Aoki Toshihiko

Pak J. S.

Yorozu Shinsuke

Kanazawa University Repository for Academic Resources

Ann・Sci.KanazawaUnivVol.25,pp､17-29,1988L2-vectorfieldsonanon-compactRiemannianmanifoldQ当withboundaryT.Aoki,J.S.PakandS・Yorozu(ReceivedAprill2,1988)171．TheresultsofthestudyofconformalandKillingvectorfieldsonacompactRieman-nianmanifoldwithboundaryhadbeenlistedinYano'sbook[3].IthasbeentriedbythethirdnamedauthortogeneralizetOthenon-compactcase,thatis,in[5],thenon-existenceofL2-Killingvectorfieldsonanon-compactRiemannianmanifoldwithboundarywasdiscussed.ThepurposeofthepresentpaperistoinvestigatetheprOpertiesandthenon-exis-tencesofL2-conformalandKillingvectorfieldsonanon-compactRiemannianmanifoldwithbomdary.WeshallbeinC｡。-category.Latinindicesrunfromlton+1andGreekonesfromlton.TheEinsteinsummationconventionwillbeused.2．Let-〃beacomplete,non-compact,connectedandorientableRiemannianmanifoldofdimensionn+1withaRiemannianmetricg.LetVbetheRiemannianconnectionon-〃associatedwithg.Wetakeanon-compactmanifoldM=aMUMsuchthatMisanon-compact,cOnnectedandopensubmanifoldof"andaM=M-Misanndimensional,compact,Connectedsubmanifoldof",whereMdenotestheclosureofMin〃'.ThenMisanorientableRiemannianmanifoldwithboundaryaMandtheRiemannianmetricinducedfromgon-〃.WedenotebygtheinducedRiemannianmetriconMandbyVtheRiemanniancOnnectiononMassociatedwithgltistrivialthatMiscompleteasametricspacewiththedistancedeterminedbyg..AteachpointpofaM,thereexistsacoordinateneighborhoodsystem{U,(xi)}ofpin〃'SuchthatUnMisrepresentedbyxn+'≧0andUnaMisrepresentedbyXn+'=0.Suchacoordinateneighborhoodsystemiscalledaboundarycoordinatedsystem.Andlet{UnaM,(x")}betheinducedcOordinateneighborhoodsystemonaM.If{U,(xi)}and{V,(yi)}areboundarycoordinatesystemssatisfyingUnV≠‘,thenwehave18ayn+'(2.1） aXn+'>0andT.AOKI,J.S.PAKandS.YOROZUayn+'=0ax"(forany")onaMnUnV.SincetheJacobianofthecoordinatetransformationof{U,(xi)}and{V,(y')}ispositive,theJacobianofthecoordinatetra鵬formationof{UnaM,(x")}and{VnaM,(y)"}ispositive.ThusaMisorientable.Leti:aM－→Mbetheinclusion.Let{U,(xi)}beaboundarycoordinatesystemofpEaMin"and{U',(u")}beacoordinateneighborhoodsystemofpinaMsuchthatU'CUnaM.Thentheinclusionimayberepresentedlocallyby.(2.2)xi=x!