Ann・Sci・KanazawaUniv・
Vol.23,pp.1‑14,1986
OnaCounter‑ExampleoftheConjecture ofDodziuk‑SingerbyM.T・Anderson.
④︑
守 一
HajimeKawakami*)andHaruoKitahara (ReceivedMay6.1986)
LetMbeacompletesimply‑connectedriemannianmanifoldwithnon‑positivesec‑
tionalCurvatureofevendimensionm.Dodziuk‑Singerconjecture[D]meansthat HE(M)=0ifp≠m/2anddimH3ノ2(M)=oo・Here,H;(M)isthespaceofsmoothL2‑
harmonicforms.TheaffirmativesolutionofDodziuk‑SingerconjectUreimplies,by
meansoftheL2‑indextheoremforaregularcoverofM.F・Atiyah[A],thepositive solutionofthewell‑knoWnE.HopfConjecture;If(M,<,>)isacompactriemannian
manifoldofevendimensionmwithnegativesectionalcurvature,then
(‑1)mﾉ2%(M)>0
Recently,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=‑.
TlmoIEM
co"ゆん"
‑1s"c"
一
ThemanifoldMinhisproofisawarpedproductH2p(‑a2)×『Sm‑2p(1)where H2p(‑a2)(resp.Sm‑2p(1))isthehyperbolic(resp.spherical)spaceformofConstantcur･
vature‑a2(resp.1),andf(x):=sinhs(x),sisthedistancefromxtoafixedtotally geodesichyperplaneH2p‑'inH2p(‑a2).
WeShallpointouthisimportantmistakesfromthefollowingpointofviews.First, thefollowingtheoremonthewarpedproductsiswell‑known;
*)AstudentoftheGraduateSchool,KanazawaUniversity.
2 HajimeKAWAKAMI,HaruoKITAHARA
r m o R E j " B ‑ O . ( R . L ・ B i s h o p ‑ B . O ' N e i l l [ B ‑ O ] ) Z , " 〃 α " " " 6 2 " " @ 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
□
T h e n , t h e o r e m B ‑ O i m p l i e s t h a t h i s m a n i f o l d H 2 p ( ‑ a 2 ) × 『 S m ‑ 2 p ( 1 ) c a n n o t h a v e n e g a t i v e c u r v a t u r e f o r a l l w a r p i n g f u n c t i o n s . T h e r e f o r e , h i s s t a t e m e n t i S f a l s e .
S e c o n d , h i s w a r p i n g f u n c t i o n f i s n o t s t r i c t l y p o s i t i v e . I n f a c t , f ( x ) = 0 f o r a l l x E
H 2 p ‑ 1 ・ T h e n , h i s m a n i f o l d i s n o t r i e m a n n i a n ・ I f h i s w a r p i n g f u n c t i o n f i s m i s s
●‑ p r i n t e d
f o r c o s h s ( x ) , t h e n ‑ a z ≦ K ≦ 1 . T h e r e f o r e , h i s s t a t e m e n t i s a l s o f a l s e .
I t s h o u l d s e e m f o r a u t h o r s t h a t h e c o n s t r u c t e d t h e m a n i f o l d . M a s f o l l o w i n g s ; F i r s t , h e n o t e d H 2 p = R × c ｡ s h ( H 2 p ‑ ! f o r t E R a n d { 0 } × H 2 p ‑ ' i s a t o t a l l y g e o d e s i c h y p e r p l a n e i n H 2 p ( ‑ a 2 ) . S e c o n d , h e c o n s i d e r e d H 2 p ( ‑ a 2 ) a s ( 0 , o o ) × c o s h ! H 2 p! u ( ‑ o o , 0 ) × o o s h ［ H 2 p ‑ ' u { 0 } × H 2 p' , a n d t h e n s e t H 2 p ( ‑ a 2 ) = ( 0 , o o ) × c ｡ s h s ( x ) H 2 p ‑ 1 , s ( x ) i s t h e d i s t a n c e f r o m x t o H 2 p' . I n f a c t , e v e r y d i f f e r e n t i a l f o r m i n h i s p r o o f i s i n v a r i a n t u n d e r r e f l e c ‑ tionthroughH2p1.Butheforgotthe{O}×H2p‑｣‑factor.
