Finsler metrics with properties of the Kobayashi metric on convex domains
Abstract
Pang, Myung-yull
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Publicacions Matemátiques, Vol 36 (1992), 131-155 . Abstract FINSLER METRICS WITH PROPERTIES OF THE KOBAYASHI METRIC ON CONVEX DOMAINS MYUNG-YULL PANG The structure of complex Finsler manifolds is studied when the Finsler metric has the property of the Kobayashi metric on convex domains : (real) geodesics locally extend to complex curves (extremal disks) . lt is shown that this property of the Finsler metric induces a complex foliation of the cotangent space closely related to geodesics . Each geodesic of the metric is then shown to have a unique extension to a maximal totally geodesic complex curve E which has,properties of extremal disks . Under the additional conditions that the metric is complete and the holomorphic sectional curvature is -4, E coincides with an extrema¡ disk and a theorem of Faran is recovered : the Finsler metric coincides with the Kobayashi metric . 1 . Introduction The Riemann mapping theorem says that all simply connected domains in C, different from C are biholomorphically equivalent . It is a well known fact that this theorem does not hold for domains in T' for n > 1, and the classification of bounded domains up to biholomorphism has been an important problem in several complex variables . One approach to understanding the structure of bounded domains is to study biholomorphically invariant métrics such as the Kobayashi or Carathéodory metrics [K]] [BD] [L3] [Pa] . In [L1] and [L2], Lempert showed that these metrics are extremely well-behaved in the special case when the domain is strictly linearly convex and has smooth boundary : In this case, the two metrics coincide, and the infinitesimal form FK of the Kobayashi metric falls into a special class of smooth Finsler metrics with constant holomorphic sectional curvature K = -4 . Since the notion of a strictly linearly convex domain is not a biholomorphically invariant concept, it
132 M .-Y . PANG is natural to ask how far Lempert's results can be extended to a more general (biholomorphically invariant) complex manifolds . One approach to this problem is to study FK from a more invariant point of view . The first step is to characterize the properties of an abstract Finsler metric F on an abstract complex manifold M' which are necessary for Lempert's results to hold . A second, and more difficult, step is to determine when the Kobayashi metric of a bounded domain in (U" has these properties . In [F], Faran analyzed the local structure of (complex) Finsler manifolds and obtained a set of local invariants by applying Cartan's method of equivalente . He proved that vanishing of certain local invariants forces F to coincide with the Kobayashi metric of the underlying manifold M provided that F is a complete metric with K = -4 . However, from the complex process of constructing these local invariants it is not easy to see how these invariants naturally arise from the properties of Kobayashi metrics obtained from Lempert's work . Thus, one would like to formulate a somewhatmore direct description of the local structure ; that is intuitively more appealing . In this paper, we give such description from the point of view of the calculus of variations by examining the local properties of the Kobayashi metric on strictly linearly convex domains, and derive equivalent conditions to the vanishing of the Faran's invariants from a simple property of Kobayashi metric (Property 1 .3) . In order to describe the local structure of the Kobayashi metric, we give brief review of Lempert's work . We define the infinitesimal Kobayashi metric FK on a complex manifold M as follows : For each v E T x M, x E M, let f be a holomorphic map from the unit disk 0 C C into M such that f (0) = xand f(0) = A f v for A f > 0 . The magnitude FK (v) of v with respect to the infinitesimal Kobayashi metric FK is defined to be the infimum of where the infimum is taken over all such f . If f actually attains the infimum (Le . FK(v) = ár ), then f is called extremall . It can be easily seen that the metric FK is invariant under the action of the group of biholomorphisms of M . Lempert showed that, if M = D C C V` is a bounded strictly linearly convex domain with smooth boundary, then FK is a smooth complex Finsler metric [L1], [L2], Le . FK is smooth outside the zero section of TD and satisfies the following conditions : (1 .1) FK(v) > 0 f'or v , : 0, FK(z .v) = Iz1FK(v) for z E C, and (1 .2) FK(vi + V2) .< FK(v1)+FK(v2) for vl, v2 E T .D, x E D, where equality in (1 .2) holds only when vl and v2 are colinear . Moreover,