(u").WedenotebyBthedifferentialoftheinclusioni,thatis,(23)B=(B:)=gf)Theinducedmetric==(3@mp)onaMisgivenby(2.4） g｡p=B%Bigij,whereg=(g,j).WemaychoosetheunitouternormalvectorfieldNonaM.LetSbetheRiemannianConnectiononaMassociatedwithg.TheequationsofGaussandWeingartenarestatedasfollows:(2.5)V#xi*T=j*(5xY)+h(X,T)･N(2.6)vi・xN=i*(-AX)foranyvectorfieldsXandVonaM,wherehdenotesthesecondfundamentalformofaM～withrespecttoNandAisdefinedbyh(X,V)=g(AX,T).3．LetAs(")(resp.As(aM))bethespaceofalls-formsonM(resp.aM).Letd(resp.H)denotetheexteriorderivativeonAs(M)(resp.As(aM)).Anoperator6(resp.3)isdefinedby(3.1)6=(-1)m*d*(resp.3=(_1)m*H*)onAs(M)(resp.As(aM)),wherem=sn+s+n(resp.m=Sn+n+1)and*denotesthestaroperator.言｜’"H="(")弩I△－＝釦登!△一曽!△=II(身p)(9.8）uoIsseJIdxe8u!"olIoJeAEIIe"uelIGL･(ng)JoxplemesIeAuIe叩SSJOuep(｢1S)9JIeq"!!4!!3=!APUe!xe/eA=IA1eSeM･IXp『身!IS=!xpg=身AqpeugepsIXo11enp身mIoJ-Ie叩lWuO!"e/e!身=XPIe!JJoPeAEdIoaWeuOLPIeg･xO1oeAJfuejI"(4v)2=(4)(2o)(N)"="u(f．8)(里*′)〃＝(塁)(ぬ）Aq(We)Ivs皇Opue(We)0Vヨ伽((We)IVヨ幻eugepe"@(We)IVヨ皇pue(W)rVヨ〃血gﾕo且．（1,LIIS=<J,@S>Jfq剛冨、I[E]u!1onpoJIdｴ膠I膠，．!層,｡!叩叫……｣I([9])I*!iLI!s¥w/='*<L@s>W/=く､L@S>>犯叩eAEqe"@slJIoddnsloedmoo叩"Wuo､LPuESsplegdIosue]-(Z@O)JfuejIo,J･WuospleIJjIosuel-("@")Joaoedse叩O1pepuelXes!ぐくC>>犯叩耶EmeleM･z(<<!>>=llllJesaM･<<!>>]onpodIdaIEIEoseq3oWOedseJIW"(W):VJouoPeldmoOe叩eq(W)s.Z'I13'I･(W)I+:Vヨラpue(W):Vヨ身AuEJoJく諏萱＞＝く§即>＞（8.8）eAEqa"uaII､L.(W):Vヨ〃‘身血EIoJ－4*V身畷ﾉ=I*<"$">唾ﾉｰく〃身" (Z．8)eAEIIe"$eId-mexeJIo,J･s1JIoddnsJoEdmooq""smJIoJJopesoduIoo((We)sV･dsaI)(W)sVJosOEdSqnse叩se40uep((We):V.dsaI)(W):VejIsH･((We):VJIo)(W):VuoJonpodIdIEIEosIEqoISe叩eqぐく:>>491Pue!<$>J(qPeJouePsI((We)sVJIo)(W)sVuo4onpoJIdIEIEosIEooIeq､L61 八ﾕepunoqII1!"PIoJ!uemue!uUemeIHJoedmoo-uoueuospIg卿エ01oaA-z'I●20 T.AOKI,J.S.PAKandS.YOROZUnヶ=&N!C(t5)"=AgB》畠▽,Vj5!－▽j▽ifi=Rjif!whereRj1denotethecomponentsoftheRiccitensorfieldofV([3],[5]).Moreover,wedefineD(dS),AfandDXasfollows:-(D("))i=V!(")=－▽i▽j易(3.6）（△写),=(d"+6d5)i=一▽j▽ja+R{&(DX)i=vjvjfi+RIglwhereRI=gikRkj([3],[5]).LetXbeavectorfieldonMand5theduall-form.ThenXIa"hasthefollowingexpression:(3.7）～X=/*X+(nf)･NonaM,forsomevectorfieldXonaM.DEFINITIoN3.1([3]).AvectorfieldXonMiscalled舷"ge"伽J(resp."owwdzI)toa"ifn5=0(resp.t5=0)fortheduall-form5toX・By(3.7),weremarkthattf=0isequivalentto/*X=0DEFINITIoN3.2.AvectorfieldXonMiscalledaco"""αノzﾉg伽γ此〃ifLxg＝2ルg,whereZ,denotestheLiederivativeoperatorand入isafunctiononM.DEFINITIoN3.3.AvectorfieldXonMiscalleda臓肋@g"gc加γ此〃ifZ,xg=0Bylocalexpression,avectorfieldXonMisaconformal(resp.Killing)vectorfieldif(3.8)Via+Vjgi=2入gij(resp.Via+Vja=0),L2-vectorfieldsonanon-compactRiemannianmanifoldwithboundary 