Finsler metrics with properties of the Kobayashi metric on convex domains

Publicacions Ma emá iques, Vol 36 (1992), 131-155 . Abs ac FINSLER METRICS WITH PROPERTIES OF THE KOBAYASHI METRIC ON CONVEX DOMAINS MYUNG-YULL PANG The s uc u e o complex Finsle mani olds is s udied when he Finsle me ic has he p ope y o he Kobayashi me ic on con- ex domains : ( eal) geodesics locally ex end o complex cu es (ex emal disks) . l is shown ha his p ope y o he Finsle me ic induces a complex olia ion o he co angen space closely ela ed o geodesics . Each geodesic o he me ic is hen shown o ha e a unique ex ension o a maximal o ally geodesic complex cu e E which has,p ope ies o ex emal disks . Unde he addi- ional condi ions ha he me ic is comple e and he holomo phic sec ional cu a u e is -4, E coincides wi h an ex ema¡ disk and a heo em o Fa an is eco e ed : he Finsle me ic coincides wi h he Kobayashi me ic . 1 . In oduc ion The Riemann mapping heo em says ha all simply connec ed do- mains in C, di e en om C a e biholomo phically equi alen . I is a well known ac ha his heo em does no hold o domains in T' o n > 1, and he classi ica ion o bounded domains up o biholomo phism has been an impo an p oblem in se e al complex a iables . One app oach o unde s anding he s uc u e o bounded domains is o s udy biholo- mo phically in a ian mé ics such as he Kobayashi o Ca a héodo y me ics [K]] [BD] [L3] [Pa] . In [L1] and [L2], Lempe showed ha hese me ics a e ex emely well-beha ed in he special case when he domain is s ic ly linea ly con ex and has smoo h bounda y : In his case, he wo me ics coincide, and he in ini esimal o m FK o he Kobayashi me ic alls in o a special class o smoo h Finsle me ics wi h cons an holomo phic sec ional cu a u e K = -4 . Since he no ion o a s ic ly linea ly con ex domain is no a biholomo phically in a ian concep , i 132  M .-Y . PANG is na u al o ask how a Lempe 's esul s can be ex ended o a mo e gene al (biholomo phically in a ian ) complex mani olds . One app oach o his p oblem is o s udy FK om a mo e in a ian poin o iew . The i s s ep is o cha ac e ize he p ope ies o an ab- s ac Finsle me ic F on an abs ac complex mani old M' which a e necessa y o Lempe 's esul s o hold . A second, and mo e di icul , s ep is o de e mine when he Kobayashi me ic o a bounded domain in (U" has hese p ope ies . In [F], Fa an analyzed he local s uc u e o (complex) Finsle mani olds and ob ained a se o local in a ian s by applying Ca an's me hod o equi alen e . He p o ed ha anishing o ce ain local in a ian s o ces F o coincide wi h he Kobayashi me ic o he unde lying mani old M p o ided ha F is a comple e me ic wi h K = -4 . Howe e , om he complex p ocess o cons uc ing hese lo- cal in a ian s i is no easy o see how hese in a ian s na u ally a ise om he p ope ies o Kobayashi me ics ob ained om Lempe 's wo k . Thus, one would like o o mula e a somewha mo e di ec desc ip ion o he local s uc u e ; ha is in ui i ely mo e appealing . In his pape , we gi e such desc ip ion om he poin o iew o he calculus o a ia ions by examining he local p ope ies o he Kobayashi me ic on s ic ly linea ly con ex domains, and de i e equi alen condi ions o he anish- ing o he Fa an's in a ian s om a simple p ope y o Kobayashi me ic (P ope y 1 .3) . In o de o desc ibe he local s uc u e o he Kobayashi me ic, we gi e b ie e iew o Lempe 's wo k . We de ine he in ini esimal Kobayashi me ic FK on a complex mani old M as ollows : Fo each E T x M, x E M, le be a holomo phic map om he uni disk 0 C C in o M such ha (0) = xand (0) = A o A > 0 . The magni ude FK ( ) o wi h espec o he in ini esimal Kobayashi me ic FK is de ined o be he in imum o  whe e he in imum is aken o e all such . I ac ually a ains he in imum (Le .  FK( ) = á ), hen is called ex emall . I can be easily seen ha he me ic FK is in a ian unde he ac ion o he g oup o biholomo phisms o M . Lempe showed ha , i M = D C C V` is a bounded s ic ly linea ly con ex domain wi h smoo h bounda y, hen FK is a smoo h complex Finsle me ic [L1], [L2], Le . FK is smoo h ou side he ze o sec ion o TD and sa is ies he ollowing condi ions : (1 .1) FK( ) > 0  'o  , : 0,  FK(z . ) = Iz1FK( )  o  z E C,  and (1 .2) FK( i + V2) .