Kawakami Hajime

Kitahara Haruo

Kanazawa University Repository for Academic Resources

Ann・Sci・KanazawaUniv・Vol.23,pp.1-14,1986OnaCounter-ExampleoftheConjectureofDodziuk-SingerbyM.T・Anderson.?、守一HajimeKawakami*)andHaruoKitahara(ReceivedMay6.1986)LetMbeacompletesimply-connectedriemannianmanifoldwithnon-positivesec-tionalCurvatureofevendimensionm.Dodziuk-Singerconjecture[D]meansthatHE(M)=0ifp≠m/2anddimH3ノ2(M)=oo・Here,H;(M)isthespaceofsmoothL2-harmonicforms.TheaffirmativesolutionofDodziuk-SingerconjectUreimplies,bymeansoftheL2-indextheoremforaregularcoverofM.F・Atiyah[A],thepositivesolutionofthewell-knoWnE.HopfConjecture;If(M,<,>)isacompactriemannianmanifoldofevendimensionmwithnegativesectionalcurvature,then(-1)mﾉ2%(M)>0Recently,M.T.Anderson[An]hasannouncedthattheconjectureisfalse,thatis,Ajv.[An]凡γα岬”≧2,0<p<籾α"dzz>|"z-2pl,"肋α≧1,"g形α形s"ply-co""gc花α池》"α"""〃"α"加姑〃z(ﾉ肋sgc伽"αJc"γz花z加泥一α2≦K≦j加オ伽沈Hp(M=-.TlmoIEMco"ゆん"-1s"c"?ThemanifoldMinhisproofisawarpedproductH2p(-a2)×『Sm-2p(1)whereH2p(-a2)(resp.Sm-2p(1))isthehyperbolic(resp.spherical)spaceformofConstantcur･vature-a2(resp.1),andf(x):=sinhs(x),sisthedistancefromxtoafixedtotallygeodesichyperplaneH2p-'inH2p(-a2).WeShallpointouthisimportantmistakesfromthefollowingpointofviews.First,thefollowingtheoremonthewarpedproductsiswell-known;*)AstudentoftheGraduateSchool,KanazawaUniversity.2HajimeKAWAKAMI,HaruoKITAHARArmoREj"B-O.(R.L・Bishop-B.O'Neill[B-O])Z,"〃α"""62""@a""α〃〃’α"沈姑α"α〃f>062α〃も花"施肋〃"c加犯o〃〃？脆〃肋g”αゆg"'""cｵ〃×′Ⅳz"肋"@e"cds2=6ﾒsM2+/26たAI2hzzsc"γZﾉ”加忽K<0が伽〃ん"i"gco""伽"S加脇：(1)d伽〃=1,0γ肋2s“伽"αJc"γz】α〃"Qf〃た〃電zz""9．(2)/"sMctlyco""2%.(3)(a)〃〃Ⅳ=1,07'(b)"esec加"αノC"γ"""Kル<0"/ルαSα加加加z"";KJv≦Oが/"eS"ot"2ﾉeα”加加"”．Q""g恋小が〃たCOWZ池彪α"α〃×，Ⅳ加s""""c2"'2)α伽'g,"e"co"dMD"s(1),(2)α〃（3）〃0脇□Then,theoremB-OimpliesthathismanifoldH2p(-a2)×『Sm-2p(1)cannothavenegativecurvatureforallwarpingfunctions.Therefore,hisstatementiSfalse.Second,hiswarpingfunctionfisnotstrictlypositive.Infact,f(x)=0forallxEH2p-1・Then,hismanifoldisnotriemannian・Ifhiswarpingfunctionfismiss-printed●forcoshs(x),then-az≦K≦1.Therefore,hisstatementisalsofalse.Itshouldseemforauthorsthatheconstructedthemanifold.Masfollowings;First,henotedH2p=R×c｡