Aubry–Mather theory for Lorentzian manifolds Stefan Suhr.pdf

该【Aubry–Mather theory for Lorentzian manifolds Stefan Suhr 】是由【探春文档】上传分享，文档一共【42】页，该文档可以免费在线阅读，需要了解更多关于【Aubry–Mather theory for Lorentzian manifolds Stefan Suhr 】的内容，可以使用淘豆网的站内搜索功能，选择自己适合的文档，以下文字是截取该文章内的部分文字，如需要获得完整电子版，请下载此文档到您的设备，方便您编辑和打印。:...(2019)21:71https:///-019-0707-xJournalofFixedPointTheorycSpringerNatureSwitzerlandAG2019andApplicationsAubry?–,’sgraphtheo-,theLorentzianHedlundexamples,?,53C22,–Mathertheoryisawell-establishedpartofthestudyofTonelliLa-grangianandTonelliHamiltoniansystems,see[11,21].–,wedirecttheattentiontowardsanAubry–Matherthe--acterofAubry–-manifold,withm≥3,–Mathertheoryhavebeenstudiedinspe-cialcases,pact2-manifoldsin[31]andgloballyconformally?atLorentziantoriin[33].TherelatedHamilton–Jacobiequationhasbeenstud-iedonLorentzian2-toriin[18].MaximalgeodesicsinLorentzian2-toriwithpoleshavebeenstudiedin[27,28].Wewillgeneralizeresultsfrom[3,4,8,21]tothenaturallygivenclassofso-calledclassAspacetimes,see[34].AcompactLorentzianmanifold(M,g)isofclassAifitis(1)timeorientable,.,itgivesrisetoacontinuoustimelikevector?eld,(2)itisvicious,.,everypointliesonatimelikeloopand(3)theAbeliancoverisgloballyhyperbolic,seeDe?,0123456789().:V,-vol:..––Mathertheoryarethestabletimecone,see[34],andthestabletimeseparation,,see[13],ormoregenerallyMather’sβ-function,see[21].Themainresultsincludetheexistenceandmultiplicityofmaximaler-godicmeasures,,theexistenceofcalibrationsforclassAspacetimes,,therelationbetweenmaximalmeasuresandcalibrations,,andversionsofMather’sgraphtheorem,,,,:,themainresultsareintro-,,?xaRie-?eldg∈Γ(T0M)2isaLorentzianmetricifforeveryp∈M,thebilinearformonTMpissym-metricandnon-(M,g)istimeorientableifthesetofcausaltangentvectors{v∈TM|v=0,g(v,v)≤0}isnotconnected.-ponentof{v=0,g(v,v)≤0}.ponentarecalledfuture--(M,g)isaspacetimeifitistime--tailsaboutLorentziangeometryrefertothestandardtextbookreferences[14,26]and[5].Formorerecentdevelopmentsincausalitytheory,see[24].??(M,g)isofclassAif(M,g)isviciousandtheAbeliancoverπ:(M,g)→(M,g)?nedasM:=M/[π1(M),π1(M)]whereMdenotestheuniversalcoverofMand[π1(M),π1(M)]-mutatorsubgroupofthefundamentalgroupπ1(M).Therefore,thegroupofdecktransformationsisisomorphictoH1(M,Z).:..Aubry?(M,g),hestabletimecone,T,tobetheclosureoftheconeoverthehomologyclassesoffuture-pointingloops,see[34]>0,setTε:={h∈T|dist.