Ifwereplacehiswarpingfunctionfbycoshr(x),r(x)isthedistancefromxtoa fixedpointOEH2p(‑a2),thenwehave
Ｊ４Ｄ凸 賊伽
州 花勺１ ａ二一 形Ｋ ２三一 娩銭 Ｌ一 應形
伽州 ＩＣ 却刎 沈加 ＩＣ の詑
伽州 伽伽
ハ〃 ｋ掘
夕〃 咽・︾
卿伽 伽伽一 Ｌ〃 ・州〃 ︾ Ｔ伽
R E M 4 J R K . H . D o n n e l l y a n d F . X a v i e r [ D ‑ X ] h a v e p r o v e d t h a t , f o r a n y p < ( m ‑ 1 ) / 2 , 0 ≦ E
＜ 1 ‑ 4 p 2 / ( m ‑ 1 ) 2 , a c o m p l e t e , s i m p l y ‑ c o n n e c t e d r i e m a n n i a n m a n i f o l d w i t h s e c t i o n a l c u r ‑ vature‑1≦K≦−1+eofdimensionmhasneitherL2‑harmonicp‑formsnorL2‑har‑
monic(m‑p)‑fOrms.
NowweretumTheoremB‑O,andwanttohaveacounterexampleofthecon‑
j e c t u r e i n t h e s a m e w a y a s M . T . A n d e r s o n . U n f o r t u n a t e l y , a u t h o r s d o n ' t k n o w e x a m ‑
OnaCounter‑ExampleoftheConjectureofDodziuk‑SingerbyM.T.Anderson 3
plesofacomplete,simply‑connectedmanifoldofnegativecurvatureandoffinitevol‑
ume・Infact,theredoeSnotexistacomplete,Simply‑Connectedriemannianmanifolds Ofconstantnegativecurvature‑landoffinitevolume.
Thenifwelacktheassumptionofsimply‑connectednessofM,wehaveTheorem 2,3withaslightreversions.
■ 〃７１ ●幻ロレ︽″レ 伽州 三〃 １均彩 娩一 ９α 形種− 嘩脆 ｜ 此 伽
︐ａ ｌｌ〃 助加 一節 加Ｓ ｌ肋 ン・Ｚ ａ
伽 棚岬
少 Ⅸ
ワ坐犯 囎・︾
卿蝿唖 γ師一一 凡岬伽 ２肌〃
伽
鋤 Ｔ岬伽 舳伽 醜 ０ 脚Ｓ
●ｍ″〃１ 花一 α三一 形Ｋ 伽竺 ︐α 副一 α花 的伽 ・地如
〃 しＣ 却刎 沈蜘 ＩＣ ン詑 伽肋
加肱 く沈 介産〃 服地
〃〃 型・唖
加州
卿・惚 伽〃ぬ
＆岬俳 唖帆〃 伽伽
Fromtheabovetheorems,itseemstoauthorsthattheConjectureofDodziuk‑
Singerisunsolvedyet.
1．L2‑cohomologyspaces
Let(M,<,>)beariemannianmanifoldofdimensionm.LetA*(M):=ZAp(M) bethespaceofdifferentialformsonM．Let
d d d d d d
O→R→Ao(M)→A1(M)→A2(M)→…→Ap(M)→Ap+'(M)→…
bethedeRhamcomplexonM.
LetA8(M)bethespaceofsmoothp‑formsonMwithcompactsupport.Wedefine theglobalinnerproductく,>onA8(M):
く " , " > : = f < " , " > d v ｡ , " = 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
■
4 HajimeKAWAKAMI,HaruoKITAHARA
' 4 の = 0 } i s a c l o s e d s u b s p a c e i n L ; ( " ) . I n f a c t , b y t h e e l l i p t i c i t y o f 4 , e v e r y w e a k solution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
〃 ( " ) . T h e r e f o r e , " ( " ) i s a c l o s e d s u b s p a c e . " ( " ) i s c a l l e d t h e s p a c e o f L 2 ‑ h a r ‑
monicp‑forms.
S i n c e " ( " ) i s d e n s e i n L ; ( " ) , d h a s t h e s t r o n g c l o s u r e " . H e r e a f t e r , w e u s e t h e
samelettersforeveryoperatorandit'sclosure・LetDdbethedomainofd.Ifthere
isan"EL;+I(")andの鹿→の,伽胞→〃,thencM(Ij="andのEDd.Itiswellknown([de R ] , ［ G ､ 2 . 3 ］ ［ C h e ] ） t h a t＝ 〃 i n t h e s e n s e o f d i s t r i b u t i o n s , 』 . e 、
くり, ＞＝＜の,">fOralljE"(")
ifandonlyifのEDd,dtMj="."#("):="(")/mmiscalledthereducedL2‑co‑
homologyspace,where
〃(〃)：＝{のEDdl"=0}
and
Bf("):=(のEL;(")￨の＝","EDd}.