sh(H2p-!fortERand{0}×H2p-'isatotallygeodesichyperplaneinH2p(-a2).Second,heconsideredH2p(-a2)as(0,oo)×cosh!H2p~!u(-oo,0)×oosh［H2p-'u{0}×H2p~',andthensetH2p(-a2)=(0,oo)×c｡shs(x)H2p-1,s(x)isthedistancefromxtoH2p~'.Infact,everydifferentialforminhisproofisinvariantunderreflec-tionthroughH2p~1.Butheforgotthe{O}×H2p-｣-factor.Ifwereplacehiswarpingfunctionfbycoshr(x),r(x)isthedistancefromxtoafixedpointOEH2p(-a2),thenwehave?????????????? ????????????????????????》??????????》???REM4JRK.H.DonnellyandF.Xavier[D-X]haveprovedthat,foranyp<(m-1)/2,0≦E＜1-4p2/(m-1)2,acomplete,simply-connectedriemannianmanifoldwithsectionalcur-vature-1≦K≦－1+eofdimensionmhasneitherL2-harmonicp-formsnorL2-har-monic(m-p)-fOrms.NowweretumTheoremB-O,andwanttohaveacounterexampleofthecon-jectureinthesamewayasM.T.Anderson.Unfortunately,authorsdon'tknowexam-OnaCounter-ExampleoftheConjectureofDodziuk-SingerbyM.T.Anderson 3plesofacomplete,simply-connectedmanifoldofnegativecurvatureandoffinitevol-ume・Infact,theredoeSnotexistacomplete,Simply-ConnectedriemannianmanifoldsOfconstantnegativecurvature-landoffinitevolume.Thenifwelacktheassumptionofsimply-connectednessofM,wehaveTheorem2,3withaslightreversions.■????｜???????????????《?????，???????????????????????》??????????? ?????????????? ???，?????? ??????????????? ??????????????????????Fromtheabovetheorems,itseemstoauthorsthattheConjectureofDodziuk-Singerisunsolvedyet.1．L2-cohomologyspacesLet(M,<,>)beariemannianmanifoldofdimensionm.LetA*(M):=ZAp(M)bethespaceofdifferentialformsonM．LetddddddO→R→Ao(M)→A1(M)→A2(M)→…→Ap(M)→Ap+'(M)→…bethedeRhamcomplexonM.LetA8(M)bethespaceofsmoothp-formsonMwithcompactsupport.Wedefinetheglobalinnerproductく,>onA8(M):く",">:=f<",">dv｡,"=L"A*",where*istheHodgestaroperator.Let6betheformaladjointofd＜血,〃＞＝＜の,6">forのor"in"(")Let4:=-(d8+M)betheLalacianactingon/1*(M).H'(M):={のE〃(")|4の=0}iscalledthespaceofharmonicp-forms.LetLf(")bethecompletionof鮒(")withrespectto<,>.ThenitisaHilbertspace.Hf("):={のEL;(")回■4HajimeKAWAKAMI,HaruoKITAHARA'4の=0}isaclosedsubspaceinL;(").Infact,bytheellipticityof4,everyweaksolutionのissmooth.Then"(M)="(")nL;(").Andifwetakeanysequenceの息in"(")convergingtoのEL;("),thenwehave･くの,4">=0forany"E"("),forO=く』の点,">=くの慮,』"＞→くの,">(た→oo).Bytheellipticityof4のe〃(").Therefore,"(")isaclosedsubspace."