(h,?T)≥εh},where.denotesthestablenormwithrespecttogR,see[13]:M×M→Rthetimeseparationof(M,g)andwithy?x∈H1(M,R)thedi?erenceofx,y∈M,-(M,g):T→Rsuchthatforeveryε>0,thereisaconstantC(ε)<∞with(1)|l(y?x)?d(x,y)|≤C(ε)forallx,y∈Mwithy?x∈Tεand(2)l(λh)=λl(h),forallλ≥0,(3)l(h+h)≥l(h)+l(h)forallh,h∈Tand(4)l(h)=limsupl(h)forh∈?Tandh∈→(2)and(3).Wecallafuture-pointingcurveγ:[a,b]→Mamaximizerifγmax-imizesarclengthoverallfuture-pointingcurvesconnectingγ(a)withγ(b).Fortheconvenienceofnotationwecallγ:[a,b]→Mamaximizerifone(henceevery)-pointingcurveγ:R→M(orM)isamaximizeriftherestrictionγ|[a,b]isamaximizerforevery?niteinterval[a,b]?:[a,b]→Maswellasofπ?γ:1ρ(γ)=ρ(π?γ):=(γ(b)?γ(a)).b?aAsequenceofcausalcurves{γi:[ai,bi]→M}i∈Nisadmissible,ifLgR(γi)→∞fori→∞.–SeifertTheorem[1,32]impliesthatforanyh∈Tthereexistsanadmissiblesequenceofmaximizers{γn:[an,bn]→M}n∈Nsuchthatρ(γn)→h,whereγnisanylifttoM.:..:[an,bn]→M(n∈N)ofmaximizerssuchthatbn?an→∞andsupposethatρ(γn)→h∈T?.Thenwehave:Lg(γn)→l(h),bn?anforn→∞.|?T≡,ifl|?T\{0}>0,>0suchthath∈,fornsu?cientlylarge,1g1gL(γn)?l(h)≤L(γn)?l(ρ(γn))+|l(ρ(γn))?l(h)|bn?anbn?an1=d(γn(an),γn(bn))?l(ρ(γn))+|l(ρ(γn))?l(h)|.bn?anThe?rsttermisboundedbyC(ε).Thesecondtermconvergesto0bybn?.LetT?:={α∈H1(M,R)|α|≥0}(M,g)∈?T?suchthatα?1(0)∩T∩H1(M,Z)R=?.Then,wehave,l|?1≡(0)∩TForthede?nitionofH1(M,Z)R,∈?T?andtotallyirrationalwithrespecttoH1(M,Z)–pletenessofthegeodesic?,however,evenif(M,g)pactorclassA,thege-odesic?owof(M,g),onecanproveplete,see[9].Therefore,anattempttodescribetherelationshipbetweenthequalita-tivebehaviorofmaximalcausalgeodesicsandtheconvexitypropertiesofthestabletimeseparationlusingthegeodesic?owof(M,g)?cationTM∪{∞}ofTM,asdescribedin[21],andextendthegeodesic?owto∞bysettingΦ(∞,t)≡∞.Thisencountersthefollowingproblem:inthepresenceofpletegeodesics,someinvariantmeasureswillconcentrateat∞,,-eterizingthegeodesic?owof(M,g)toa?owΦinawaythatevery?∈TM,denotewithγv:(αv,ωv)→Mtheuniqueinextendiblegeodesicof(M,g)with˙γv(0)=.:..Aubry?(M,g)beapseudo-Riemannianmanifold,Φgitsgeo-?neΦ:TM\Z×R→TM\Z,(v,t)→γ?(t),vwhereγ?visthetangent?eldtotheconstantgR-arclengthparameterizationofγvwith|γ?|=|v|.Then,Φisasmooth?ow,calledthepregeodesic?owofv(M,g)?and?RtheLevi-Civitaconnectionof(M,g)and(M,g),?eldT:=???=v∈TMRandconsidertheuniqueg-geodesicγv:(αv,ωv)→Mwith˙γv(0)=-notewith?γv:R→MtheconstantgR-arclengthparameterizationofγvwith|γ?|≡|v|,where?γ:=dγ?.Notethat˙γ=|γ˙v|γ?.Then,wehave:vvdtvv|v|v0=?γ˙=?Rγ˙+T(˙γ,γ˙)γ˙vvγ˙vvvv2|γ˙v|R1=2?γ?vγ?v+T(?γv,γ?v)?2gR(T(?γv,γ?v),γ?v)?γv.|v||v|Consequently,?γvsatis?esthefollowingdi?erentialequation:R1?γ?vγ?v=2gR(T(?γv,γ?v),γ?v)?γv?T(?γv,γ?v).