FINSLGR MGTRICS ANDTHC KOBAYASHI MGTRIC 133 he proved the following theorem : Theorem (Lempert) . Suppose that D is a bounded strictly linearly convex domain with smooth boundary . (1) There is a unique extremal map corresponding to each v E TD . (2) All the extremal maps are proper isometric imbeddings, and can be smoothly extended to the closed unit disk 0 . (3) The extremal disks f (,~i) passing through a pointt xE D form a complex foliation of D - {x} . (4) Extremal disks are (the only) one-dimensional holomorphic retracts of D . One of the key ideas in describing the geometry of D is the construction of the holomorphic retract of D onto the extremal disk f (A) . Lempert proved that the field of holomorphic tangent planes of áD on f (¿9A) can be holomorphically extended to the interior of the disk f ( 0 ), and defines a holomorphic field of complex hyperplanes on f (A) that are transversal to f (A) . In other words, there is a well defined (n - 1)-dimensional holomorphic vector bundle p : E , f (A) over the extremal disk with fibers defined by the hyperplanes in T' . The union of the hyperplanes contains the domain D, and the holomorphic retract is defined by the restriction to D of the projection mapp . The existente of such holomorphic retracts has further implications . For example, it forces every (locally length minimizing, connected) geodesic curve of FK to be contained in an extremal disk . The properties of the Kobayashi metric that interests us are the following : Corollary . Let f : A - D be a extremal map for v E T f D . (1 .3) The extremal disk f (A) coincides with the union of geodesic curves through x tangent to a common complex line in T x M . (1 .4) There is a canonical splitting TD,f(o) = T(f (0)) ® E where E is an (n - 1)-dimensional holomorphic subbundle of the restriction TD,f(o) of the tangent bundleTD to f (A) . We wish to generalize the condition (1 .3) to an abstract complex Finsler metric F defined on an'n-dimensional complex manifold M . Let exp, denote the exponential map from a neighborhood of 0 E T, M ¡rito M defined by the geodesics of F . For each tangent vector v E T~M, the image exp (U,) of a small neighborhood U, of 0 in the complex line 0 v defines a surface in M . A reasonable generalization of the condition (1 .3) is the following : (1 .5) For all v E TM, the surface exp (U,) is a complex curve (1dimensional complex submanifold) in M .
134 M .-Y . PAlvc This condition was first introduced by Royden [R], and it is, in fact, equivalent to a condition given by Faran [F] . The purpose of this paper is to study the local structure on of Finsler metrics satisfying condition (1 .5), and further, to show that many properties of the Kobayashi metric of convex domains extend to this more general class of complex manifolds . One of our major results is a construction of a biholomorphically invariant family of complex curves which enjoys many of the properties of extremal disks in convex domains . To see how such complex curves are constructed, recall, from the calculus of variations, that the metric F uniquely determines a vector field X on the cotangent space of M, called the geodesic vector feeld, such that the integral curves of X are mapped into geodesics of F by the projection map 7r : T*M > M . Let Z be the vector field on Tó M = {v E T*MIv =~ 0} generated by the circle action of unimodular complex numbers defined by multiplication on TO* M . Theorem A . The following conditions are equivalent (Theorem 4 .7) : 1 . The surface exp (U ) C M is a complex curve for all v E TM . 2 . [X, JX] = KZ for some smooth function r, on Tó M . 3 . The distribution D = e X ® (E Z C T(TO M) is involutive . If any of the above conditions is satisfied then each complex curve exp (U ) extends uniquely to a maximal, totally geodesic, immersed complex curve E --~ M (Theorem 4 .9) . The sígnificance of the condition 2 and 3 in the theorem is as follows : The condition 2 provides a computational methods to check whether F satisfies the property (1 .5) . The condition 3 implies that, by Frobenius Theorem, the distribution D defines a 2-dimensional complex foliation .Fo of Tó M . The complex curve E is constructed by projectiog each leaf of the foliation Fo by the projection map 7r onto M . The curves E share many local properties in common with the extremal disks described in the Lempert's theorem . For example, by Theorem A, any real geodesic curve of F is contained in one of the curves E . Furthermore, a generalization of the property (3) in Lempert's theorem holds : the complex curves E passing through a point x form a complex foliation of some neighborhood of x . A less trivial result is the following generalization of property (1 .4) : Theorem B . For each complex curve E, there is a canonical splitting TM¡£ = TE ®P-E, where T 1 E is an (n - 1)-dimensional holomorphic subbundle of TMI£ . (Theorem 4 .9 .)