(")iscalledthespaceofL2-har-monicp-forms.Since"(")isdenseinL;("),dhasthestrongclosure".Hereafter,weusethesamelettersforeveryoperatorandit'sclosure・LetDdbethedomainofd.Ifthereisan"EL;+I(")andの鹿→の,伽胞→〃,thencM(Ij="andのEDd.Itiswellknown([deR],［G､2.3］［Che]）that”＝〃inthesenseofdistributions,』.e、くり,‘＞＝＜の,">fOralljE"(")ifandonlyifのEDd,dtMj="."#("):="(")/mmiscalledthereducedL2-co-homologyspace,where〃(〃)：＝{のEDdl"=0}andBf("):=(のEL;(")|の＝","EDd}.Let(",<,>)beacompleteriemannianmanifoldofdimension"z.Wedefineasmoothfunction":R-→RbyO≦"(ｵ)≦1(#ER),"(#)=1(ｵ≦1),"(i)=0(ｵ≧2),anddefinethesequenceoffunctions""by""(x):="(p(")")(#EJV)wherep(x)isthegeodesicdistancefrom%tosomefixedxoE".（1.1)．Z,E""H FbγのE〃(M),回aOnaCounter-ExampleoftheConjectureofDodziuk-SingerbyM.T・Anderson||C加点Aのll≦(c")||のll,||伽点A*の||≦(c")||@ll.5PγoQ/.p(")isalocallyLipschizfunction,andatpointswhichitsderivativesexist,<3ip,ajp>≦77@.Thenwehave|伽陶'2≦(c")2.TJEoREM(1.2).〃Misco"此彪伽〃別(")={のEL;(M)|"=6(z)=0}PγoQf・Noteth誠〃(")="(")nL;(M).Ifdtz)=Suj=0,then4の=0.Conversely,ifのEHy("),z"鹿のE鮒(M)andso0＝く4の,Z(ﾉ点の>==一＜血,[加点』の＞一くda),z"MMQ)>一く6の,”鹿6の＞＋＜6の,*(‘加陶A*の)>.Letting〃→co,bymeansofLemma(1.1),wehave||"ll2+IIMI2=0Therefore,wehave"=aa)=0.WeconsiderdasanunboundedoperatoronL;(")withthedomain"(").SinceAf(M)isdenseinL;("),thereistheadjoint4*andsinceZiisanegative-definitesynunetricoperator,thereistheclosure4.LetD:*(resp.")bethedomainof4*(resp.』).ThenwehaveD:*=D:$Df,whereDA:={のED4*|4拳の＝畑,4>0}.Z,EMMA(1.3).([R-S],〃"se"dg/5""Opem功γggE""ec加溶、(ﾉ肋加s"2ﾉg●pp.186-137)LetA6e""yclbs""agzz"〃e-〃ｼ"蝿Sywcw2eWC,0〃αHM/69〃”αce.Zｿ""A=A*がα"αo"な〃伽γgα””oegg"z"z"esi"DA*.LEMMA(1.4).(Yau[Y])S@Wos"加t〃たcowzp"".〃のisczZz-p-X)7'"zs"cﾉZ/加メ』*の＝畑かsowzgA>0,"g"(Iノメsj【た""""zeγり．⑮6 HajimeKAWAKAMI,HaruoKITAHARAP7℃Q/.Sinceく〃,の>=くめ,j*の>=入くめ,の>forall.E"("),のisaweaksolu-tionofanellipticequationandsoのissmooth?????? ??》《。??????????????，????????????8缶Then,wehave||"陶血''2＋||”陶伽''2＝一入＜”ん2の,の＞－2く”ん‘加左4の,〔加＞＋2くの,z"h"ﾉ々八曲＞≦21く加々‘加陶Aの,"＞|＋21くの,Z""加々/16の>|≦211&加愈||-||の||(IIz"""||+||z"点伽ll),andsoIIz"た"||+llz"庫伽||≦411z"All"||の||(Here,weusetheinequality:%2+y2≦c(|"|+lyl)implieSthatl"|+lyl≦2c.)Letting々-うm,wehavem=6の=0.Therefore,we.haveの＝え－14*の＝0．Lemma(1.4)impliesthatD:*=",i.e.4*=4.（1.5)．』たα〃esse"勉砂s〃-α伽伽ォOpe7'n加γ0〃TfEoRE""zzz"加脇αCO"ゆた彪池"zα""zα〃qCoRoLLARY （1.6)・〃のα"a4のα沌加‘(肌肋g〃血α"‘伽α花吻避伽↓α"α助e"zS.aS29"e"Cgの息加鮒(〃s"c〃伽zオの点→の,』の庭→4の,αtzﾉ"→〔わ，加点→伽伽L2-Sg〃S9．