SUT Journal oま Ha土he血at ic.s..、, (Foコロnerly TRU Hathematics) Volume 26, Number 1 (1990), 55−67
.THE PROPER VALUE OF△FOR 1・FORM AND
FUNCTION IN CERTAIN.』 一 ”‘
COMPACT KAE肌ERIAN MAN皿OLDS 、、
KAZUHIKO TA.KANO
1(Recgi▼ed February 1, 1990;.Revised April 18, 1990)
Abstrsct. For any proper 1.for血and proper function of△corresponding tothe proper value λ,we show that the gb.special KiMng 1−form with k plays essentiahbles三n certain compact Keehlerian msnifelds when A actUally takes the possible minimal value. A」lfS 198θsubiect classifications. Primary 53C55;Secondary 53C65. Key脚?ds and夕九7α”ε8. Kaehlerian皿anifold, Lsplacian operator, Pmper form,由special Killing 1−form with・k, K−confbma1 Killing 2一伽m・ §1.IntIod皿ction. Let M「be an n−dimensio丑訓Riema皿ian ’manifold. Throughout this papel, we assume that manifolds・are・con皿ected and of dass C°°. Denoteエespectively・ by g飾Rki‘九,and Ri‘=Rhj‘b the皿etric te皿sor, the curvature tensor and the Ricci te丑soI of」M「in terms of local coordinates{¢‘}, where Latin indices run over the.range{1,∴.,n}. AKaehleエia皿manif・1d K2m withmet永g・f・eal⊃(五tnensil・皿n=2m・iS・a・Riemannia皿 manifo1《1 admitting a parallel tepsor五ddφ∼such that . ウ φ‘?φ㌶ ニーδ!, φ」輌=一φi∫, where we putφ∫‘=φ∫?侮. Let us associate to a p−fblm%(2η膓i≧p≧2)a scalar qp(u)by a・(・)一:@i…㌔・‘・一㌔†φ加φ」’・・」・3・−i?U・∼・一恒φ九ψ’・・」・・一ら・・∼ll・・r‘・》 If thele exists a positive constantんsuch that ・ −Rk5ih uig7’ uih.≧んσ2(%) : ・ . ビ holds for .any 2−fbrm%on。K2m evelywhele, the皿K2,n is said to be ofσ一positive cuτva− ture operator. In a co血pact Kaehlelian ma㎡fold K2冊ofσ一positive cu茸冊ture opetator, S』㌔・hiban・([5D・hb能d th・t“2m・dm』・6・1・・ed PT・pe・1−f・1m⑭f△・9・・e・p・亘d・ i皿ξtO the pr・peエv曲e(m+1)克, th・h匝aφ一sp・Clal・K皿桓91−{brm with k. M6re(頑, he plove{l that a compact simply co丑hected Kaehlelia皿ina皿ifbld K2nt 6fσ一positive curvat皿xe operato写.is isq皿etric with the complex proje.: ▲ive space Opm(k)with the Eubi皿i−Study m・tTi・・fC。・・t・・品・1・rP・・Phig Secti・pal・u・頑・e k・ . We consideエaKaehleΣia皿rnanifold K2M satisfying t・he condition(*):一・ (*)F・(%)≧}(冊+1)k国2 f・r any・1−f・・m・u,
︷
F2(w)≧▲克(4m l w 121十1Φ殼ノ13−41Aw l 2) fbエany 2−fbl血、セρ..55
56
THE PROPER V▲LUE OP △ FO駐 1−FORH ▲m》 F㎝CT工ONwhele先返apositive co皿stant.(s㏄【8D・The abo▼e condition(*)桓alwayミsatis6ed血a
Kaehlerian・manifold・of・a−P・siti▼e curvature dperator. Remalk. If K鍋{s of constant holomolphi¢s㏄tional cl11▼ature,. the equality signs in the condition(*)hOld. Recently, we have proved the fo皿owing theorems(18D. THEOREM A・Inαcompact Kaeゐlem’an manifold sat‘吻‘π9坑e coπ♂伍oπ(*), for any ・cわ8e♂P7りpef・2.form wφ∫△co7写esponding teオ克eρハoper valueλ, tρeゐαηεωザΦw・d・一・繊働・4嚇紘・πλ≧鵠ぱ為,
↓2ノザlw”απ輌θゐe8 identically, tゐeπλ≧(m十1)〃,ωザm>5andλ=(m十1)k, tゐεπw品αclosed K二coηプiortnal Killin92イbm・