< FK( 1)+FK( 2)  o l, 2 E T .D,  x E D, whe e equali y in (1 .2) holds only when l and 2 a e colinea . Mo eo e , FINSLGR MGTRICS ANDTHC KOBAYASHI MGTRIC  133 he p o ed he ollowing heo em : Theo em (Lempe ) . Suppose ha D is a bounded s ic ly linea ly con ex domain wi h smoo h bounda y . (1) The e is a unique ex emal map co esponding o each E TD . (2) All he ex emal maps a e p ope isome ic imbeddings, and can be smoo hly ex ended o he closed uni disk 0 . (3) The ex emal disks (,~i) passing h ough a poin xE D o m a complex olia ion o D - {x} . (4) Ex emal disks a e ( he only) one-dimensional holomo phic e- ac s o D . One o he key ideas in desc ibing he geome y o D is he cons uc ion o he holomo phic e ac o D on o he ex emal disk (A) . Lempe p o ed ha he ield o holomo phic angen planes o áD on (¿9A) can be holomo phically ex ended o he in e io o he disk ( 0 ), and de ines a holomo phic ield o complex hype planes on (A) ha a e ans e sal o (A) . In o he wo ds, he e is a well de ined (n - 1)-dimensional holomo phic ec o bundle p : E , (A) o e he ex emal disk wi h ibe s de ined by he hype planes in T' . The union o he hype planes con ains he domain D, and he holomo phic e ac is de ined by he es ic ion o D o he p ojec ion mapp . The exis en e o such holomo phic e ac s has u he implica ions . Fo example, i o ces e e y (locally leng h minimizing, connec ed) geo- desic cu e o FK o be con ained in an ex emal disk . The p ope ies o he Kobayashi me ic ha in e es s us a e he ollowing : Co olla y . Le : A - D be a ex emal map o E T D . (1 .3)  The ex emal disk (A) coincides wi h he union o geodesic cu - es h ough x angen o a common complex line in T x M . (1 .4)  The e is a canonical spli ing TD, (o) = T( (0)) ® E whe e E is an (n - 1)-dimensional holomo phic subbundle o he es ic ion TD, (o) o he angen bundleTD o (A) . We wish o gene alize he condi ion (1 .3) o an abs ac complex Finsle me ic F de ined on an'n-dimensional complex mani old M . Le exp, deno e he exponen ial map om a neighbo hood o 0 E T, M ¡ i o M de ined by he geodesics o F . Fo each angen ec o E T~M, he image exp (U,) o a small neighbo hood U, o 0 in he complex line 0 de ines a su ace in M . A easonable gene aliza ion o he condi ion (1 .3) is he ollowing : (1 .5)  Fo all E TM, he su ace exp (U,) is a complex cu e (1- dimensional complex submani old) in M . 134  M .-Y . PAl c This condi ion was i s in oduced by Royden [R], and i is, in ac , equi alen o a condi ion gi en by Fa an [F] . The pu pose o his pape is o s udy he local s uc u e on o Finsle me ics sa is ying condi ion (1 .5), and u he , o show ha many p op- e ies o he Kobayashi me ic o con ex domains ex end o his mo e gene al class o complex mani olds . One o ou majo esul s is a con- s uc ion o a biholomo phically in a ian amily o complex cu es which enjoys many o he p ope ies o ex emal disks in con ex domains . To see how such complex cu es a e cons uc ed, ecall, om he calculus o a ia ions, ha he me ic F uniquely de e mines a ec o ield X on he co angen space o M, called he geodesic ec o eeld, such ha he in e- g al cu es o X a e mapped in o geodesics o F by he p ojec ion map 7 : T*M > M . Le Z be he ec o ield on Tó M = { E T*MI =~ 0} gene a ed by he ci cle ac ion o unimodula complex numbe s de ined by mul iplica ion on TO* M . Theo em A . The ollowing condi ions a e equi alen (Theo em 4 .7) : 1 .  The su ace exp (U  ) C M is a complex cu e o all E TM . 2 . [X, JX] = KZ o some smoo h unc ion , on Tó M . 3 . The dis ibu ion D = e X ® (E Z C T(TO M) is in olu i e . I any o he abo e condi ions is sa is ied hen each complex cu e exp (U  ) ex ends uniquely o a maximal, o ally geodesic, imme sed com- plex cu e E --~ M (Theo em 4 .9) . The sígni icance o he condi ion 2 and 3 in he heo em is as ollows : The condi ion 2 p o ides a compu a ional me hods o check whe he F sa is ies he p ope y (1 .5) . The condi ion 3 implies ha , by F obenius Theo em, he dis ibu ion D de ines a 2-dimensional complex olia ion .Fo o Tó M . The complex cu e E is cons uc ed by p ojec iog each lea o he olia ion Fo by he p ojec ion map 7 on o M . The cu es E sha e many local p ope ies in common wi h he ex emal disks desc ibed in he Lempe 's heo em . Fo example, by Theo em A, any eal geodesic cu e o F is con ained in one o he cu es E . Fu he mo e, a gene aliza ion o he p ope y (3) in Lempe 's heo em holds : he complex cu es E passing h ough a poin x o m a complex olia ion o some neighbo hood o x . A less i ial esul is he ollowing gene aliza ion o p ope y (1 .4) : Theo em B . Fo each complex cu e E, he e is a canonical spli ing TM¡£ = TE ®P-E, whe e T 1 E is an (n - 1)-dimensional holomo phic subbundle o TMI£ . (Theo em 4 .9 .) FINSLER