P7℃Qr.ForのE〃("),lldtzjll+IMI=－＜の,4の>,andsoII"||2+||6の||2≦||の|lll4の||・Bythecontinuity,theirestimatesholdalsoonD:*=D：．Theexistenceofsuchasequenceの魔in"(")followsfromthedefinitionofD4'andtheaboveest加ates.B凸OnaCounter-ExampleoftheConjectureofDodziuk-SingerbyM.T,AndersonT"OREM(1.7).(ThegeneralHodgeTheoremofKodaira)([deR])r加oγ伽go"αノ戊形cts"""co"zposi加泥加姑o〃α"y"g"""""〃”α"加〃L；(")=Mf+'(")$"-'(")$Hr(").Z~Ms"co"zposj肋〃ルo姑α応0允γ〃(")nL;(")CoRoLLARY(1.8).(Gaffney{G1])O〃αco"zp彪彪池沈α""ね宛沈α"加脇〃(M)=Hr,#(")("s研必鰄幼αces)7PγDQf.NotethatW(M)=(d"-'("))Ln(Mf+'(M))4,where()Listheorthogonalcomplementof()inL;(").ByTheorem(1.7),thenaturalmap6:"(")→別,#(")issurjectiveand@|(cZ"-'("))Hsanisometricinjection.Moreover,disaclosedopera.tor,kerdisaclosedsubspaceinL;(M).ByTheorem(1.5),wehaveく”,り＞=くの,6">forのEDd,"ED8Therefore,fisinjective2.TheLaplaciansofWarpedProductLet(","M2)and(lV,"N2)beariemannianmanifoldofdimensionwzand"re.spectively.Wedefineawarpedproduct(〃×,IV,"2)asariemannianmanifoldwithametricds2:="M2+f2(x)@ZsN2where/isapositivefunctionon",calledawarpingfunc.tion.Let〃×ﾉIVbeawarpedproduct.Weshalldiscusstheinfluenceof/onthedeRhamcomplex,inparticular,theLaplacian.**)LetL;''(〃×ﾉIV)bethecompletionofAO''9(〃×Ⅳ):="(")ﾉ1"(N)inL2'+9(〃×奴).Then,L5(〃×Ⅳ)canbegeneratedby$p+9=7L2p'9(〃×")**)SZuker[Z)computedtheLaplaciansonawarpedproduct.ButhisComputationscontainerrors.｜，｜／I日｡8 HajimeKAWAKAMI,HaruoKITAHARAWenotethatthevolumeelementofthewarpedproductis,intermsofdiJoんand伽0人，(たﾉoノ=f通吻oん"ﾉoINThen,.wehaveZ,EﾉMMA(2.z).Fbγ伽"E"(")ﾉMf("),伽ムー"0γ"zIIM"||Egi"2"by'"''=／'’'劃'"'"蝿-‘'伽仙｡んTheexteriorderivativedassociatedtotheproductstucturemaybeWrittend=伽⑧1"+(-1)'1"②｡脚on"'｡(〃×Ⅳ).Sincethefollowingcalculationsarelocalinnature,wemaysupposethat"andjVareorientable.TherelationbetweentheHodgestaroperators*,*",and*jvisasfol-lows;(2.2）＊＝(－1)9(鋼-')*ﾉ*〃 on"'9(〃×Ⅳ),where*/:=Fb*Mand&:=/"-29.Letd*andd"*betheformaladjointofdandcZNrespectively.Thenwehave(2.3)d*("")=(_1)'(*71cj*")"+(_1)2f-2"(jv*"forの〃E"'9(〃×Ⅳ）LEMMA(2.4).Fbγ伽〃E鮒'｡(〃×jV),[Zr*("")=dh,*("")一("－29)(4d(,｡g,)｡)A",”舵”(2.5)cif*:=(-1)p*7'd*,PγひQr.TheequalitiesI’｜I’’8●OnaCounter-ExampleoftheConjectureofDodziuk-SingerbyM.T・Andersond,*("")､-1)p凡-'*"-'d(Fb*,,'｡)"