THEOREM B. L・t.κ2m‘eα・・mp・ct・Ka・ゐ’・・‘⑰ 四en we九α暫e疏e lb〃owingS: ρノ皿e‘帖e璽包α在£y manザbld 8α‘isfu輌n9‘みe cond‘tion(*). 1▽du l 2≧克[〈δ血,%〉十〈(ア♂包,蚕忽〉−2〈dA.du,Φ%〉]−2(m十1)ん21%12 holds∫for any 1・1向m%, tσゐere‘ゐe eguality signゐolds if and op『yザt九ε1−form of‘5φ・special Killingψitゐ‘. ヒ ↓2/ij K2M a伽‘‘・αC・Cわ8eば卿,er 1・f・rm U Of△C・”℃sp・ndi.ng t・砺ρ・・ρeずカα』 (7π十1)克,ぴεπ%‘8毒ゐeφ一θρec‘α∼Killing 1−form tρ‘毒九克. THBOR■M C・Let」K2m beαco”rpact simp ly connected Kaehleri’an manijold satisfuing重ゐe COπイ‘‘‘0π(*)・If K2m admits a coclosed pro,er 1・ノb丹πq∫△corresponding‘O tみe proper ralue(m十1)‘,疏eπK2m‘θ‘30冊e‘ず‘C tρ‘疏cpm(克).. Fロrthermore, in{6], we ha▼e co皿si《leτed the plopel▼altieλof 1−fbエm%s皿ch thatΦdu=0桓acompact Kaehledan maコifold K2m satiSfying the co皿ditio皿(*)and have
obtai皿ed“s lower 1》ound・We also discussed the case w1[e皿λactua皿y takes the possible minima1▽alue and showed that the K−conformal Ki皿ing 2−fbrm and the aspecial Ki伍ng 1−form with k play essenti訓エoles in this field. h§3some operators for the differential P−form h a Kaehle車n manifold K2,n ale recalled a皿d we gi▼e the aefinitions of theφ一specia[Ki皿ing 1−f(》lm Withみand the K− conforrnal Killing 2−form. We c・nsider the p・・peτ▼alue・A・f 1−form u in a c・mpact Kaehlerian鵬nif・ld・K2m satiSfyi皿g the co皿dition(*)and show that its lower botind co丘tCide‘With the lower proper value of Theorem A..We show that’the g6−special ’Ki皿ing 1−form with k Plays esse丑tial roles in§4・h§5, we disc皿ss the proper value of the plopex function. The author woUld like to express hiS hearty thanks to Professor S. Yl皿aguchi・a取d Professor N. Abe for their help血l advice. He also.wo皿ld like to admowledge the encour− agement of ProfessoX S・「fachibana・ .§2・Pxd丘n桓頭es・We宝eprese皿t ten50Xs by the血components With respect to the
K編丁▲K▲加 ”57
n蜘「al basisD and’@use ‘he’
@Sntn血輌.c°”e
q.,F°⇔鑓・・en‘ill P−fo皿 ..
1 . . 旭=碗晦・一㌔♂ 2鞄く… 〈♂¢8, with skew symme垣c coe伍cients勉‘1..,i., the coeMcients of its exteτior differential du a皿d the exteτioΣco《lifferentialδ秘ale given by ,十1 傾)・、..,・?.、一Σ(−1)a+1▽・.・、、..乱..㌔+1, α=1 (δセt)‘2..,i.=一▽hOf・h‘2..ぷ., wh・・e▽九=9み」▽」,▽」d・n・t題th・・P…t・・㎡・⑰㎡・nt d・riv・ti▽・, and;。 m㎝・sい。 be deleted. R)r p−forms%a皿d v the inner Prod皿ct〈%,”〉, the Ien{紬s l旭lan《l l▽旭l aエegiven by 〈%lv〉−i!u・・−i、v’1・・.ie,国2−〈包,・〉・ 1▽・12=〉▽鵬一ら▽ゐu’・一ら・ . De丑oti皿g by△=dδ十δd the Laplacian operator,.we ha▼e△∫=一▽,▽, f fbr a functio皿
∫and
(2・1) (△%)・、1.㌔=一▽’▽,%・、..ぷ.+恥)・、…・. as the coeMcients of△%, whele H(%)‘1...i. ale the coe]吊cients of H(%)gi▼e皿by (2.2) 丑(%)ξ=R‘, H(旭)‘1..,iP=Ul
ΣR
(P・=1), i.’u・、…・…i,+ΣR、.“”eq、一.t…...,i. a<b (n≧カ≧2). hthe sec・皿d・term・n the right−ha皿d・Side・f the・1ast・ab・▼e equati・皿, the subscript・皿d 8are伍the positio皿s of ia and‘6 respecti▼dy, and We shall use血Ular arrangements of hdices without any special notice.(2.1)mぴy be vvritten as follows: (2・3) △%=一▽,▽.%十H(%). The quadxatic fb血Fp(%)of%is define《l by Fp(%)=<.H(%),u> 一(ρ」1)!(争漣一ら㎡・、一ら+P;1阜ぷ♂・・㌔磯A), and it appeals i皿the fb皿owhlg wel1−k皿own .formUla which iS va五《I fbr any. pfoエm%:(…) 1△(1・12)一く△・,・戸1▽・臼(・).’