METRICS AND THE KOBAYASli1 METRIC  135 The signi ican e o he unc ion , in he Theo em A is i s ela ion o he holomo phic sec ional cu a u e o F . No e ha each complex cu e in M is na u ally equipped wi h a He mi ian me ic induced by F, and, he e o e, has an associa ed Gaussian cu a u e . Following he de ini ion by Wong and Royden [W] [R], we de ine he holomo phic sec ional cu a u e o F a by he Gaussian cu a u e o he cu e exp (U  ) . Theo em C . I F is a complex Finsle me ic sa is ying he condi ion (1 .5) hen he holomo phic sec ional cu a u e o F is de e mined by he unc ion Finally, using he esul desc ibed abo e, wc show ha , unde he condi ion ha F is comple e a,nd , = -4 he complex cu es E coincide wi h he ex emal disks . This esul was p o ed ea lie by Fa an [F] . No e ha , om Lempe 's esul , he Kobayashi me ic FK on a s ic ly linea ly con ex domain D C en has cons an holomo phic sec ional cu - a u e -4 . (This is a di ec consequence o he ac ha e e y ex emal map : A -> D is an isome y wi h espec o he Poinca é me ic and FK such ha ( 0 ) is locally de ined by exp (U  ) .) 4 .24 Theo em [F] . Suppose F is a comple e complex Finsle me ic on a complex mani old, M wi h cons an , holomo phic sec ional cu a- u e -4 sa is ying h,e p ope y (1 .5) . Then F = FK, )hc e FK is he Kobayashi me ic on M . The pape is o ganized as ollows : In Sec ion 2, we de elop basic ools and p o e some basic ac s abou complex Finsle mani olds . Sec ion 3 is an in oduc ion o Legend e olia ions and i s applica ion o complex Finsle mani olds . In Sec ion 4, we p o e he main heo em using he esul s o Sec ions 2and 3 . Th oughou he pape , M deno es an n dimensional complex mani old and F a complex Finsle me ic ; on M . The 'ollowing no a ions a e used : (1) The indices a, band c ango om 1 h ough 2n, and o ., (5, y ango om 1 h ough 2n - 1 .  Summa ion con en ions a e in o e h oughou . (2) (X I , . . ., xn, x n+1 . . ., x2n) deno e he eal coo dina es on M ob- ained om a holomo phic coo dina es x, + ixn+ , o = 1, . . ., n . (3) (xl  X n x n+1  x 2n u l  un u n+l  u2n) deno e he coo - dina es on T*M induced by (xl, . . ., xn, x n+1 ,. . ., x 2n) . (4) Fo F E C°°(T*M), Fa, FaL, . . . deno e ' g " - ,  9z "  and so on . du° (5) I V is a ec o ield on a mani old M, e' V : M- M deno es he 1-pa a le e amily o di eomo phisms gene a ed by V . Thus, o 136  M .-Y . PANG Acknowledgmen s . I would like o hank T . Duchamp o his help and encou agemen , and o in oducing me o his subjec . I also wish o exp ess my hanks o J . Bland o con e sa ions . In his sec ion, we p o e some gene al ac s abou complex Finsle me ics . A complex Finsle me ic on he co angen bundle o M is a map F : T*M - R sa is ying p ope ies (1 .1) and (1 .2) . When M is equipped wi h acomplex Finsle me ic, we will call M a complex Finsle mani old . 2 .1 The Geodesic Vec o Field and Complex S uc u e . In o de o de ine he exponen ial map, we in oduce he geodesic ec o ield on T*M . Recall ha T*M is na u ally equipped wi h a 1- o m de ined by he equa ion (2 .2) and ha he 2- o m d( is a symplec ic 2- o m on T*M (Le . a smoo h closed 2- o m on T*M sa is ying he non-degene acy condi ion (do)2,, =~ 0) . The geodesic ec o ield X on T* M is uniquely de e mined by he condi ion (2 .3)  XJdS = -FdF . In pa icula , lle iden i y XF = 0 holds . In e ms o coo dina es, we Na e (2 .4) cach xE M, ~--~ e x is an in eg al cu e o V s a ing a x (Le . e ° x = x and  d I e x = Ve  .) . 2 . Complex Finsle Me ics X=F Fa _ó _ _ áF _ 8 ~ . áxa  Óx a áua a=1 2n _ E ua dxa, a-1 To desc ibe how X is ela ed o he complex S uc u e, conside co- o dina e exp ession o he complex S uc u e on T*M . The complex S uc u e J on M is exp essed as -i) (2 .5)  J 5 = ,In' ~  whe e  (Jb) _ (1  0  , (2 .6) FINSLER METRICS AND THI ; KOBAYASIII METRIC  137 and I is he (n x n)-iden i y ma ix . The na u al complex s uc u e on T*M is hen gi en by a  b a  _a __  ba I gX a I can be di ec ly checked ha he complex s uc u e de ined by his is independen o he choice o coo dina es . By abuse o no a ion, we will deno e his complex s uc u e on T*M by J . F om he dc ini ion abo e, i is clea ha J o 7 * = 7 * oJ . No e ha condi ion (1 .1) p o ides a compa ibili y condi ion o F and he complex s uc u e, which can be exp essed as ollows : Le Y be he adial ec o ield on Tó M gene a ed by he , ac ion o IR 1 :>y nul iplica ion o é, E I1 Z . I can be easily checked ha Y and Z sa is y he ela ion Thus, i F sa is ies he condi ion (1 .1), we ha e F(e` ) = F( ) a,nd F(e' ) = e'F( ), and he e o e he ollowing iden i ies hold : (2 .8)  ZF = 0,  YF = F . No e ha he coo dina e cxp essions o Y and Z a e (2 .9)  Y = u°  09 ,  Z= -Ja U'' a'9 a . The e o e, condi ions (2 .8) a e equi alen o (2 .10)  .1' 71" F a = 0,  F,, u' =F . 