21where5istheduall-formtoX.DEFINITIoN3.4([4]).AvectorfieldXonMhas""Z,2-"07,ｿ〃if5EL2''(M)nA'(M)fortheduall-form'toX・AvectorfieldXonMwithfiniteL2-normiscalledanL2_2ﾉ“加γ〃〃．Weremarkthat,in[4],anL2-vectorfieldiscalledavectorfieldwithfiniteglobalnorm..AnL2-vectorfieldXonMsatisfiesIIXII<oO．4.ForeachpointpofM,wedenotebyp(p)thedistancefromptoaM.SinceaMiscompactandconnected,piswell-defined.Weset(4.1)B(2r)={pEMIp(p)≦2r}foranyr>0.Afunction"onRsatisfiesthefollowingproperties:?? ???????? ．??? ?????、?(4.2）ThenwedefinelocallyLipschitzcontinuousfunctionsG)ronMby(4.3)G)r(p)="(p(p)/r)r=1,2,･･･foranypEMLEMMA4.1([1],[4]).T""e"is応α加s鰯zﾉg〃""z6"Cdepe"〃"go"〃0"鮎s"c"伽Z/IIG)rA"||:(2d≦(n+1)Cr-211"lli(2r)||"A*"||&(2d≦(n十1)Cr-211"||:(2r)lld公②〃||&(2j≦(n+1)Cr-211"ll:(2d/b"｡"""EAs(M),z"""gll･!|&(2d=γ“yEAs(M”g〃||･Il&(2r)く･,･>>B(2r)=ﾉB(2r)<･,･>*1.22 T.AOKI,J.S.PAKandS.YOROZUWeremarkthat,forany"EL2's(M)nA(M),GDr"EA:(M)andG)r〃－－→〃asr－→oointhestrongsense.LEMMA4.2.凡γ""y"EAI("),〃〃0肱加zオα(の：")＝のfd"+2G)rd"A" α、e、o〃M6(G)f")=G);助一*(2G)rdQ)rA慾〃)Lz.g.0"M,〃〃"e'".g.''"""s#泣加OSjgZﾉgW@"んe形'《5.Bythedirectcalculation,wehavePRoPosITIoN5.1．LetX比αzﾉ“加γ此〃0〃Mα”を肋g‘加zﾉ1油γ"toX.Zｿ'g〃党加ﾉ汰肋α＃．(5.1）<DX､1-n=,)D4,:X>+<Lxg+n¥,("g,L(."g+五二丁(6(""))･g>=_6(i(G);X)(Lxg)+五呈T("泌剛ByStokes'theorem,wehave/B(2r)d(*(｡；")=/aB(2r)":"(N)*1.一Sinced(*(の訪))=-*6(の訪)andaB(2r)=aMU{，E〃|p(p)=2r},(5.1)implies(5.2） ~/B(2r)6(":")*1=/aM"(N)*1"ByProposition5.land(5.2),wehavePRoPosITIoN5､2.Z,e#X6gα〃“加γ此〃0〃Mα"αご肋gd3"/1物γ"jpX.T"e〃が〃0随s肋αオく□X-(1一石杲試D㈱,":X>>B(2r)L2-vectorfieldsonanon-compactRiemannianmanifoldwithboundary 23+くLxg+T=T(")g,L"fr=ﾉaM(Z,Xg+志俺)･g)(X,N)*l.(6(の諺)).g>>B(2r)ThuswehaveTHEoREM563.Z,"X〃α〃Z,2-"ec加γ此〃0〃Mα”占幼edMzIl-X"w@zoX.X"α〃Z,2-co"か'7"αﾉzﾉ“加γ此〃0〃M〃α""o"ly"Z物g〃???????〈?????????，???｜??｜?