Let(",<,>)beacompleteriemannianmanifoldofdimension"z.Wedefinea smoothfunction":R‑→Rby
O≦"(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
a
OnaCounter‑ExampleoftheConjectureofDodziuk‑SingerbyM.T・Anderson
￨ ￨ C 加 点 A の l l ≦ ( c " ) ￨ ￨ の l l ,
￨ ￨ 伽 点 A * の ￨ ￨ ≦ ( c " ) ￨ ￨ @ l l .
5
P γ o Q / . p ( " ) i s a l o c a l l y L i p s c h i z f u n c t i o n , a n d a t p o i n t s w h i c h i t s d e r i v a t i v e s e x i s t ,
<3ip,ajp>≦[email protected]
￨ 伽 陶 ' 2 ≦ ( c " ) 2 .
TJEoREM(1.2).〃Misco"此彪伽〃別(")={のEL;(M)￨"=6(z)=0}
P γ o Q f ・ N o t e t h 誠 〃 ( " ) = " ( " ) n L ; ( M ) . I f d t z ) = S u j = 0 , t h e n 4 の = 0 . C o n v e r s e l y , i f
のEHy("),z"鹿のE鮒(M)andso
0＝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
￨ ￨ " l l 2 + I I M I 2 = 0
Therefore,wehave"=aa)=0.
WeconsiderdasanunboundedoperatoronL;(")withthedomain"(").Since A f ( M ) i s d e n s e i n L ; ( " ) , t h e r e i s t h e a d j o i n t 4 * a n d s i n c e Z i i s a n e g a t i v e ‑ d e f i n i t e s y n u n e t r i c o p e r a t o r , t h e r e i s t h e c l o s u r e 4 . L e t D : * ( r e s p . " ) b e t h e d o m a i n o f 4 *
(resp.』).Thenwehave
D:*=D:$Df,
whereDA:={のED4*￨4拳の＝畑,4>0}.
Z,EMMA(1.3).([R‑S],
〃"se"dg/5""Opem功γ ggE""ec加溶、(ﾉ肋加s"2ﾉg
●pp.186‑137)LetA6e""yclbs""agzz"〃e‑〃ｼ"蝿Sywcw2eWC, 0〃αHM/69〃 αce.Zｿ""A=A*がα"αo"な〃伽γgα o
egg"z"z"esi"DA*.
L E M M A ( 1 . 4 ) . ( Y a u [ Y ] ) S @ W o s " 加 t 〃 た c o w z p " " . 〃 の i s c z Z z ‑ p ‑ X ) 7 ' " z s " c ﾉ Z / 加 メ
』*の＝畑かsowzgA>0,"g"(Iノメsj【た""""zeγり．
⑮
6 HajimeKAWAKAMI,HaruoKITAHARA
P7℃Q/.Sinceく〃,の>=くめ,j*の>=入くめ,の>forall.E"("),のisaweaksolu‑
tionofanellipticequationandsoのissmooth
参 加 ４ ン左 加シ
州︾︽ ︒〃冊
の嘩娼呼
々 がん血抑 心蜘抑鰄蜘
︐ わく一一一 恥一一一一二
く 入
8
缶
Then,wehave
￨ ￨ " 陶 血 ' ' 2 ＋ ￨ ￨陶 伽 ' ' 2
＝一入＜ ん2の,の＞−2く ん 加左4の,〔加＞＋2くの,z"h"ﾉ々八曲＞
≦21く加々加陶Aの,"＞￨＋21くの,Z""加々/16の>￨
≦ 2 1 1 & 加 愈 ￨ ￨ ‑ ￨ ￨ の ￨ ￨ ( I I z " " " ￨ ￨ + ￨ ￨ z " 点 伽 l l ) ,
andso
I I z " た " ￨ ￨ + l l z " 庫 伽 ￨ ￨ ≦ 4 1 1 z " A l l " ￨ ￨ の ￨ ￨
( H e r e , w e u s e t h e i n e q u a l i t y : % 2 + y 2 ≦ c ( ￨ " ￨ + l y l ) i m p l i e S t h a t l " ￨ + l y l ≦ 2 c . )
Letting々‑うm,wehavem=6の=0.Therefore,we.have
の＝え−14*の＝0．
Lemma(1.4)impliesthatD:*=",i.e.4*=4.