FINSLER METRICS AND THE KOBAYASli1 METRIC 135 The significante of the function r, in the Theorem A is its relation to the holomorphic sectional curvature of F . Note that each complex curve in M is naturally equipped with a Hermitian metric induced by F, and, therefore, has an associated Gaussian curvature . Following the definition by Wong and Royden [W] [R], we define the holomorphic sectional curvature of F at v by the Gaussian curvature of the curve exp (U ) . Theorem C . If F is a complex Finsler metric satisfying the condition (1 .5) then the holomorphic sectional curvature of F is determined by the function Finally, using the result described above, wc show that, under the condition that F is complete a,nd r, = -4 the complex curves E coincide with the extremal disks . This result was proved earlier by Faran [F] . Note that, from Lempert's result, the Kobayashi metric FK on a strictly linearly convex domain D C en has constant holomorphic sectional curvature -4 . (This is a direct consequence of the fact that every extremal map f : A -> D is an isometry with respect to the Poincaré metric and FK such that f ( 0 ) is locally defined by exp (U ) .) 4 .24 Theorem [F] . Suppose F is a complete complex Finsler metric on a complex manifold, M with constant, holomorphic sectional curvature -4 satisfying th,e property (1 .5) . Then F = FK, v)hcre FK is the Kobayashi metric on M . The paper is organized as follows : In Section 2, we develop basic tools and prove some basic facts about complex Finsler manifolds . Section 3 is an introduction of Legendre foliations and its application to complex Finsler manifolds . In Section 4, we prove the main theorem using the results of Sections 2and 3 . Throughout the paper, M denotes an n dimensional complex manifold and F a complex Finsler metric ; on M . The f'ollowing notations are used : (1) The indices a, band c rango from 1 through 2n, and o ., (5, y rango from 1 through 2n - 1 . Summation conventions are in forte throughout . (2) (X I , . . ., xn, x n+1 . . ., x2n) denote the real coordinates on M obtained from a holomorphic coordinates x, + ixn+ , for v = 1, . . ., n . (3) (xl X n x n+1 x 2n u l un u n+l u2n) denote the coordinates on T*M induced by (xl, . . ., xn, x n+1 ,. . ., x 2n) . (4) For F E C°°(T*M), Fa, FaL, . . . denote ' g " - , 9zr " and so on . du° (5) If V is a vector field on a manifold M, e' V : MM denotes the 1-pararrleter family of dif eomorphisms generated by V . Thus, for
136 M .-Y . PANG Acknowledgments . I would like to thank T . Duchamp for his help and encouragement, and for introducing me to this subject . I also wish to express my thanks to J . Bland for conversations . In this section, we prove some general facts about complex Finsler metrics . A complex Finsler metric on the cotangent bundle of M is a map F : T*M - R satisfying properties (1 .1) and (1 .2) . When M is equipped with acomplex Finsler metric, we will call M a complex Finsler manifold . 2 .1 The Geodesic Vector Field and Complex Structure . In order to define the exponential map, we introduce the geodesic vector field on T*M . Recall that T*M is naturally equipped with a 1-form defined by the equation (2 .2) and that the 2-form d( is a symplectic 2-form on T*M (Le . a smooth closed 2-form on T*M satisfying the non-degeneracy condition (do)2,, =~ 0) . The geodesic vector field X on T* M is uniquely determined by the condition (2 .3) XJdS = -FdF . In particular, tlle identity XF = 0 holds . In terms of coordinates, we Nave (2 .4) cach xE M, t ~--~ e tv x is an integral curve of V starting at x (Le . e °v x = x and d I etv x = Ve .) . 2 . Complex Finsler Metrics X=F Fa _ó _ _ áF _ 8 ~ . áxa Óx a áua a=1 2n _ E ua dxa, a-1 To describe how X is related to the complex Structure, consider coordinate expression of the complex Structure on T*M . The complex Structure J on M is expressed as -i) (2 .5) J 5 = ,In' r~ where (Jb) _ (1 0 ,