andα(凡*"の)=tZR9A(*"d)+氏α(*,,'d)implythat(2.6)d,*("")=(-1)'Fb~'*"~'(dFM*")"+(鋪‘)〃AndR=/"-29impliesthatcZFM*Mj=("-29)/"-29-1(ZM*"の，andthefirsttermoftherighthandside<2.6)isequalto(-1)'("-29)*"-'(d(log/)A*"d)".Settingd(log'):=2乃のjandj:=ji,…ゎのlIA･･･Aの‘’intermsofalocalorthonormalcoframing{の'}on",wehave*"(cJ(log/)A*"妙）（鮮二簗,_‘)鋤偽'ん〃－=(-1)("-p)(p-')乃のi,...ipSg"=(-1)("-p)(p-')4d(I｡g,)j,whereZ(.)meanstheinteriormultiplicationby(.).Then,wehave*"'(d(log/)*"｡)=(－1)'-'f"(,｡g,)j,whichimpliesthatthefirsttermoftherighthandsideof(2.6)isequalto一("－2q)(C"(,｡g/)j)A".9Therefore,wemaycomputetheLaplacian4:=(f*d+"*actingon州〃×Ⅳ）｜’目●10HajimeKAWAKAMI,HaruoKITAHARATHEoREM(2.7).』＝4＋/~24"+2(-1)'@d(,｡g,)Cjv+2(-1)p+'/~2E"ぬ*"”"” 4：的＊＊′：α"deN走娩2＝"ぬ*+fif*dh">=(-1)'*,-1dh,*"=/〃-29*“域“た油γ”"姉"“絢犯ムydh"(log/)Moreover,wehave,TIEoR"(2.8)4=4,+(-1)p(dZF-6Fcf)0〃〃("),"ん"eF:="d(log/)REMARK(2.9).F､:=nd(log/)=-(themeancurvatureofJVinM×ﾉJV)Weconsiderthecomplexon〃×ﾉ"(2.10)0→R→ﾉ10(〃×災jV)望〃(〃×族ﾉv)g"(〃×奴)望…望〃(〃×JJV)三…whichissometimescalledthe<$basic"deRhamcomplex([K.1,2]).If〃×ﾉNiscom-pleteand"iscompact,thenwehaveCOROLLARY(2.ZZ).T"co"ゆ〃(2.10)saj域es伽凡j"cαだ‘加z肋加伽L2-Zgs"''CO肋"zo"3′が〃×烈走α’7り血砿〃､α"加脇地/isco"s麺雌In.fact,thePioncaredualityholdsinthe{wbasic''cohomologyifJVisminimal,df=O([K-T].[K2]).3．ThesectionalCurvaturesofawarpedproduct.Let(〃×，JV,*2)beawarpedproduct.WehavetheorthogonaldecompositionofthetangentbundleT(〃×，Ⅳ,"2);8む一OnaCounter-ExampleoftheConjectureofDodziuk-SingerbyM.T・Anderson 11T(〃×，Ⅳ,"2)==DMGD",whereDMandDjvareimegrabledistribUtionsonM×，Ⅳ.TheleavesofD"(respDN)areisometricto(",広"2)(reSp.(N,f2曲"2)).PROmsIYYoﾉv(3.1).([B=Ol)"esgchO"6zIc"γzﾉzMeK(")"(〃×'凡広2)isgiz)e〃妙K(")=K"(XY)IIXAYll"2－f(x){<"W>N((▽")2/)(XX)-2<1/;W>"((▽")2f)(XY)+<J/;V>N((▽")2/)(Y;Y)}+/2("){KW(1/;W)-ll97'rzd/||"2}<Wl砿ⅦW>,"he形▽(.)(7cm.Ki.))た幼gcozﾉα池"t庇"""e(7esp."esec肋"αJc""ﾉ"""Qf(.)),α"団（▽")2/iS"Hbssitz'@qf/ 1Hereafter,letH"(一α2)andJV"(-1)bethehyperboliCspaceformofdimensionll@andacompact(orfinitevolume)hyperbolicspaceofdimension".Andlets(x)bethegeodesicdistancefromXtoafixedpointOin"2p(一α2),andset/("):=cos"s(").Exz4MPLE1."2'(一α2)×，Ⅳ郷-21(_1).Thenwehave,一α2≦K(")≦－1P7'DQf・Let"begeneratedbyorthonolmalframing{X,V}.Then.wehavecoshasK(")=一a sinhassinhs一coshs'whichimpliesthatifs→oo,K(")→一aandifs→0,K(")→－1.Ontheotherhand,if"cDN(resp.D"),K(")=-1(resp.