58=
THE PROPER ▼▲LUE OF・△}FOR 1−FORIl▲m FUNCT工ON
If a nonzelo垣b叫u satisfies△%=.加with a.positive constantλ, i巳ca皿ed a
P・・pe・fbxm(ぜ△coロespond桓g t・the pπ・pe・value・A.K,。監、鷲瓢゜漂2竺h興uei, we shaJ’岬eτaKaeMe麺ma興
The血ndamental{bmφ垣he 2−form given by
・ 1 . . . φ=豆φ」‘d♂〈d『8・ . Now, we wartt to recal1 some operators for di ffere皿tia1 f(》rms in K2m. De皿ote byヂρ t’?E・et・f雄戸㎜・・n K2m・Th・・pe・・t。・s
r.:アP→タP+1, 0:アP→ク7−1, 委:戸→.プ?are・defined・respectively bプ . ド’ 内 ’. .
P (ru)・。..・.一Σ(71)αφ㌔’▽・・、。..』。...、,,・ α=0 (c%)‘…エ・=φ1’▽・%’‘?;・’i・, , ’ . 、. (叫、..㌔一Σヂ㌔?eq、..・..㌔ . ’ 』 6=1 . fbl a皿y p{brm u. FOI O−fblm旭o, we・defi皿ed O廻⑲=QUo=0. We denote by L(ヱesp. A)the exterior(resp.加αior)product with the fUndamental 2−fbxmφ, then the operators L:フmP→ヂ7+2 and A:戸→フ「P−2 ale wlitten by l Lu」.φ∧旭,..A%ニ(−1),*五*泌 f(》ra’吹Dform包, whexe*means the dilal operator. A is・t亘・6al on O a丑d 1−f()rrns. These }ocal explessio丑s ale《le五ned by ・「 , , ・ .. . ⑭…、..・,=φ脳、..・.一Σφ・醐…・.・・,一Σφ・・.・・、…・…ii+Σφ・。・.・・、…・…・..,i,, 4=1 b=1 α<b . 1・..・\∵(A・)・・,み.=豆φ”・”・…㌔・’ ‘.h、..
亜bl the opelatols abo▼e, we have丘01n[1]a皿d【2】 (3.1) (d!壷一委d!)勉=−r賜, (3.3) (rQ一φ]F)%=du, (3.5) (δr十rδ)%=0, (3.7) (δC十〇δ)包二〇, (3・9) (dC十〇d)勉「. q, (3◆1i)(AΦ一QA)旭=0, {3.13)(Llr・1{匝Z})%=0,. (3.15)(1’0十(アr)%:=△%, (3.17)】P2u=0. . ’ (3.2) (δ委一φδ)%=−0勉, 、(3・4).(0曼・一倭C)勉=δ%, (3.6) (61r十rd)%=0, (3.8) (d!A.−Ad)%=一(ア1比,三 (3.10)./(]F4v−Ar)%=δ%, (3.12)(C五一1}0)%=_dηL, ⑬.ユ4)(△委「.倭△)旭=.0,、 (3.16)σ2%==0,K..・T▲IUπ0.