2 .11 Lemma . The ollowing iden i ies a e sa is i ied : (2 .12)  [Z, X] = -JX,  [Z, JX] = X, [Y, X] = X,  [Y, JX] = JX,  [Y, Z] = 0 . P oo : The compu a ions in he p oo o his lemma a e based on he iden i ies (2 .8)-(2 .10) and he ollowing basic iden i ies de i ed om hem : _a _ _a __  _a ____a  a  b a I Y~ ax a ] - LZ~ Oxa ]  ~~  LY~ au a ]  a7l .a  [ Z ' a_ua  _ - ,%a a7/,b 138  M .-Y . PANG Now compu e [Z, X] using coo di la es : [Z, X] _ {  1 :~ GiX _G Z F  Fa  a  _ aF  a ax a  axa aua a=1 = F E { (ZFa) a axa _ ax aF a~Z á .a) }- Obse e ha (ZFa)  Z (au )  a BuF ) + [Z' aáa, F- (Ja a , b ) F  Ja,Fb . The e o e, using he skew-symme y o I ba, we ob ain =(YF) ) J (J 6 Fb ax a + 'Ia axa W , ) a=1 - 7 { F 2n (Fa _a  - (9F  o9 )} = DF  (ax a aU a a=1 =-JX . To show he iden i y [Z, .IX] = X, do e haa .Cz,I = 0, and compu e [Z, JX] = GZ(JX) = J(GZX) = J(-JX) = X . To p o e he iden i y [Y, X] = X, no e lla YF a , = Y ( O F , ) = aua (YF) + Y, aa y  F  F a - Fa = 0 . Using his iden i y, compu e [Y, X] as ollows : [Y, X] =GyX = Gy  F  (Fa a  - aF  a Y!  axa  axa au a a=1 F a 0  OF 0 ) axa  axa au a a=1 _ F' Gy  (F .  a  _ aF  a ) } --- axa  ax a (gU a a=1 _a __OF_a (Íxa  ax a (gua a 2n  ()  (  ) } _aF _a _aF _ l Y ax a a~ a + axaau- ~y a=l ~(aa F ) aáa  ááaa} =X . FINSLGR MCTRICS ANDTHG KOBAYASI-II ML'TRIC  139 The iden i y [Y, Z] = 0 is clea since he ac ions o e and e - " commu e, and he iden i y [Y, JX] = JX ollows om he compu a ion : [Y, JX] = [Y, -Gzx] = -Gz[Y, X] + [GZY, X] = -GzX = JX . 2 .13 Lemma . The ec o ields JX and [X, JX] sa is y he iden i- ies : (2 .14) ((JX) = dF(JX) = 0,  and  (([X, JX]) = dF([X, JX]) = 0 . In pa icula , JX and [X, JX] a e angen o he submani old SFM C To M . P oo :: To p o e he lemma, ecall ha we ha e he iden i ies Cx( = X -id( =-F dF, X F= 0 and C(Z) = 0 . Also, ecall om iden i ies (2 .8) and (2 .12) ha Z F = 0 and JX = [X, Z] . Using hese iden i ies and he ac ha Gx is a de i a ion, compu e as ollows : ((JX) = (([X, Z]) = «£x Z) = £x«(Z» - (£x C) (Z) = FdF(Z) = 0 dF(JX) = dF([X, Z]) = XZ F - ZX F = 0 Using hese iden i ies again, we comple e he p oo o he lemma : (([X, JX])=( (£x(JX))=Gx{((JX)} - (Gxo) (JX)=FdF( .IX)=0 dF([X, JX]) = X(JX) F - ( .IX)X F= 0 . 2 .15 The Exponen ial Map . The exponen ial map is de ined simi- la ly as in he case o He mi ian mani olds . To de ine i , we in oduce he dual complex Finsle me ic F : TM , R, sa is ying condi ions (1 .1) and (1 .2), and de ine geodesics as cu es wi h locally leng h minimizing p ope y wi h espec o F . We b ie ly e iew some concep s o he cal- culus o a ia ions . Fo mo e de ails abou he calculus o a ia ions see [GF] and [S] . To de ine F, le ToM = { E TM1 :~ 0}, and de ine : a bundle map xP :TóM~ToMby (2 .16)  T (w) = 7  (X  ,)  o  w ETó M . In coo dina es, we ha e 7L (2 .17)  xPx(w) _  F (w) F,, (w) 8 á .,, a= I . .1 14 6  M .-Y . PANG Bu om Le nma 3 .8, [X, JX] does no ha e a componen in he di ec- ion o X and Xl, and hus [X, JX]  , E L  , . Again, om Le nma 3 .8, i ollows ha S i (X) = 0, and we ob ain (4 .3)  JX = XII  and  [X, JX] = Si Z <, . F om he s uc u e equa ion (3 .6), i can be easily shown ha he iden i y Ex Z a = X~ + i ' (X) Zp holds . This iden i y and he second iden i y o (4 .3) gi es Ex [X, .JX] = Ex (Si Z a = (XSi ) Z x + Si (,ex Za) _ {(X S-) Z,,+SpZ ~(X)} Z < ,, +Sl X ' . Bu , Again, since we p o ed (Ex [X, JX]) , E RX  , (D 1FI(Xl)  , ® L,D, componen s o he ec o Ex [X, JX] in he di ec ion o X a o a =~ 0 has o anish . The e o e, i ollows ha Si = 0 o a > 1 and [X, JX] _ Si Z, and he condi ion A2 ollows . To p o e he con e se, suppose ha [X, JX] = Z holds o some , E C°°(S M) . We p o e ha he su ace de ined by C ,( , s) = 7 (e x e - S z iu) is a complex cu e o all Zu E T*M by showing ha J `~  -(-T, 0) is angen o he cu e C  , o small T < 0 .  