－α2).Therefore,wehavethetheaboveestimate.E狸〃fPLE2.H2p(一α2)×'S"-2p(1).Thenwehave-α2≦K(")≦14.ProofSofTheorems9寺12 HajimeKAWAKAMI,HaruoKITAHARAWeshallproveTheorem2.OurproofisaslightreversionofoneofM.T.Ariderson[An].":="21(－α2)×，Ⅳ''z-2p(-1)isacomplete,notsimply-connectedriemannianmanifold.Let{X}bealocalorthonoImalframingonH2'(－α2)ofeigenvectorsofV2fand{VI}alocalorthonormalframingonN銅-21(_1).Thenthe2-framings{XI4",{XI1卿and{1/i41/j}diagonalizethecurvaturetensorR:42(")→42(")withcorre-spondmgsectionalcurvature-α21-α(cothzzs)(tanhs),-1.ForのE"("2'(一α2)),4の＝‘"2,の+(-1)'[dZF一6Fd]のwhereF=(加－21)(J(log/).Anditholdsthat"2'(－α2)=="2(一α2)×gH2p-2(一α2)whereg:"2(一α2)→R,g(%):=cosh@zp("),p(")isthegeodesicdistancefromafixedpointOE"2(一α2)to%.Bythisdecomposition,F､istangenttothe&(－α2)-factor.Letの：＝の〃,｡E"("2(－α2)),77E"-I("2p-2(－α2)).If"isanyharmonic(P-1)-formon"21-2(一α2),then4の=0ifandonlyif(4.1）〃一はF－‘胡の=OOn"("2(－α2))Setting.:="andusingtheconformalequiValenceof"2(－α2)withj2:={(%,8)|xeER,8E(-'r/2,'r/2)},wehavethat(4.1)isequivalentto(4.2）ケ況伽2＋32""82＋"(8)3"脚＝0，"(8):=(1"i)(a/i/38),/i:=ル2(-@2)=1/2{(q]ﾉα一β'ﾉα]/bos｣ﾉαβ},q:=1+sin8,8:=1-sin6.Notethat"=OonaM2.Wemaysuppose,withoutlossofgeneraliry,that(加-21)>0andso">0.(4.2)hassolutions,smoothupto".Ifweconformallyidentify"2(一α2)withB2(1)withtheflatmetricjwemayobtainaninfinitedimensionalspaceofsolutionsof(4.2).llllisaconformalinvariantonformsinthemiddled肋ension.Forのasabove,we､’1gー志haveOnaCounter-ExampleoftheConjectureofDodziuk-SingerbyM.T.Andersonj;'"''-L"×…,'の|2/"-2p伽仙｡ル-"｡'(jv"-")L…_21'|21"|γ緬-21"｡‘卿｡伽恥婁一≦,｡！(Ⅳ…)"｡I(B"-2('))L""/…'伽‘13Here,wehaveusedtheconformalequivalenceofHk(一α2)WithBk(1),"=2,2p-2andhavesupposedthat"isaharmonic(p-1)-formwithl771_≦1withrespecttotheflatmetriCOnB2p-2(1),e.g."=(1/(p-1)/)dxIA･･･八dxp-｣ThenwehaveLJ'"<cr~%",andsoif(m-2p)/a<1,wehave〃の''<”TheoremlcanbealsoobtainedbyreplacingNm-2p(-1)byS"-2p(1)SO(m,1)/SO(m)isacomplete,simply-connectedriemannianmanifoldwithacom-pactquotient八SO(m,1)/SO(m),whereFisagroupactingdifferentiablyandproperlydiscontinuouslyonSO(m,1)/SO(m).LEMMA(4.3).TyWe歯αγg〃""glyco"z"ctOPe"setC"SO("@,Z)ISO("@)""I℃＝SO("',Z)/SO(wc).IfwereplaceNm-2p(-1)bytheaboveC,M:=H2P(-a2)×fCisanincomplete,①simply-connectedriemannianmanifold・vol(C)≦vol(C)impliesTheorem3.Co""ENT.Ifwemayfindacomplete,smply-connectedriemannianmanifoldwithnon-positivecurvature,andfinitevolume,theconjectureofDodziuk-SingerisfalsebyM.T.Anderson'sconstruction.14 