59
Moxeovel桓a compact K2m, it fb皿ows丘o血{1】tha1 、 (3.18) (Lu,,%)=(ω,A%) for w∈ヂ7 a皿d%∈夕予+2,層 (3.19) .{委w,』%)聖=一(ψ,Φu)1 ・二.9for w,%∈’∫P, .‘ (3.20) (Pw,旭)=(切,0%) fbl w∈アP aロd%∈1p+.1, whexe(,)denotes the global hmex product.L・t・一鋤‘b・・K皿・g1−f・・m,甑・1一蝕m・…hth・t th・▼㏄t・r・fi・ld・#一(め
・btain・d丘・m・by輌・・f th・頑i・gi・輌五㎡t輌曲・m・t・y・
▽5吻十▽iUi=;O:AK過9・−f・・m・iS・d・・d・Of・peci麺砕江麺i訊・・
1 ▽・▽」聾・+ik(9・5・・−9刷一φ・∫毎+φ繭+2φ吻一〇, wheleんis a CO丑stant and we P皿t毎」(IU)‘. A2−form a〃is said to be I(−co皿fblmal Ki皿ing, if it satiSfies ▽・W」・+▽iWk・=2ρ・9ザρ・9」・−Pi gk・+3(P為φ5‘十砺φ為‘)・ where we p皿t 1 ρ‘=−h+2{6w)‘, p‘=(¢P)・・ It・is easy to see f}om the above equation that 1 (3.21) dΦw=2rwり.. §4.The p宝opel▼alue of the proper 1−fblm. At first, we plepare thdb且owing Lemrdas飴「1・tc・u・e・ .,.、. ∴. .・、. ’ ,
L・MM・4.1.1。 K・冊, w,』ノ。。。輌W 1.fo。m。 ㌧’
(…) .:A砕』・,∂畷㌍・・l
LEMMA・4.2.加・卿・ct・K・・ゐ㎞・・m・晒ぼ頁2m, w・ゐ鋤か・・y・1.f。rm・u
(4.2) 1|Φ刺2=2(lldall2−ll酬2).=2(llr包ll2−llδ・ll2),
(4.3) II輌12−llC・ll2,
⑭.、 ll酬12−1酬∵.... .、・.,L
l匝{彦動112r恒吻ll2,・.・ 、(4.5)(4・6) llA輌ll2一胸12,
60
THE PROPER V▲LUE Oセ ム’FOR 1−PORIf AND FUNCTION
ψゐe肥1卜ll den・勧疏e g∫・bal length. Pxoof It{b皿ows丘om(3.1)∼(3.5),(3.8),(3.9),(3.15),(3.19),(3.20)and△=砺十δdthat
(Qdu,《¢du)=(堀%,饅包)+.(釘包,r%). =一傾,翻委%)一佃,Φr%) r2[ll刺2−(du, rQu)] =2{lld電島ll2−(Cda㌧4}%)】 =2{1μ%ll2十(dCu,亜%)】 =2[ll♂%‖2十((ア%,δΦ%)] =2(lld:毘ll2−llO%ll2) =2(llr%ll2−llδ%ll2), (A4%, Adu)ニ(dA%+0%, d加+(ア旭) =‖o%ll2and
(dlu, dΦu)=一(包,4}δ♂Φ%) =一(%,(δ委+c)踵%) =一(勉,δ(dQ十r)Φ%十Cd{Pu) =(%,《1δu十δば%−rc%)=(%,△%−ro%) .