No e ha Gx £xZ = Ex [X, Z] = Ex (JX) = Z, and le 771 = e-Tx u) . Using he de ini ion o he Lic de i a i e, we compu e 2 2 (e* xZ(e X ,))  x{ .CXZ}~~ Xw)) = .e* x {GXGXZ}(e xm) = K ( e x 7u ) {e* x z ( ,IX  ,) } . The e 'o e, i we lc W( ) = e* xz( e x,b) E T z (S*M), W( ) sa is ies a second o de o dina y di ie en ial equa ion W"( ) = ( )W( ) wi h ini ial condi ions W(0) = Z,z and W'(0) = JX,T, . Consequen ly, we ha e W( ) E span{Z,b, JXú,} o small , and in pa icula , W(' ) _ e*Tx z ( , x,D) E sean{Z,7 JX,J . Subs i u lng 7 = e-Tx7 , (4 .4)  W (T) = e * ' x Z  , E span{Zw, JX  ,} . Using he iden i y 1 (T, 0) = 7 * X  , and 7 * o J =J o 7 * , we ob ain a a w (T, 0) = ás  { 7 (e 'x esz7 V) } s=o = 7 * e' x Z  , = k 7 * (JX),í, = k J 7 * (X,D) = kJa a w (T, 0) FINSLER METRICS AND TIiE KOBAYAS1i1 METI IC  147 o some k E IR, and hence J ~áe (T, 0) is angen o he su ace de ined by C  , . (ü) I [X, JX] = Z holds o some c E C' (S* M), hen he in olu- i i y o D ollows om he iden i ies (4 .5)  [Z, X] = -JX,  [Z, JX] = X  and  [X, JX] = KZ . The con e se o his is an immedia e conséquence o he he iden i y (3 .10) . 4 .6 Rema k . (1) No e ha , in he p oo o Theo em 4 .2, he wo equa ions S i (X) _ 0 and SQ = 0 o /3 > 1 a e equi alen o he single condi ion [X, JX] = Z wi h c = Si .  The condi ion S i (X) = 0 can be in e p e ed as a compa ibili y condi ion o F and he complex s uc u e . Fo example, i F is he induced no m o a Kaehle mani old, his condi ion is sa is ied . In his case, he condi ion Sp = 0 o ,(j > 1 pu s es ic ions on he cu a u e o he Kaehle me ic . One special case o his is when M is a Kaehle mani old wi h cons an holomo phic sec ional cu a u e . In his case, i can be e i ied using esul s in [P] ha Sá = cb í 3 o some cons an e . (2) Condi ions equi alen o Al-A3 o Theo em 4 .2 we e in oduced by Royden [R] and Fa an [F] . The condi ions Al-A3 in Theo em 4 .2 can be equi alen ly s a ed as condi ions on Tó M . 4 .7 Theo em . The ollowing condi ions a e equi alen : B1 .  exp(U  ) is a complex cu e o all E TOM . 132 . [X, JX] = Z onTó M o some e E C°°(TO M) . 133 . The dis ibu ion CX ® 0Z C T(TO M) is in olu i e . Mo eo e , hese condi ions a e equi alen o condi ions Al-A3 in Theo- em 4 .2 . P oo : No e ha he condi ions Al and Bl a e clea ly equi alen . To p o e he heo em, we show (i) ha A2 implicas B2, and (ii) ha A3 implies B3 . The con e ses o hese a e i ial o p o e . (i) Suppose ha [X, JX] = c Z holds on SFM o some c E C°° (S M) . Ex end K o Tó M by c( w) = z K(W) o all > 0 and w E SI*M . We claim ha [X, JX] = c Z on Tó M . To show his, we show ha bo h W = [X, JX] and W = n Z on Tá M mus sa is y he o dina y di e en ial equa ion GyW =2W . No e ha he in eg al cu es o Y 14 8  M .-Y . PANG a e he adial li ios in Tó M . Thus, i bo h [X, .IX] and c Z sa is y he equa ion, hey nus coincide because he iden i y [X, JX] = Z gi es he same ini ial condi ion a poin s on SFM . To show ha he ec o ield [X, JX] sa is ies he di e en ial equa ion, ecall, o Lemma 2 .11, ha G y X =X and GyJX=JX . Using hese iden i ies and he ac ha Gy is a, de i a io , we compu e Gy [X, JX]=[GyX, IX] +[X, GyJX]=[X, JX]+[X, JX]=2[X, JX] . To show ha he ec o ield , Z sa is ies he di e en ial equa ion, ecall ha he ec o ield Y on Tó M is gene a ed by he ac ion o II3, by mul iplica ion o e' . Using homogenei y o K, we ob ain he ollowing iden i y : Fo w E TO M, d ~~ d , =o (Le . GyZ = [X, Y] = 0), we compu e d  {e2 (w) } = 2~ ;( ~) .=o Using his iden i y and he ac ha he ec o ields Y and Z commu e { .Cy ( Z)} = (Y ,) Z= 2(K Z) . (ii) The ; p oo ha he condi ion A3 i nplics he condi ion B3 im ne- dia c ;ly ollows o hc ; iden i ies in Lc n na 3 .8 . 4 .8 To ally Geodesic Complex Cu es . We call a complex cu e E o allly geodesic i , o any angen ec o o he complex cu e E and geodesic segn cn y : (-e, e) - M such ha ^c (0) = , y, ( ) is co ai ed in he complex cu e o small . The main esul o his see io is l a , unde condi ion (1 .5), he geodesics o F can be uniquely ex ended o i nme sed complex cu es ha a e o ally geodesic subman- i olds o M . In ac , l ese cu es a e p ecisely hc ones de i ed by he complex cu es exp (U  )'s in condi ion (1 .5) . Theo em 4 .9 . I he condi ion (1 .5) holds, he complex cu e exp (U  ) can be uniquely ex ended o a maximal o ally geodesic complex cu e : E - M imme sed in M . Mo eo e , he e is a canonical (n-1)- dimensional holomo phic ec o subbundle T'E o *(TM) ans e sal o E . P oo :: Recall om Theo em 4 .2 ha D= span{X, JX, Z} is an in- olu i e dis ibu ion . By he F obenius heo e , his i plies ha S* M is olia ed by 3-di ne sional maximal in eg al sub nani olds o D . Le (4 .10) FINSLGRMETRICS AND THC KOBAYASII1 MGTRIC  149 E be a lea o his olia ion .