（1.5)．』たα〃esse"勉砂s〃‑α伽伽ォOpe7'n加γ0〃
TfEoRE"
"zzz"加脇
αCO"ゆた彪池"zα""zα〃
qCoRoLLARY （1.6)・〃のα"a4のα沌加 (肌肋g〃血α" 伽α花吻避伽↓α"α
助e"zS.aS29"e"Cgの息加鮒(〃s"c〃伽zオの点→の,』の庭→4の,αtzﾉ"→〔わ，加点→伽伽L2‑
Sg〃S9．
P 7 ℃ Q r . F o r の E 〃 ( " ) , l l d t z j l l + I M I = − ＜ の , 4 の > , a n d s o I I " ￨ ￨ 2 + ￨ ￨ 6 の ￨ ￨ 2 ≦ ￨ ￨ の ￨ l l l 4 の ￨ ￨ ・ B y
t h e c o n t i n u i t y , t h e i r e s t i m a t e s h o l d a l s o o n D : * = D ： ． T h e e x i s t e n c e o f s u c h a s e q u e n c e
の魔in"(")followsfromthedefinitionofD4'andtheaboveest加ates.
B
凸
OnaCounter‑ExampleoftheConjectureofDodziuk‑SingerbyM.T,Anderson
T"OREM(1.7).(ThegeneralHodgeTheoremofKodaira)([deR]) r加oγ伽go"αノ戊形cts"""co"zposi加泥加姑o〃α"y"g"""""〃 α"加〃
L；(")=Mf+'(")$"‑'(")$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)
7
PγDQf.NotethatW(M)=(d"‑'("))Ln(Mf+'(M))4,where()Listheorthogonal
complementof()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,"ED8
Therefore,fisinjective
2.TheLaplaciansofWarpedProduct
Let(","M2)and(lV,"N2)beariemannianmanifoldofdimensionwzand"re.
spectively.Wedefineawarpedproduct(〃×,IV,"2)asariemannianmanifoldwitha metricds2:="M2+f2(x)@ZsN2where/isapositivefunctionon",calledawarpingfunc.
tion.Let〃×ﾉIVbeawarpedproduct.Weshalldiscusstheinfluenceof/onthede Rhamcomplex,inparticular,theLaplacian.**)LetL;''(〃×ﾉIV)bethecompletionof
AO''9(〃×Ⅳ):="(")ﾉ1"(N)
inL2'+9(〃×奴).
Then,L5(〃×Ⅳ)canbegeneratedby
$p+9=7L2p'9(〃×")
**)SZuker[Z)computedtheLaplaciansonawarpedproduct.ButhisComputations
containerrors.
｜ ，
｜／
I
日
｡
8 HajimeKAWAKAMI,HaruoKITAHARA
Wenotethatthevolumeelementofthewarpedproductis,intermsofdiJoんand
伽0人，
(たﾉoノ=f通吻oん"ﾉoIN
Then,.wehave
Z,EﾉMMA(2.z).Fbγ伽"E"(")ﾉMf("),伽ムー"0γ"zIIM"￨￨Egi"2"by
' ' " " ' ' ' = ／ ' '' ' 劃 ' " ' ' " 蝿 ‑' 伽 仙 ｡ ん
TheexteriorderivativedassociatedtotheproductstucturemaybeWritten d=伽⑧1"+(‑1)'1"②｡脚
on"'｡(〃×Ⅳ).
Sincethefollowingcalculationsarelocalinnature,wemaysupposethat"andjV areorientable.TherelationbetweentheHodgestaroperators*,*",and*jvisasfol‑
lows;
(2.2）＊＝(−1)9(鋼‑')*ﾉ*〃 on"'9(〃×Ⅳ),
where*/:=Fb*Mand&:=/"‑29.Letd*andd"*betheformaladjointofdand cZNrespectively.Thenwehave
(2.3)d*("")=(̲1)'(*71cj*")"+(̲1)2f‑2"(jv*"
forの〃E"'9(〃×Ⅳ）
LEMMA(2.4).Fbγ伽〃E鮒'｡(〃×jV),
[ Z r * ( " " ) = d h , * ( " " ) 一 ( " − 2 9 ) ( 4 d ( , ｡ g , ) ｡ ) A " ,
舵
(2.5)cif*:=(‑1)p*7'd*,
PγひQr.Theequalities
I
｜
I
8
●
OnaCounter‑ExampleoftheConjectureofDodziuk‑SingerbyM.T・Anderson
d,*("")､‑1)p凡‑'*"‑'d(Fb*,,'｡)"
and
α(凡*"の)=tZR9A(*"d)+氏α(*,,'d)
implythat
(2.6)d,*("")=(‑1)'Fb '*" '(dFM*")"+(鋪 )〃
AndR=/"‑29impliesthat
cZFM*Mj=("‑29)/"‑29‑1(ZM*"の，
andthefirsttermoftherighthandside<2.6)isequalto
(‑1)'("‑29)*"‑'(d(log/)A*"d)".