=(u, CTu) =llr%ll2, whidl yidd that(4.2)∼(4.4)are true. It is easy to see that the rest hold by▼irtile of (3.4)a皿d(42)∼(4.4).LEMMA 4.3.1れαcompαc毒καeゐ’eパαπmanifeld K玩,∫ごアany p”per 1・∫b牝m%o∫△
correspOnding毒o fゐεp7りpεP valueλ‘ゐε∫bπ0電∂‘πgノ’ormulqsゐ0「」: ’ (4・7) ’ (Φδdu, dAdu)=一(蚕δ4%, Odu)=一λllOul12,(4・8) 11δdφ・ll2一λ11r・ll2,
(4・9) (Φδd委u,(IAdφ%)=一(¢δdQ%, Od委u)=一λll6%‖2,(・…) 1剛1→1・・酬・一・ll…ll・,
(・…) ll・卿1!一・ll壷剛1・・
Proof We obtain i皿mediately(4.7)by(3.1),(3.5),(3.7)∼(3.9),(3.19),(3.20)a泊d△=d6+δd. N血g acco皿t・of△=dδ+δd,‘we have
llδdul▲2=λlldul12.K.T▲K▲NO・一’・・1:「
61
The皿cha丑ging包i皿to委%in theε山o▼e equatioロa孤《Lowihg to(4・4)ジwe 96t(4.8).’It拍1雛ぽ;鵠th°’日by壷t皿e°f(3司∼(3外(3・15),(3’16)・(生2),⑭・㈹,
hth・・e舳・d…f趣・ecti・・, w・…m・th・t・K2m・・t域・li・th・・c・・diii・・(・)㎝d is comp㏄t. The positi▼e cons‘ant先h(*)wil1 be fU【ed thlo皿gho皿t papeτ. h[6],we Plo▽ed the fbllo㎡丑9 fheorem. . . THBOR取l D・1}bずα〃oper 2−form w q∫△corresponding toαρ7りpeず拠玩eλ‘παcomρact π2m satisfuin9坑e condit‘on(*),坑e‘πε璽題α況ダ .・1回12≧}1酬2−m…、【II・wll・+・(亜δw, (IAw)一(・・w7・切】』
+{・{輌ll・+lliwll・一司1酬 ゐolds・ij‘ゐεeguality signゐeld●‘πオゐεabOVe‘negualit.y,彦ゐeπtゐe 2−♪om⑩‘8 K;「confol∀πα「 Killing. . ・ Let u be a proper 1−form。f△coエr巴pon曲1g t。 the pxopex曲eλ. Then changing whlto♂%a皿d{煙%in Theoτe皿D,τespecti▽ely and owing to(3.14), we get 1 λ11dull2≧一 {11δdu‖2十2(《臣δ4%, dAdu).一(委δ血, C4%)l m十1 +;酬刺・+ll呪・ll・一・1』‖・1,・ll酬2≧−mし1‖1・…ll・+・(委δd蚕包,立ば《5包)一(・・…幽)1・
+;・{・mll剰1・+1匝西・ll・一・llA・酬 and b㏄a皿se of(4.2)∼(4.9), the above two eq皿atio草s are rewritten as fb皿ows:(4・12) [2(仇+2)λ一伽+1)(2m+1)‘1酬2−3[2λ一(m+1)‘川σ・ll2≧0,
(4・13) [2伽+2)λ一(m+1)(2m+1)克川「ザ三3[2λ一(m+1周llδ・ll2≧・・
Also, changhng w麺to dCru in Theolem D and makhlg use of(3.14),(4.1),(4.10),(4.11),△0ニ0△皿d△r=r△,we have l
(4.14) {2(”駕十2)λ一(冊十1)(2”葛十1)ゐ川委ば%‖2≧0. Hence we have PROPOSITION 4.4. Inαcompαc毒Kaehlert’an M’ anifoldκ2mθᣑ∂ノシ‘πg毒ゐe coπ♂“‘oπ(*),忽㌶慧繊’㌶工1Z△‘9”−s?°”4‘π四毒ゐ1卿P『ず”α九e戎ザ委4We8城‘4伽
(m十1)(2m十1) λ〉 克. 2(協+2)62
THE PROPER ▼▲LUE O野△./,FOR 1・−PORN ▲ND FUNCTION ・P零oo£Erom(4.14).and Theorem D. we have..・, t−..t.: …−r(・)江賄d・es・』・・i・allY・Vani・h,』・hと・λ≧m吉鵠+1ゐ,.層.
−2)ギλ三.me・’−1・)+;靱ρ・拠姪・岬K−・・輌m垣K過・2−f°・m・、
We c…ider・・th・・.’・eas…f(2)・W・dS・um・・thatrA’≒竺き器笥‡Uk・’・Th・n・it・f・ll・w・丘om(4.12)a丑d(4.13)that O%=Oa皿dδμrO. By▼iヰUe.of(3.15)a皿d O包=0, we
obt血 F・一一ば伽一d△・一λdu一雲(1)(2m+1nt+2)1)晒 ,
whi《血mea皿s that du iS a closed K−confbτmal K皿皿92−fbm. Also, Changing w into du麺 (3.21)a皿《ltah皿g acco皿nt of(3.1),.we have ..、 . t ・、. .. 口%=0.If we operateδto the above equatio皿a皿d regard to(3.5),δ%=Oa且d△=dδ十δd,
the皿we get r%=0, which denotes by y詮t皿e of(3ユ5)a皿d Oμ=.Othat A%=0. This…t・adi・・t・t・λ一誓器笥+1ゐ・H・・c−・血dλ>ml袈穿1随・皿(1)・nd(2)・∵・
Let.us.・pr◎ve the fbUowings・; 、。・. ・ LEMMA 4.5.1カ.1ζ2仇, we’ 薰≠魔?@ 1 (4.15) ‘ ・ 4}drδu:=0;『 φ’acO%=0,(4・16) ∴ .・.・一岬㍑=0.