HajimeKAWAKAMI,HaruoKITAHARAReferences[A)M.F.Atiyah,Ellipticoperators,discretegroupSandvonNeumannalgebras,Asterisque32-33(1976),43-72[An)M.T.Anderson,L2gharmonicformsandaconjectureofDodZiuk-Singer,Bull.ofAmer.Math.Soc.13(1985),163-165.[B-O]R.LBishop-B.O'Neill,Manifoldsofnegativecurvature,Trans.Amer.Math.Soc.145(1969),1-49.(Che]J・Cheeger,OntheHodgetheOryofRiemannianpseudomanifoldS,Proc・Sympos.PureMath.A.M.N.36(1980),91-146.[D)J.Dodziuk,Lz-hatmoniCformsonrotationallysymmetricRiemannianmanifolds,Proc.ofAmer.Math.Soc.77(1979),395-400.[D-X]HDonnelly-F.Xavier,OnthedifferentialformspectrumofnegativelycurvedRiemamianmanifolds,Amer.J.Math.106(1984),169-185.[K-T)EW.Kamber-PhTondeur,DualityforRiemannianfoliations,ProcSymp.PureMath.40,Amer.Math.Soc.,1982,609-618[K1]H.Kitahara,Remarksonsquare-integrablebasiccohomologyspacesonafoliatedriemamianmanifold,KodaiMath.J.2(1979),187-193.[K2]H.Kitahara,DifferentialGeometryofRiemannianFoliations,mimeographednote,1983.[G1)M・Gaffney,Theharmonicoperatorforexteriordifferentialforms,Proc.Nat.Acad.Sci.U.S.A.37(1951),48-50.(G2]M.Gaffney,AspeciaIStokes'stheoremforcompleteRiemannianmanifolds,Ann.ofMath.60(1954),140-145.[G3]MGaffney,TheheatequationmethodofMilgramandRosenbloomforopenRiemamianmanifolds,Ann.ofMath.60(1954),458-466.[deR]G・deRham,Variet6sDifferentable,HermannParisl960[R-S]M.Reed-B.Simon,MethodsofmodernmathematicalphysicsV6111FourierAnalysis,AcademicPress,NewYork.1975.[S)RS.Stricharty,AnalysisonLaplacianonthecompleteRiemannianmanifold,J:ofFunc.Analysis.52(1983)48-79.(Y)S.T.Yau,Somefunction-theoreticpropertiesofcompleteRiemannianmanifoldandtheirapplicationstogeOmetry,IndianaUniv・Math.J.25(1976),659-670.[Z)S.Zuker,L2=cohomologyofWarpedProductsandArithmeticGroups,Invent.math:70（1982),169-218.＄一?》r

On a Counter-Example of the Conjecture of Dodziuk-Singer by M.T.Anderson

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On a Counter-Example of the Conjecture of Dodziuk-Singer by M.T.Anderson

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