-D, hen he e is a well de ined S I C e ac ion on E since Z is angen o E . Le us deno e he quo ien space E/S I by E . I is no di ñcul o see ha E is a complex cu e wi h he complex s uc u e induced om he complex s uc u e o E C T* M, and ha he e is a holomo phic imme sion such ha he ollowing diag am commu es : inclusion E~S m E J M Recall ha he complex cu e exp (U  ) is he su ace de ined by (4 .11)  ( ,, s) - exp ( esw) = n (e Xe- .yi4,( )) . F om his, i is clea ha he complex cu e : E - M is locally de ined by exp (U ) since e' x c -9z i E E . F om his, i clea ly ollows ha : E - M is o ally geodesic . To de ine he ans e sal holomo phic subbundle T - L E o * (TM), no e ha 7 z E is a p incipal ci cle bundle . De ine he ibe T,LE o T l Ea xEEbyT~E={ E *(TM)Jw( * )=0 o al¡ wEE - ,,} . I is clea ha T l E is a holomo phic ec o bundle wi h dimension n-1 . To show ha T~E is ans e sal o E, suppose E TLE n TE . No e ha xP(w) is angen o E o all w E E y since : xP( ) = 7 * X  , and X is angen o E . Mo eo e , 'Y(ui) :,¿ 0 because w (xP(w)) = w(7 * X  ,) = FF a ua =F 2 = 1 z~ 0, whe e uJ =Ea -l , a dx a . Since , * Y' (w) E T, ; E, we ha e * = zkP(w) o some z E (E . Bu , ecall ha E T L E, and he e o e, )( ) = 0 . This implies ha = 0 because 71) (V) =w (zXP(w)) = z {w (xP(w))} = z . 4 .12 The Holomo phic Sec ional Cu a u e . I F sa is ies he p ope y (1 .5), he e is a na u al way o de ine holomo phic sec ional cu a u e K o F . In his sec io'n, we show ha K is de e mi ied by he smoo h unc ion K in he condi ion A2 o Theo em 4 .2 . Holomo phic sec ional cu a u e K o a complex Fi isle ne ic F has been s udied by Wong and Royden [W] [R] . To de ine K( ) o a uni ec o E T, M (Le . F( ) = 1), no e ha each complex cu e U C M angen o has a canonical complex Finsle ne ic, de ined by he es ic ion F,1 , u : TU -> IR . In ac , because U is o complex 15 0  M .-Y . PANG dimension one, i can be easily seen ha he me ic F1 h u is a no m induced by a He mi ian me ic g on U : I F( ) = 1, F ((a + i/l) ) = l a + i,31 F( ) =  a2 -+Q2 . In [W], Wong de ined he holomo phic sec ional cu a u a K( ) as he sup emum o he Gaussian cu a u a o g a x E U, whe e sup emum is aken o e all complex cu es angen o . In he special case when F is he no m induced by a He mi ian me ic, his de ines he usual holomo phic sec ional cu a u a o He mi ian me ic . Obse e ha , i F sa is ies he p ope y (1 .5), hen by Theo em 4 .2, he e is a na u ally de ined o ally geodesic complex cu e o he o m exp (U  ) angen o . In [R], Royden showed ha he Gaussian cu a u a o he induced me ic g a x on his complex cu e a ains he g ea es alue and, hence, i de ines he holomo phic sec ional cu a ú e K( ) . Thus, he ollowing heo em holds : 4 .13 Theo em . Le , N be a complex submani old o M, and le K' be he holomo phic sec ional cu a u e o he induced me ic FITN on N . Then he inequali y K'( ) < K ( ) holds e e y uni ec o angen o N . In pa icula , i U C M is a complex cu e angen o a uni ec o E TM, he Gaussian cu a u a o he induced, me ic g de ined by FITU is bounded om abo e by K( ) . The unc ion K can be ega ded as a unc ion on he quo ien space S* MIS' . Recall om (2 .12) and(4 .5) he iden i ies [Z, X] = -JX, [Z, JX] = X and KZ= [X, ,IX], and compu e (ZK)Z =G Z(KZ) = £z [x, JX] - [L ZX, Jx] + [x, L Z (Jx)] = [-Jx, Jx] + [x, x] = 0 . F om his ide i i y, we conclude ZK= 0 . This implies ha he unc ion K is in a ian unde he ci cle ac ion o unimodula complex numbe s de ined by mul iplica ion on TO M, and hence he unc ion K can be ega ded as a unc ion on S*M/S 1 . 4 .14 Theo em . I K is he holomo phic sec ional cu a u a o F, o e e y uni ec o E TM . K ( ) = K o -D( ) Be o e we hegin he p oo , no e ha by Theo em 4 .9 he complex cu e U C M has a unique ex ension o a maximal o ally geodesic complex cu e : E -+ M, and ha he e is a ci cle bundle : É , E o a E . FINSLP ;I1 METRICS ANll TIIG KO 3AYAS1ll METRIC  151 4 .15 Lemma .  