󰀏󰀏󰀏21||x−y||x−y|1−α≤C(α)r1−αsupAr|∂m+1Gρ󰀏󰀏󰀏21|≤ρC(α)r1−αP(K0)maxij󰀋󰀂Gij󰀂Cm+2(D)󰀌supArm+1󰁛l=0|rl−m∂l+1v󰀏󰀏󰀏21|(B.5)Form=0,apply(B.4)and(B.5),wehave:󰀂Gρ(v)󰀏󰀏󰀏21󰀂0,α;N−1,δ≤C(α)(K0+1)maxij󰀂Gij󰀂C2(D)ρ󰀂v1−v2󰀂2,α;N,δ.(B.6)Similarly,fortheQpart,bytheMeanValueTheorem,wehave:|Qρ(v)󰀏󰀏󰀏21|=ρ·󰁛ijk|xkQijk(ρx,ρ∂v)∂ijv󰀏󰀏󰀏21|≤ρ󰁛ijk|xk|·|Qijk(ρ∂v1)∂ijv2−Qijk(ρ∂v2)∂ijv1+Qijk(ρ∂v2)∂ijv1−Qijk(ρ∂v2)∂ijv2|≤ρ󰁛ijk|xkQijk(ρx,ρ∂v)󰀏󰀏󰀏21|·K0+|xkQijk(ρx,ρv2)||∂2v󰀏󰀏󰀏21|≤ρ󰁛ijk|xkQijk|C1(D)(K0|∂v󰀏󰀏󰀏21|+|∂2v󰀏󰀏󰀏21|)66|Qρ(x,v)󰀏󰀏󰀏21−Qρ(y,v)󰀏󰀏󰀏21||x−y|α≤ρ·󰁛ijk|xkQijk(ρx,ρ∂v)∂ijv󰀏󰀏󰀏21−ykQijk(ρy,ρ∂v)∂ijv󰀏󰀏󰀏21||x−y|αTobebrief,inthenumeratordenotexkQijk(ρx,ρ∂v1)∂ijv1(x)asQ(x,v1)∂2(x,v1),simi-larlyforyandv2.Wethenhave:|xkQijk(ρx,ρ∂v)∂ijv󰀏󰀏󰀏21−ykQijk(ρy,ρ∂v)∂ijv󰀏󰀏󰀏21)|=A+B+C+D,whereA=Q(x,v)󰀏󰀏󰀏21·󰀃∂2(x,v1)−∂2(y,v1)󰀄,B=Q(x,v2)󰀕∂2(x,v)󰀏󰀏󰀏21−∂2(y,v)󰀏󰀏󰀏21󰀖,C=∂2(y,v1)󰀕Q(x,v)󰀏󰀏󰀏21−Q(y,v)󰀏󰀏󰀏21󰀖,D=(Q(x,v2)−Q(y,v2))󰀕∂2(y,v)󰀏󰀏󰀏21󰀖.Respectively,wehave:|A||x−y|α=|xkQijk(ρx,ρ∂v)󰀏󰀏󰀏21|·|∂2(x,v1)−∂2(y,v1)||x−y|α≤K0󰀂xkQijk󰀂C1(D)(|∂v󰀏󰀏󰀏21),|B||x−y|α≤r·󰀂xkQijk󰀂C0(D)·|∂2(x,v1−v2)−∂2(y,v2−v2)||x−y|α,|C||x−y|α≤K0r|Q(x,v)󰀏󰀏󰀏21−Q(y,v)󰀏󰀏󰀏21||x−y||x−y|α≤C(α)K0supAr|∂xkQijk(ρx,ρ∂v)󰀏󰀏󰀏21|≤C(α)K0(1+K0)rN−1󰀂xkQijk󰀂C2(D)(|∂v󰀏󰀏󰀏21|)+|∂2v󰀏󰀏󰀏21|),|D||x−y|α≤C(α)r1−α󰀂xkQijk󰀂C1(D)(|∂2v󰀏󰀏󰀏21|).Insummary,fortheQterm,wehave:󰀂Qρ(v)󰀏󰀏󰀏21󰀂0,α;N−1,δ≤C(α)(K20+K0+1)maxijk󰀂xkQijk󰀂C2(D)ρ󰀂v1−v2󰀂2,α;N,δ.