S e t t i n g
d(log'):=2乃のjandj:=ji,…ゎのlIA･･･Aの
intermsofalocalorthonormalcoframing{の'}on",wehave
* " ( c J ( l o g / ) A * " 妙 ）
（ 鮮 二 簗 , ̲) 鋤 偽 ' ん 〃 −
= ( ‑ 1 ) ( " ‑ p ) ( p ‑ ' ) 乃 の i , . . . i p S g "
= ( ‑ 1 ) ( " ‑ p ) ( p ‑ ' ) 4 d ( I ｡ g , ) j ,
whereZ(.)meanstheinteriormultiplicationby(.).Then,wehave
* " ' ( d ( l o g / ) * " ｡ ) = ( − 1 ) ' ‑ ' f " ( , ｡ g , ) j ,
whichimpliesthatthefirsttermoftherighthandsideof(2.6)isequalto 一 ( " − 2 q ) ( C " ( , ｡ g / ) j ) A " .
9
Therefore,wemaycomputetheLaplacian4:=(f*d+"*actingon州〃×Ⅳ）
｜
目
●
10 HajimeKAWAKAMI,HaruoKITAHARA
THEoREM(2.7).
』 ＝ 4 ＋ /2 4 " + 2 ( ‑ 1 ) ' @ d ( , ｡ g , ) C j v + 2 ( ‑ 1 ) p + ' /2 E " ぬ * "
" 4：
的＊
＊ ′ ： α"deN走娩2
＝"ぬ*+fif*dh">
= ( ‑ 1 ) ' * , ‑ 1 d h , * "
=/〃‑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 . 1 0 ) 0 → R → ﾉ 1 0 ( 〃 × 災 j V ) 望 〃 ( 〃 × 族 ﾉ v ) g " ( 〃 × 奴 ) 望 … 望 〃 ( 〃 × J J V ) 三 …
whichissometimescalledthe<$basic"deRhamcomplex([K.1,2]).If〃×ﾉNiscom‑
pleteand"iscompact,thenwehave
COROLLARY(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
I n . f a c t , t h e P i o n c a r e d u a l i t y h o l d s i n t h e { w b a s i c ' ' c o h o m o l o g y i f J V i s m i n i m a l , d f =
O([K‑T].[K2]).
3．ThesectionalCurvaturesofawarpedproduct.
L e t ( 〃 × ， J V , * 2 ) b e a w a r p e d p r o d u c t . W e h a v e t h e o r t h o g o n a l d e c o m p o s i t i o n
ofthetangentbundleT(〃×，Ⅳ,"2);
8
む
一
OnaCounter‑ExampleoftheConjectureofDodziuk‑SingerbyM.T・Anderson 11
T(〃×，Ⅳ,"2)==DMGD",
whereDMandDjvareimegrabledistribUtionsonM×，Ⅳ.TheleavesofD"(resp DN)areisometricto(",広"2)
( r e S p . ( N , f 2 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/
1H e r e a f t e r , l e t H " ( 一 α 2 ) a n d J V " ( ‑ 1 ) b e t h e h y p e r b o l i C s p a c e f o r m o f d i m e n s i o n l l @ a n d a c o m p a c t ( o r f i n i t e v o l u m e ) h y p e r b o l i c s p a c e o f d i m e n s i o n " . A n d l e t s ( x ) b e t h e g e o d e s i c d i s t a n c e f r o m X t o a f i x e d p o i n t O i n " 2 p ( 一 α 2 ) , a n d s e t / ( " ) : = c o s " s ( " ) .
Exz4MPLE1."2'(一α2)×，Ⅳ郷‑21(̲1).Thenwehave, 一α2≦K(")≦−1
P7'DQf・Let"begeneratedbyorthonolmalframing{X,V}.Then.wehave
coshas
K(")=一a sinhas
一coshs' sinhs
whichimpliesthatifs→oo,K(")→一aandifs→0,K(")→−1.Ontheotherhand,if
" c D N ( r e s p . D " ) , K ( " ) = ‑ 1 ( r e s p . − α 2 ) . T h e r e f o r e , w e h a v e t h e t h e a b o v e e s t i m a t e .