The be s o he ci cle bundle 7 z : E - E de ines a Legend e olia ion . :L wi h espec o a na u al con ac , 1- o m de ned by he pull-back o 77 o E . The s uc u e equa ions o his Legend e olia ion a e he pull baks o E o he equa ions : (4 .16)  dB' =- 7A~',  d =0 1 n ;',  d~' =K17A0 1 P oo . : The s uc u e equa ions (4 .16) a e ob ained by pulling back he equa ion (3 .4) o É . No e ha , since JX= XI and Z = ZI, we lla e Ba (JX) = ~' (Z) = 0 o cx > 1 . The e o e, on E, we lla e dB' = -77 A ;' + G'110 1 A ~' d i=0 1 A~' d l =SigAB 1 +Q 1 10 1 A~ l . Bu , om hc ; iden i ies (4 .5), i easily ollows ha G' 1 = Q 11 = 0 a d Si = , . Hence, we ob ain he s uc u e equa ions (4 .16) : F om he equa ions, i is clea ha he pulí back o É o l is a con ac o m since ) A d = A B' A  '  0on E . P oo .. To p o e i e heo e n, choose E T,,M and le U C M be a o ally geodesic complex cu e such ha x E U a d E T,;U . Also, le : E , M be he unique ex ension o U desc ibed in Theo em 4 .9, and le Éu deno e he es ic ion o he ci cle bundle 7 : É -~ E o U . Obse e ha U is also a submani old o S M/S' since U C E= E/S' C S M/S' . We claim ha he Gaussian cu a u a o g o U is Klu E C_ (U), wl c e K is ega ded as a unc ion o S¡,M/S' . The heo e n ollows om he claim . To sea his, p o eed as ollows : No e ha by commu a i i y o he diag am (4 .10), 7 = o 7 z . Hence, i w E Eu, hen xP(u ) = 7 ,X w = * o (7 ),X  , . The e o e, he ap S M , SpM sends Eu in o he zeni a ge bundle S .U o g . Since T is a bundle map o a U such ha T(es'u» = e`T(w), kP maps u di eomo phically on o S .U, o equi alen ly, we lla e a bundle ap 1 Is,u = <PIs,u : SU - Eu o a U . Thus, since , is co s an along ibe s o Eu a d E T,,U, we lla e , o <D( ) = K(x) . B,y hc ; clai , K(x) is he Gaussian cu a u a o g a x E U, and he iden i y K( ) = Koq>( ) ollows . To p o e he clai n, we deno e i e Gaussian cu a u a o g on U by ic E C'(U), and show K = k . Recall ha , ' o P oposi io 3 .5, hc ; Legend e olia ion on Sg*U i as he s uc u e equa ions (4 .17)  dé' =-~A : ',  di7=8'A~',  d~'=k~AB', 15 2  M .-Y . PANC whe c : {H1, ~, `1} is he in a ian co ame on S* U . The ; p oo o he claim is done by es S ablishing he equi alen e o he Legend e olia ion on S . .U wi h ,Fz . This is p o ed by showing ha he e is a di eomo phism 0 : Eu ~ S9 U such ha (4 .18)  0*~= 7,  and  7 o0=7 . I ollows ha O*Hl = 0 1 and ?P*~ 1 = ~1, and in pa icula , he pull-back by 0 o he equa ions (4 .17) a e he s uc u e equa ions (4 .16) o .Fz . The iden i y k o 0= , ollows . We de ine O(w) o w E Eu C T*M as he by pull-back o w o he angen space o U . In coo dina es, we ha e (4 .19) l  n+1  1  n n+1, . . ., 2 a ( :L ) 0, . .,O,X'  ,0, . .,0¡U , . . .,26  ,4L  2L  ~(x l xn+1 u1 ,9xn+1) whe e (x 1 , . . ., .xn, xn+1 ,. . ., x2,) is aken so ha U C M is locally de ined by . ,°' = 0 o a =~ 1, n + 1 . To comple e he p oo ; i emains o show ha (1)  naps Eu di eo no pllically o o S .*U, and (2) "  ~ = 71 (1) : To show ha 0 is a di eo no phism on o Sg*U, ecall ha T maps Eu on o SU . Hence, i w E Eu, hen he ec o s xP(w) and T(Jw) _ -JT(w) 'o m an o hono mal ame o T, ;U o some x E U . The ol- lowing compu a ion shows ha he co ec o s 0(7u) and 0(Jul) = ,Iz/1(w) o o he dual co ame o {xP(w),T(Jul)} : Using he ide l i ies (2 .10), compu e ; {1h(1u)} (IP(w)) = w(7 * X w ) = FF <l U, a = F 2 = 1 {O(Jw)} (  0%711)) = {Jo(w)} (-JT(w)) = {Y'( 1 V)} (XP(lu)) = 1 {O(IV)} (IpG%w)) = {O(lu)} (-JP(ul)) = {-J~J(7U)} (`P(1 )) = {-Jw}(7 *X ,) = uaJ," F F, = 0 . Hence z/>(Elx) C S .*U . Since V) is a bundle map p ese ing he ci cle ac ion, i casily ollows ha 0 is a di eo lo pllis n . (2) : To p o e lle iden i y z)*~ = TI, ecall ' onl (2 .2) ha l = ul dx 1 + un - " dx n+1 . Tlle e ó e, onl (4 .19), ' %1 * = ul dx 1-}- u "+1 . dxn+l . On he o he hand, he con ac 1- o o 77 on E is de ined by he pull-back o 71 _ 1 :2n .=  o E . Bu , since dx° = 0 o a =,~ 1, n+ 1 on E, we ha e 77 = 7x 1 d x 1 + u n+1dxn+1, Hence he iden i y 0*~ = 17 ollows . a The ollowing co olla y is a consequence o he p oo o Theo em 4 .14 : FINSI,I ;R MI . :TI ICS AND 'CII1 ; KOBAYASIII MI TILIC  153 4 .20 Co olla y .  The Gaussian cu a u a o he induced me ic .q on E ás ,,É . 4 .21 Rela ion o he Kobayashi Me ic . In his sec ion, we p o e a e sion o a heo em o Fa an which s a es ha anishing o ce ain local in a ian s o es F o be he Kobayashi me ic o M, p o ided ha F is comple e and sa is ies he condi ion K = -4 (sea In oduc ion) . In he e sion o he heo em p esen ed he e, he condi ion o anishing o in a ian s is eplaced by he equi alen condi ion (1 .5) . Recall o i Lempe 's esul desc ibed in he in oduc ion ha , i D C T' is a bounded s ic ly linea ly con ex domain wi h smoo h bounda y, hen c e y ex emal disk : A -> D is an iso ne ic imbedding (Le . *FK coincides wi h he Poinca é no m en ,), and ha (A) is a maximal o ally geodesia co nplex cu e ; in D . Since hc ; Poinca é nie ie has Gaussian cu a u a -4, he holo no phic ; sec ional cu a u c o hc ; Kobayashi nc ic FK is -4 . O i he o he hand, i F is a ny co nplex Finsle me ic : on a co nplex mani old M, he condi ion o cons an holo no phic ; sec ional cu a u a K =-4 imposes a es ic ion on he me ic F . In aca, we show ha , i F is any comple e co nplex Finsle me ic ; wi h he p ope ies K = -4 and (1 .5), hen F nus coincide wi h he Kobayashi me ic . To show his, we need hc ; ollowing le ima due o Ahl o s [A] [K] : 4 .22 Gene alized Schwa z Lemma . Le (N, g) be a, 1-dimensional . He mi ian mani old such ha he Gaussian cu, ' a u 'e is bounded, abo e by a nega i e cons an -C . Fo ' any holo no phic mal) : A , N, he inequali y (4 .23) Il . * ll .