(B.7)Combining(B.6)and(B.7),weconcludetheprooffork=2.Assumethestatementistruefork=m−1.Fork=m,wehave:󰀂Tρ󰀏󰀏󰀏21󰀂m−2,α;N−2,δ+󰀂Tρ󰀏󰀏󰀏21󰀂0,α;N−1,δ≤󰀂Tρ󰀏󰀏󰀏21󰀂m−3,α;N−2,δ+󰀂Tρ󰀏󰀏󰀏21󰀂0,α;N−1,δ+sup0<r≤δrm−N󰀳󰁅󰁃supAr|∂m−2Tρ󰀏󰀏󰀏21|+supx∕=y∈Arrα|∂m−2Tρ(x,∂v)󰀏󰀏󰀏21|−∂m−2Tρ(y,∂v)󰀏󰀏󰀏21||x−y|α󰀴󰁆󰁄67Hence,bytheinductionassumption,itissufficienttoshowthat:sup0<r<δrm−N󰀳󰁅󰁃supAr|∂m−2Tρ(v)󰀏󰀏󰀏21|+supx∕=y∈Arrα|∂m−2Tρ(x,v)󰀏󰀏󰀏21|−∂m−2Tρ(y,v)󰀏󰀏󰀏21||x−y|α󰀴󰁆󰁄≤C(α)P(K0)Mρ󰀂v2−v1󰀂m,α;N,δ.FortheGterms,apply(B.4)tom−2,wehave:sup0<r<δrm−N󰀕supAr{|∂m−2Gρ(v)󰀏󰀏󰀏21|}󰀖≤ρP(K0)maxij󰀋󰀂Gij󰀂Cm−1(D)󰀌sup0<r<δr−NsupAr󰀫m−2󰁛l=0rl+1|∂l+1v󰀏󰀏󰀏21|󰀬(B.8)Apply(B.5)tom−2,wehave:sup0<r<δrm−N󰀻󰁁󰀿󰁁󰀽supx∕=y∈Arrα|∂m−2Gρ(x,v)󰀏󰀏󰀏21−∂m−2Gρ(y,v)󰀏󰀏󰀏21||x−y|α󰀼󰁁󰁀󰁁󰀾≤ρC(α)P(K0)maxij󰀋󰀂Gij󰀂Cm(D)󰀌sup0<r<δr−NsupAr󰀫m󰁛l=0|rl+1∂l+1v󰀏󰀏󰀏21|󰀬(B.9)FortheQterms,bytheLeibnizrule:∂m−2(Qρ)=∂m−2(ρxkQijk,ρ∂ijv)=ρ󰁛a+b=m−2󰀕m−2a󰀖∂a(ρxkQijk,ρ)∂ij(∂bv)Foreachterm∂a(ρxkQijk,ρ)∂ij(∂bv),denote˜Qijk=∂a(ρxkQijk,ρ)and˜v=∂bv,ap-ply(B.7),wehave:sup0<r<δr1−N󰀳󰁅󰁃supAr|∂m−2˜Qijk󰀏󰀏󰀏21|+supx∕=y∈Arrα|∂m−2˜Qijk(x,v)󰀏󰀏󰀏21|−∂m−2˜Qijk(y,v)󰀏󰀏󰀏21||x−y|α󰀴󰁆󰁄≤C(α)P(K0)maxijk󰀂˜Qijk󰀂C2(D)ρ󰀂˜v1−˜v2󰀂2,α;N,δ.Multiplybyrm−1onbothsidesoftheinequalityabove.Bydefinitionofthe󰀂·󰀂k,α;N,δnorm,wehaverm−1󰀂˜v1−˜v2󰀂2,α;N,δ≤󰀂v1−v2󰀂m,α;N,δ,andhencewehave:sup0<r<δrm−N󰀳󰁅󰁃supAr|∂m−2Qρ(v)󰀏󰀏󰀏21|+supx∕=y∈Arrα|∂m−2Qρ(x,v)󰀏󰀏󰀏21|−∂m−2Qρ(y,v)󰀏󰀏󰀏21||x−y|α󰀴󰁆󰁄≤C(α)P(K0)maxijk{󰀂xkQijk󰀂Cm(D)}ρ󰀂v1−v2󰀂m,α;N,δ.(B.10)Combining(B.8),(B.9)and(B.10),weconcludethek=mstep.□68