for any 1・≠「orm u. ・ . 』・ ・.ζ t/: ;. .. “ LE瑚[M[A 4.6.1カαcρmραcfκαe脳eパαππ己anifo’d頁2m,∫bアany proper 1・∫ρ仇%o∫△ CO竹℃3ρ6π{“句・toオゐe pmpeア”α1題6λ’tゐe fo∼あwing formulasゐold: (4.17) ミ ・ξ :−r : ・‖drδu,H 3=‖Adriδ%‖2=ごλ2 Hδ旭川2,・(4・18)_’・.、...、 ll郷μ r rλ311δ・ll2,... .「 、・、
(4.19) (倭δ孤δ%,瑚L《江δ%)=一(Φ6drδu, cacδu)=一λSlIδ包II2,(4・2°)....㌧一、1μCcll2−1|∧在9・ll2三λ2110・ll2,一・ 、.
(4.21) 11δdl’Oull2=λ311Cul12, . ’ (4・22).・.fΦδ4ro%,44ばrσ%〕ニー(’¢6arσu3 cerOu)ヒ=→3HOull2.(生23) 1酬1・一醐・ll・二・ll酬1・,「『 −
(生・4) 岬二ll・三・ll軸・.「
・.Plo6f. Erom(3.5)∼(3.7),(3.9),(3.15),(4.3)a孤d△=品十δd we obta口1㎞mediately (4ユ7)..Ma1血g・use of(3.1)∼(3.3),(3:5)∼(3.7),(3.9),.(3:10),(3.15)∼(3.17),(3,・20), (4.2),(4.3),(4.5),(4.7)and△=が十δば, we have the Iest. ’ ∴ 」! ・K.TAK▲NO
63, Now, cha皿gipg w i皿to dl)δu .and aSCu in Theorem D, respectively. and ow口lg .to (4.15),(4.17)∼(4.22),△OrO△an{ユムr=r△, we get [λ一(i}十1)珂11δull2≧0;.’ 』 1 ・ . 、[λ一.(m.十1)klllCull2≧0..tt.・一、He皿ce, we have .. .・ .,. 、. ..
T THE・REM 4・7・血.…殉・・t K“ゐ伝・・m・nif・ld K2nt・・ti・fU吻蓄ゐ・c・ndition(1), f・r any proper 1−∫わ㎝旭0∫△CO洲pe5,0π♂輌π9 to the Pfりperηψελ,ωεゐave . . (1/if 6u ・r Cu d…n・t・id・nti・・鞠・碗・ゐ, tE・n・」L≧(”・+1)ん, (2) ifλ=(m+1)克, th・n・drδu⑰」4rρ%・副ゐ・ψ・・d K−・・nf・rm・1・Killing 2・釦㎜,(3)ザδ…d伽励ゐ硫‘・・lly・・th・・.λ>m器笥+1先・
Ploof It is easy to see丘om the above two inequalities and Theorem D that(1)artd (2)hold. Now, we considex the case of(3). We assume thatδ%=Oa皿d O%=0. The皿it姐・ws丘・m(4.12)that
λ[2(7π十2)λ一(m十1)(27π十1)k]IIul12≧0. tt ≧Beca・se・f・All・ll2>0, w・g・tλ≧m耀笥+Uk・E・peci姐y,血the㈱・fλ一
冊器笥+1ん,w・・ee by輌・・f・The・・em・D・th・鋤杣・・1・・ed K−…f・・mal・Killi・g
2−fblm・Also, changingセ〃hto♂勉麺(3・21)and皿akhtg use of(3.1), we get Tdu=0. Opelatingδto the above equation an《l owi皿g to(3.5),δ%=Oand△=砺十δd, we obt血「・=0畑・h implies f「・m O・−Oth・t△・=0・Ths c・・t・・di・t・t・λ一m吉器笥+llk・