~ <_ c Il ll holds o allll E TA, )hc e II Ils is he no m on N induced by y and II II deno es he no ' n de ined by he Poinca é ne ic on A . We call a co nplex Finsle me ic : F comple e i he geodesic : ec o ield X is comple e ; (o equi alen ly, i e e y geodesic can be ; ex ended o a geodesic de ine(¡ en all o IR) . 4 .24 Theo em [F] . Suppose F is a comple e complex Finsle me ic on a complex mani old M wi h cons an e holomo phic sec ionall cu a u a K = -4 sa is yinq he p ope y (1 .5) . The dual me ic F coincides wi h he Kobayashi me ic FI< o M . P oo .- We e ; i y he equali y FK =F by e i ying hc ; inequali ies FK<FandF<FK . 15 4  M .-Y . PANG (i) To show ha he inequali y FI< < F holds, le E TM and ecall o n Tlico en 4 .9 ha he e is an i n ne sed o ally geodesic co iplex cu e : E , M angen o . Sinee F is a comple e me ic wi h K = -4 and E is o ally geodesic, he induced me ic g on E de ined by *F is a comple e Kaehle me ic wi h Gaussian cu a u e -4 (see Theo em 4 .14 and Co olla y 4 .20) . The e o e, he e is a holo no phic co e ing nap 2 : A - E which ¡s 'a local isome y be ween he Poinca é me ic and 9 (see chap e IX o [KN]) . By composing and 2, we ob ain a holo no phic nap o 2 : A - M ha is an isome ic imme sion wi h espec o he Poinca é me ic and F (Le . {( o 2) * F}(w) = 11wil) . Recall ha he Kobayasl i me ic FK( ) is de ined as he in i num o Ji . * il o e all complex cu e : A - M angen o . Hence, he inequali y ollows : FK( ) < 11 ( l o 2)* jj = F( )  o  E TM . (ü) To p o e he inequali y F( ) < FK( ), no e ha , o each complex cu e : : A -> M angen o E TM, he e is a He mi ian me ic me ic g on 0 de ined by *F . By Theo em 4 .13, l e Gaussian cu a u e n is bounded abo e by -4 . Applying he Genc alized Schwa z Lem na 4 .22 o hc iden i y mal) id : -> (0, g ), we ob ain he inequali y { *F}( » <_ 11wII, o equi alen ly, F( * w) <_ jjwjj o all uj E TA . In pa icula , his implies ha F( ) < li * il o any complex cu e angen o . Since FK ( ) is he in i num o 11 * 11 o e all such , he inequali y F( ) < FK( ) ollows . Re e en es [A] L .V . AnLFORS, An ex ension o Schwa z's lemma, T ans . o Ame . Ma h . Soc . 43 (1938), 359-364 . [BD] J . BLAND, T . DUCHAMP, Moduli o Poin ed Con ex Do nains  In en . Ma h ., ( o appea ) . [F] J .J . FARAN, He mi ian Finsle Me ics and he Kobayashi Me ic, Jou nal o Di e en ial Geome y 31, no . 3 (1990), 601-625 . [GF] I .M . GEAYAND, S .V . FOMIN, "Calculas o Va ia ions," P en- ice-Hall, Inc ., Englewood Cli s, 1963 . [K] S . KOl3AYAS11l, "Hype bolic Mani olds and Holo no phic Mappings," Ma cel Dekke , Inc ., New Yo k, 1970 . [KN] S . KOBAYASM, K . NOMizu, "Founda ions o Di e en ial Geom- e y," Vol . 1 a d 2, John Wiley & Sons, Inc ., New Yo k, 1963 . FINSL R METRICS AND TI-I KOBAYASIII MI TRIC  155 [L1] L . L mP R , , La me ique de Kobayashi e la ep esen a ion des domains su la boule, Bull . Soc . Ma h . F ance 109 (1981), 427-474 . [L2] L . LEMPERT, In insic Dis an es and Holomo phic Re ac s, Complex Analysis and Applica ions 81 (1984), 341-364 . [L3] L . LEMPERT, Holomo phic in a ian s, no mal o ms, and he moduli space o con ex domains, Annals o Ma h . 128 (1988), 43-78 . [P] M . PANG, The S uc u e o Legend e Folia ions, T ans . o Ame . Ma h . Soc . 320, no . 2 (1990), 417-455 . [Pa] G . PATR1zIO, Disques ex emaux de Kobayashi e equa ion de Monge-Ampe e complexe, C . R . A ad . Sci . Pa i s 305, Se ie 1 (1987),721-724 . [R] H .L . ROYDGN, Complex Finsle Me ics, Con empo a y Ma he- ma ics, P oceedings o Summe Resea ch Con e ence, Aug . 12-18, Ame . Ma h . Soc ., 1984, pp . 119-124 . [S] S . STERNBERG, "Lec u es on Di e en ial Geome y," P en ice Hall, Englwood Cli s, 1964 . [W] B . WONG, On he Holomo phic Cu a u e o Some In insic ; Me - ics, P oc . Ame . Ma h . Soc . 65, no . 1 (1977), 57-61 . Depa men o Ma hema ies Box 1146 Washing on Uni e si y S . Louis, MO 63130-4899 U .S .A . Rebu el 12 de Ma e, de 1991