(1)|v|Itiseasytoseethatg(?γ,γ?)ispreservedalong?γ.RvvvItisnotclearwhetherforanarbitraryspacetime(M,g),thepregeodesic?owΦ:TM\Z×R→TM\,forexampleifgRisa?,see[22].,wewillnotconsiderΦitself,buttherestrictionofΦtotheunittangentbundleT1Mof(M,gR).WeomittheindicationoftherestrictionanddenoteΦ|T1M×-:M→RbeaLipschitzcontinuousfunctionandμa?niteΦ-,wehave:?fvdμ(v)=?niteΦ-??niteΦ--tionvectorρ(μ)∈H1(M,R)theuniquehomologyclasssatisfying:[ω],ρ(μ):=ωdμ,T1Mforeveryclosed1-formωonM.:..,weintroducethenotionofmaximalinvariantmeasureswith??niteinvariantmeasureswithsupportentirelyinthesetoffuture-?niteΦ-invariant(orforshortinvariant)Borelmeasureswithsup-portinthesetoffuture-,wedenotewithM1,thesetofinvariantprobabilitymeasureswithsupportinthefuture-gpointinggR--?topologyandgitsextremalpointsarepreciselytheergodicmeasuresof(T1M,Φ),bytheTheoremofKrein–Milman,see[20].Forμ∈Mg,heaveragelengthofμ:L(μ):=?g(v,v)dμ(v).T1M1?NotethatLandω→ωdμforω∈Λ(TM)(M,g)ofclassA,wehaveT=ρ(Mg)andl(h)=sup{L(μ)|μ∈Mgwithρ(μ)=h∈T}.(M,g)beofclassAandletb:=dimRH1(M,R)denotethe?,thepregeodesic?owΦadmitsatleastb--?erentialgeometryandvariationalanalysis,see[15].Especiallyinthecalculusofvariations,,thegeneralde?nitionofacali-[4]and[8].In[8],calibrationsappearas“generalizedcoordinates”.Toourknowledge,thecalibrationswhichhavemadeappearancesinpseudo-Riemanniangeometryis[16,19,23].pactspacetime(M,g)withLorentziancover(M,g).Letl∈(0,∞).Wecallafunctionτ:M→Ranl-pseudo-timefunctionifforeveryp∈MthereexistsaconvexnormalneighborhoodUofpsuchthat:τ(q)?τ(p)≥l·d(p,q),Uforallq∈J+(p),wheredUdenotesthetimeseparationofthespacetimeU(U,g|U).NotethatifτisLipschitz,theinequalityτ(q)?τ(p)≥ld(p,q)-Lipschitzcontinuityofthetimeseparationontheboundary?(J+(p)).U:..Aubry?MathertheoryforLorentzianmanifoldsPage7of4271RecallthatthegroupofdecktransformationsoftheAbeliancoverisisomorphictoH1(M,Z).Therefore,givenaclassα∈H1(M,R),wewillcallafunctionf:M→Rα-equivariantif:f(x+k)=f(x)+α,k,forallx∈mandk∈H1(M,Z),where“+”denotestheactionofthe?rsthomologybythedecktransformations,,and.,.?:T?→R,α→inf{α(h)|l(h)=1}.De?∈(T?)?.Afunctionτ:M→Risacalibrationrep-resentingαifτisanα-equivariantLipschitzcontinuousl?(α)-∈α∈(T?)?andF:M→Rbeaprimitiveofπ?,thefunctionτω:M→R,x→liminf[F(y)?l?(α)d(x,y)]y∈J+(x),dist(x,y)→∞(M,gR),thedualstablenormcoincidesonH1(M,R)withtheco-massnormα?:=inf{ω∞|ω∈α},see[4].Thisposesthequestion:istheanalogousresulttrueforthestabletimeseparationandthedualtimeseparation?Wegiveapositiveanswertothisquestionon(T?)?anddiscussw