(#）PRooF.IfXsatisfies(#),then,byProposition5.2,wehave0=くLxg+五旱丁(dZ)･g,L(":x)g十万二了(地緋g>>B(2r)=IG)r(LXg+了睾T(")･g)|;(2n+4くG'r(Lxg+五二丁俺)･g),d峰②盲>B(2r).Thuswehave,byLemma4､1,|"r(Lxg+五旱T(ag)･g)|3(2r)=-4くQ)r(Lxg+荒呈T(").g),dG)r@5>B(2r)≦告IG'r(Lxg+五呈丁(JS).g)li(2r)+8(n+1)Cr-21fl:(2r)Thuswehave~'",(Lxg+万辛丁(")･g)li(2r)≦8(n+1)Cr~zl51:(2r)Asr－→",wehaveILxg+5¥T(坊)･gll2=0ThereforeXisanL2-conformalvectorfieldonM.Theconverseistrivial24 T.AOKI,J.S.PAKandS.YOROZUThefollowingtheoremisgivenascorollaryofTheorem5.3THEoREM5.4;Z,"X6gα〃L2－"“加γ漉〃0〃Mα"‘ご肋ed""/1-んγ"@joXXis""L2_筋肋噌zﾉ“加γ此〃0〃Mがα"do"〃が(5.3）□X=0α"""=0(Lxg)(X,N)=0－0〃Mo"aMLetrbethemeancurvatureoftheboundaryaMdefinedbyで＝二αh(文α,X@r),where{文α}denotestheorthonolmallocalframeonaMwithrespecttog.T"e"Ⅱ■■■■LEMMA5.5.LetX6gα〃“加γ旗〃0〃Mα"d亭助edMz/1ai'"zioX.Z伽",o"aM〃〃0肋j加オ(▽NS)(N)=万琴十で(n5)-65(Lxg)(X,N)=(VN5)(X)+h(X,X)+r(n5)2＋2.(n5)･否琴一万((n5)r)-(n5)･65,"〃g"X=j*X+(nS).N""d琴〃"0妬伽""zz/1-f)"〃加文．ByTheorems5.3and5.4andLemma5.5,wehaveCoRoLLARY5.6.Z,e#X6gα〃Z,2－2ﾉ“加γ此〃0〃M趣"gg""I(γ膠S力．〃0γ"､αﾉ）加娩g加""伽〃aMα"α今娩e〔加zI1-jb""@IoX.T"e"Xis@z"L2-CO伽γ"zaj"ec加γ""0"Mがα"α0"〃が□X-(1-で封D(")=0(▽Nf)(X)+h(X,X)=0(”で.(nS)2_号幸÷<n5)･"=01■■■■0 〃M0"aM0"aM).ヨL2-vectorfieldsonanon-compaCtRiemannianmanifoldwithboundary 25CoRoLLARY5.7．Le#X6e""L2_"gC功γ"〃o〃MMZgE"加ノ(7℃Sp."0γ7"αﾉ）加勉g加""血沙aMα"‘専娩g〔加Zﾉ1-f)"伽加X.ZZe"X恋α〃L2_K"""g"gc伽漉"0"M"α"〃0"4ノグ□X=0,dS=0ー0〃M(▽N5)(X)+h(X,X)=00〃aM(resp・で｡(n5)2=0 0"aM)■■■■■6.LetXbeaconformalvectorfieldonM,thatis,Z,xg=2川g,andletSbethedual1-formtoX・Wedefineal-form"by(6.1)"=(Vja)5jdxi.ThenwehaveLEMMA6.1.〃加姑Z加オ一助＝＜男亭,を＞－2＜▽亭,▽5>+(n+1)<d入,5>+2(n+1)<入，入＞－必=(n+1)1<N,">=<3(nf),tf>+<C(t5),tf>*(2"dG)rA*")=-2<G)rdG'r②今,▽ど>,@"""g(5),=RI畠α"d(▽ど)ij=v!giPRoPosITIoN6.2ZWe"〃〃0姑メ加オLerX6e@zcomi)""zz/2ﾉ配加γ伽〃o"Mtz""5"ec加z/1拘γ"z功XくG)r男裏G)rS>>B(2r)-2ぐくG)r▽ず,のr▽ご>>B(2r)+(n+1)<く断dL瞳)rf>>B(2r)+2(n+1)<<Q)rl,G)r入>>B(2r)-2くdG)r②を,吟▽を>>B(2r)26 T,AOKI,J.S.PAKandS.YOROZU=ﾉaM<a(n5),tg>+<C(tS),tf>*1.PRooF.Foral-form"=(Vja)fjdx!onMwehave(6.2)d(*(・訪))=-*6(･訪）＝＊（。§(一助)t*(2G)rdG>rA*")).ByStokes'theorem,wehave(6.3）ん(2nd(*(";"))=ﾉaB(2r)<N,G):">*1ByLemma6､1,(6.2)and(6.3),wehave(6.4)<<G'r男亭'G)r'>>B(2r)-2くG〕r▽音,G)r▽写>>B(2r)+(n+1)<<G)7dl,G)r5>>B(2r)+2(n+1)くQ乃.1,G)r入>>B(2r)-2<<d"7②彦,Q"▽ど>>B(2r)=/aB(2r)②；{<5(nど),tf>+<C(t5),tf>}*1SinceaB(2r)=aMU{pEMIp(p)=2r},G)r=1onaMandG)r=0on{pEMIp(p)=2r},therighthandsideof(6.4)isequaltoﾉaM<3(n5),tf>+<C(tz),t5>*1.LEMMA6､3.