E狸〃fPLE2.H2p(一α2)×'S"‑2p(1).Thenwehave
‑α2≦K(")≦1
4.ProofSofTheorems
9
寺
12 HajimeKAWAKAMI,HaruoKITAHARA
W e s h a l l p r o v e T h e o r e m 2 . O u r p r o o f i s a s l i g h t r e v e r s i o n o f o n e o f M . T . A r i d e r s o n [ A n ] . " : = " 2 1 ( − α 2 ) × ， Ⅳ ' ' z ‑ 2 p ( ‑ 1 ) i s a c o m p l e t e , n o t s i m p l y ‑ c o n n e c t e d r i e m a n n i a n m a n i f o l d . L e t { X } b e a l o c a l o r t h o n o I m a l f r a m i n g o n H 2 ' ( − α 2 ) o f e i g e n v e c t o r s o f V
2 f a n d { V I } a l o c a l o r t h o n o r m a l f r a m i n g o n N 銅 ‑ 2 1 ( ̲ 1 ) . T h e n t h e 2 ‑ f r a m i n g s { X I 4 " , { X I 1 卿 a n d { 1 / i 4 1 / j } d i a g o n a l i z e t h e c u r v a t u r e t e n s o r R : 4 2 ( " ) → 4 2 ( " ) w i t h c o r r e ‑
s p o n d m g s e c t i o n a l c u r v a t u r e ‑ α 2 1 ‑ α ( c o t h z z s ) ( t a n h s ) , ‑ 1 . ForのE"("2'(一α2)),
4の＝"2,の+(‑1)'[dZF一6Fd]の
whereF=(加−21)(J(log/).Anditholdsthat
" 2 ' ( − α 2 ) = = " 2 ( 一 α 2 ) × g H 2 p ‑ 2 ( 一 α 2 )
w h e r e g : " 2 ( 一 α 2 ) → R , g ( % ) : = c o s h @ z p ( " ) , p ( " ) i s t h e g e o d e s i c d i s t a n c e f r o m a f i x e d p o i n t O E " 2 ( 一 α 2 ) t o % . B y t h i s d e c o m p o s i t i o n , F ､ i s t a n g e n t t o t h e & ( − α 2 ) ‑ f a c t o r .
Let
の：＝の〃,｡E"("2(−α2)),77E"‑I("2p‑2(−α2)).
I f " i s a n y h a r m o n i c ( P ‑ 1 ) ‑ f o r m o n " 2 1 ‑ 2 ( 一 α 2 ) , t h e n 4 の = 0 i f a n d o n l y i f
(4.1）〃一はF− 胡の=OOn"("2(−α2))
S e t t i n g . : = " a n d u s i n g t h e
ec o n f o r m a l e q u i V a l e n c e o f " 2 ( − α 2 ) w i t h j 2 : = { ( % , 8 ) ￨ x E R , 8 E ( ‑ ' r / 2 , ' r / 2 ) } , w e h a v e t h a t ( 4 . 1 ) i s e q u i v a l e n t t o
(4.2）ケ況伽2＋32""82＋"(8)3"脚＝0，
" ( 8 ) : = ( 1 " i ) ( a / i / 3 8 ) ,
/ i : = ル 2 ( ‑ @ 2 ) = 1 / 2 { ( q ] ﾉ α 一 β ' ﾉ α ] / b o s ｣ ﾉ α β } ,
q:=1+sin8,8:=1‑sin6.
N o t e t h a t " = O o n a M 2 . W e m a y s u p p o s e , w i t h o u t l o s s o f g e n e r a l i r y , t h a t ( 加 ‑ 2 1 ) > 0 a n d s o " > 0 . ( 4 . 2 ) h a s s o l u t i o n s , s m o o t h u p t o " . I f w e c o n f o r m a l l y i d e n t i f y " 2 ( 一 α 2 ) w i t h B 2 ( 1 ) w i t h t h e f l a t m e t r i c j w e m a y o b t a i n a n i n f i n i t e d i m e n s i o n a l s p a c e o f s o l u t i o n s
o f ( 4 . 2 ) .
l l l l i s a c o n f o r m a l i n v a r i a n t o n f o r m s i n t h e m i d d l e d 肋 e n s i o n . F o r の a s a b o v e , w e
､
1
g
ー