]日[e皿ce we see that(3)hol《1s. Let us show the fb且owing. TH・・R・M 4・8・L・t K2m b・α…卿《. A故・娠・晒・η{岡・・働拘th・・c・・diti・・(・). ij K2,n・dmit・α〃・P・・1・f・・rm・u。∫△・・仇・P・π輌‘・£ゐe’ P・・p・r・valu・伽+1)いゐ・η t九e 1イform rδu and rCu are ¢−sp’e’ cial Killing with k.一・ P…f・・lt・iS・d・・y t・・ee・th・t th・1麺i 6drδu‘and・6are…e+・pecial Killi・g With d々2)・Owi・g’ tb〈3・5)・∼(3・7),(3.9)and△−dδ+δd, w・g・t みby The・rem B・(2)an δ《肛δ%=rδ△%=(7亮十1)克rδ%, δaCCu=「o△・−d6「Cu=(m十1)克「o・・ whi・h・impli・・th・t th・1−f・・m r5%・皿d rO%・・e◆・pecial・K沮桓9曲h克. Th・・e・c・mp1・t・’ proof. ’ Also, changhlgψi皿to 6dru in Theo‡em D and oWing to(4.16),(4.23)∼(4.25)and△rニ
r△,we Hnd (4.25) .、 14λ一3(2m十1)ゐ川委血‖2≧0. ⊥ P ♪ . 〉、 − . ’ . . . r . ’ He皿ce, we get 1. 9 『 ・64
皿 PROPER ▼」LLUE OF △・FO且、1−FORN ▲M》FUNCTION
THEOREM 4.9. Let K2冊6εαcompact simp’ly bonnected」Kaeみleri¢n manijわld satiSli旨ing ‘ゐ…nditi・n(*).『ij K2m admite a P・・p・・1−f・rm U ・f△・・f・rNl・Pbnding t・品pmP・・・・lu・(m+1)いゐ・n K2m i・輌・拠撤嚇.cpmω・
・ Proof. We assume that rδu=O and TOu・=0. Then, by Virtue of(3.5),(3.7),(3.9), (3.15)and(3.20), we obt血δ旭=Oand O%=0, whidl denotes丘om(4.2)and(4.25)that 旭▼輌hes identica皿yL This co皿tradicts to、 u≠0・Thus rδ笹and rCu dobS皿ot i《le皿tica皿y ▼anish. Co皿seq皿e皿tly, the fact together With Theorem C a丑d 4.8 co皿plete the proof. We sha皿prove the folloWing. LEMMA 4.10. In K2冊, weノ言π♂Jbずany 1−Jb㎜旭(4.26) A委ru=0, AΦCLu=0.
It is easy to see that (4.27) for any 2−form w. The皿we get ll亜2wll2 ・= 4111tvH2 LE】ばM【A 4.11. Inαcompact Kaeゐlerian manifold K2m, w¢ムαガeプbr any 1−Jb牝m包 (4.28) II委rul12=IlΦdull2, (4.29) ll委(ア1}笹‖2=[1壷diutl2, (4◆30) 1ドφ21、包ll2=4‖{φdu‖2, (4.31) 1|蚕2CLul12『411¢(彦%ll2. Proof It follows丘om(3.1)∼(3.5),(3.9),(3.12),(3.13),(3.19),(3.20)肌d(4.2)that
(Φr%,¢r%)=(壷r題,rΦ%)一(亜r包吻) =一(ru, ¢r¢v)一(δ¢rOf, u) =一(r%,(re 一 d)Φu)一((委δ一のr%,%) =llr司12+(ru,d¢u)一(倭δr%,%)+(σr%,%) =211r電tll2十(%, CdQtり一(rδ%,壷電り ・==211rull2−(包,dO¢uL)一(δ%, C委r匹) ニ2(llrz,ll2−11δ%ll2) ニll郵旭ll2 a皿d (ΦCL%,壷CL%)=(倭(LO−d)%,亜(LC−4)%) ’ =(LΦc包一Φ砺,必委c%一郵%) =1ρ州12, w]hich yields that(4.28)a皿《1(4.29)atre t皿e. It is’e8sy to see that theτest hold by v丘琶皿e of(4.27) ∼ (4.29).K. r▲K▲NO ..二」∵