〃〃0〃sメ畑オ<G)rd入,G)r5>>B(2r)=-(n+1)ぐく"rl,G)r入>>B(2r)－ぐくのrL率(2dGh.A*5)>>B(2r)･ThenProposition6.2andLemma6.3implyぐくG)r身考'"rg>>B(2r)=211G'｢▽今lli(2r)+(n+1)(n-1)|胸入lli(2r)+(n+1)く｡〕r入,*(2d""rA*')>>B(2r)+2＜d"r","rVf>>B(2"+/BM<3(n5),t5>+<C(tf),tf>*1L2-vectorfieldsonanon-compactRiemannianmanifoldwithboundary 27BytheSchwarzinequalityandLemma4.1,wehave21<<G)r入,*(dG)rA*5)>>B(2r)|≦211Q)rlllB(2mllda'rA*fllB(2r)≦211"rllli(2r)+(n+1)C･2-'r-2115113(2r)and21<<dG)r②琴,G"･▽ど>>B(2r)|≦ll"▽をlli(2r)+(n+1)Cr-211f||&(2r)Thuswehave<§＜"r男考,G)r5>>B(2r)≧211公▽琴||:(2r)+(n+1)(n-1)||"｢入||;(2r)-2(n+1)||公入ll3(2r)-(n+1)2C･2-1r¥211511:(2r)-IIG)｢▽ど||:(2r)-(n+1)Cr~211511i(2r)+/6M<3(n5),tf>+<C(tS),t5>*l=||G>｢▽を||;(2r)+(n+1)(n-3)||GDrlll3(2r)+/bM<3(n5),tど>+<C(t5),t5>*1-(n+1)(n+3)C･2~'r~2115113(2r)･PRoPosITIoN6.4.L"X比α〃L2-cM/b''"z"Jz)gc加γ此〃0〃Mα"αど肋gdim/1油γ’刀IDX."dimM=n+1≧4Lz"dlimsuPく②『男裏G)r5>>B(2r)<OO,"2"r→。｡limsupぐく"勿婁G)r5>>B(2r)r→o◎’28 T・AOKI,J.S.PAKandS・YOROZU≧||v5112+(n+1)(n-3)||八||2+/3M<3(nE),tf>+<C(t5),tS>*LCoRoLLARY6.5([5]).LetX6g""L2-K"""gzﾉgc加γ旋脇o"M"""亭伽‘加z/1和γ加joX."limsup<<G)r〃裏@@rf>>B(2r)<OO,"g"r→。◎limsupぐく瞳>r〃裏G)rf>>B(2r)r→‐≧||▽ど''2+ﾉaM<3(n'),t5>+<C(t5),t'>*1.IfthesecondfundamentalformofaMwithrespecttoNisnOn-negative,thenwehave/aM<C(t'),t5>*l≧0.Thus,byProposition6.4andCorollary6.5,wehaveTHEoREM6.6.(i)S"Posgj加ｵlimsupくのr男裏G)r5>>B(2r)≦0允γα〃畜匡L2''(M)nA'(M),"seco"‘ん"血加g"〃/b',"@r"aM"""sp〃勿伽〃"〃0"tW"o"""Jz)ec加γ"〃is"0"-"ggzz"zﾉe,""ddimM=n+1≧4.T""2zﾉeryL2_co伽γ7"αﾉ〃“加γ〃〃0〃MMZgg"伽/ZDaMispamﾉ肌(ii)a"OSe"tlimsupくG〕r〃裏のr5>>B(2r)≦Mﾌγα岬r→mgEL2,'(M)nA'(M)@z"ddimM=n+1≧4.Z塊g〃g"ML2-co伽γ7"α／〃“加γ此〃o〃M〃0γ"2α/jOaM走加"IMTHEoREM6.7([5]).(i)勤伽osg伽ｵlimsupく＆〕r〃裏&)r5>>B(2r)≦0/bγα"y今EL2''(M)nA'(M)a""伽sgco"dん"血'"e""ﾉrr;wzqfaM""'eSpecｵ加伽""〃o"〃"0γ7"αJ〃“加γ此〃た〃0"-""n"〃g."2"g"27@yL2_K"""gz)gc加γ"〃0〃MMZgE"""oaMis""zzJ"(1)a"Osej加tlimsupくG)r男皇Q)rf>>B(2r)≦0んγα"j5EL2,'(M)nA'(M).TWe"e"27@yL2-剛""g"ec〃禿g〃0"M"0γ伽/jOaMiS"〃〃THEoREM6､8.(i)a"osgj加fMisqf"agzz""gRiCc〃"γ沈z"",dimM=n+1≧4,α"α"eseco"‘ん"血沈g"〃/Mf@qfaM"鋤形ゆgc#勿伽〃"が0"たγ〃0ｸ'w@"J"g伽γん〃た"0"-"Zn""g.T"〃肋g"α""ZO"0"~Ze7'iOL2_com/MMz/zﾉ鉦加γ此肱0〃Mね咽E"j""oaM.(ii)a"ose"#MiSqf"ggzz""gRibcic"γ2％z〃”α"adimM=n+1≧4.T脆〃肋g形α形〃0〃0"-z"oL2-co伽γ"@a/zﾉ“功γ此肱0〃M"o""α/jOaM.THEoREM6.9([5]).(i)S"加sg肋ajMjsQr"agZZ""gRiCcic""ﾉα〃〃α"""eseco"dん"血沈g"〃ﾉﾉbγ"z"aM”肋”sl“ｵ加娩g〃"〃o""γ〃0γ"､α〃cc加γ此〃た〃0"-"ggzz"z)2.Z脆〃肋gγgα形〃0〃O"~Z"OL2_K"""gzﾉ“加γ勉肋o〃M〃"gg"伽/joaM.(ii)a"加sgZ加＃Misqf〃蝦""2ﾉeRibcic"γ21α〃".T"e〃肋g"""e"0"O"~Z"OL2_K"""g"“加γ旋姑0〃M"0""αjjOaM.ExAMPLE6.10.Wesetr=(x2+y2+z2),ﾉ2foranypoint(x,y,z)ER3andx=rcos&y=rsin&cos&z=rsin8,sin&,－Lz-vectorfieldsonanon-compactRiemannianmanifoldwithboundary 29thatis,(&,&,r)isthesphericalcOordinatesinR3.Fortwopositiveconstantnumbersa』anda2(a,<a2),weconsiderametricds2onR3suchthatds2=r2((daI)2+sin2a,(d&)2)+(dr)2ds2=r-2ﾉ3((da,)2+sin28I(d&)2)+(dr)2r≦alr≧(a,+a2)/2Then(",d")=(R3,ds2)isacomplete,non-compact,connectedandorientableRiemannianmanifold"WesetM={(&,&,r)こ〃|r≧a2},thenMisanon-compact,connectedandorientableRiemannianmanifoldwithacompactandconnectedboundaryaM={(8,,&,r)E'lr=a2}.ThenX=r-''3a/arisanL2-conformalvectorfieldonMnormaltoaM.LEMMA6.11,Z,〃W6eaR""z"""趣〃"α"加脇”"〃αR彪加”α""zα〃〃ze"cgWzz"dXα腫脇〃g"ec加γ"〃o"W"""Specj加娩gWZe"Cgw."f62α加s""g〃"C幼〃O〃WS"Cルォ加ｫX(f)≠00"W,"g"XおαcMIMMzIzﾉ“加γ此〃0"W"""Spcet加娩g"g加Cf･gw･InLemma6､11,ifgwisacompletemetriconWandfisaboundedfunctiononW,thenf･gwisacompletemetriconW([2]).ThusL2-conformalvectorfieldsonMtangentialtoaMaregivenbyexamplesin[5].REMARK6.12.1f,inTheorems6.8and6.9,wetakethatMisofnon-positiveRiccicurvatureandofinfinitevolume,thenwealsOhavethesameconclusion.REFERENCES[1]A.AndreottiandE.Vesentini,CarlemaneStimatesfortheLaplace-Beltramiequationoncomplexmanifolds,Inst.HautesEtudesSci.Publ.Math.25(1965),313-362.[2]K.NomizuandH.Ozeki,TheexistenceofcompleteRiemannianmetrics,Proc.Amer.Math.Soc.12（1961),889-891．[3]K.Yano,IntegralFormulasinRiemannianGeometry,MarcelDekker,INC,NewYork,1970.[4]S.Yorozu,KillingvectorfieldsoncompleteRiemannianmanifolds,Proc.Amer.Math.Soc.84(1982),115-120．[5]S.Yorozu,Killingvectorfieldsonnon-compactRiemannianmanifoldswithboundary,KodaiMath.J.5(1982),426-433.ToshihikoAokiDepartmentofLiberalArtsKanazawalnstituteofTechnologyNonoichi-machi,921JapanShinsukeYorozuDepartmentofMathematicsCollegeofLiberalArtsKanazawaUniversityKanazawa,920JapanJinSukPakDepartmentofMathematicsTeachersCollegeKyungpookNationalUniversityTaegu,635Korea

境界をもつ非コンパクトなリーマン多様体上のL2-ベクトル場

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境界をもつ非コンパクトなリーマン多様体上のL2-ベクトル場

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