(2 .6) FINSLER METRICS AND THI ; KOBAYASIII METRIC 137 and I is the (n x n)-identity matrix . The natural complex structure on T*M is then given by a b a _a __ ba I gX a It can be directly checked that the complex structure defined by this is independent of the choice of coordinates . By abuse of notation, we will denote this complex structure on T*M by J . From the dcfinition above, it is clear that J o 7r * = 7r * oJ . Note that condition (1 .1) provides a compatibility condition of F and the complex structure, which can be expressed as follows : Let Y be the radial vector field on Tó M generated by the , action of IR 1 :>y rnultiplication of é, t E I1 Z . It can be easily checked that Y and Z satisfy the relation Thus, if F satisfies the condition (1 .1), we have F(e`v) = F(v) a,nd F(e'v) = e'F(v), and therefore the following identities hold : (2 .8) ZF = 0, YF = F . Note that the coordinate cxpressions of Y and Z are (2 .9) Y = u° 09 , Z= -Ja U'' a'9 a . Therefore, conditions (2 .8) are equivalent to (2 .10) .1' 71" F a = 0, F,, u' =F . 2 .11 Lemma . The following identities are sattisfified : (2 .12) [Z, X] = -JX, [Z, JX] = X, [Y, X] = X, [Y, JX] = JX, [Y, Z] = 0 . Proof : The computations in the proof of this lemma are based on the identities (2 .8)-(2 .10) and the following basic identities derived from them : f _a _f _a __ f _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 compute [Z, X] using coordirlates : [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) }- Observe that (ZFa) Z (au ) a BuF ) + [Z' aáa, F- (Ja av, b ) F Ja,Fb . Therefore, using the skew-symmetry of I ba, we obtain =(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 the identity [Z, .IX] = X, dote thaat .Cz,I = 0, and compute [Z, JX] = GZ(JX) = J(GZX) = J(-JX) = X . To prove the identity [Y, X] = X, note tllat YF a , = Y ( O F , ) = aua (YF) + f Y, aa ry F F a - Fa = 0 . Using this identity, compute [Y, X] as follows : [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 identity [Y, Z] = 0 is clear since the actions of e t and e - " commute, and the identity [Y, JX] = JX follows from the computation : [Y, JX] = [Y, -Gzx] = -Gz[Y, X] + [GZY, X] = -GzX = JX . 2 .13 Lemma . The vector fields JX and [X, JX] satisfy the identities : (2 .14) ((JX) = dF(JX) = 0, and (([X, JX]) = dF([X, JX]) = 0 . In particular, JX and [X, JX] are tangent to the submanifold SFM C To M . Proof :: To prove the lemma, recall that we have the identities Cx( = X -id( =-F dF, X F= 0 and C(Z) = 0 . Also, recall from identities (2 .8) and (2 .12) that Z F = 0 and JX = [X, Z] . Using these identities and the fact that Gx is a derivation, compute as follows : ((JX) = (([X, Z]) = «£x Z) = £x«(Z» - (£x C) (Z) = FdF(Z) = 0 dF(JX) = dF([X, Z]) = XZ F - ZX F = 0 Using these identities again, we complete the proof of the 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 Exponential Map . The exponential map is defined similarly as in the case of Hermitian manifolds . To define it, we introduce the dual complex Finsler metric F : TM , R, satisfying conditions (1 .1) and (1 .2), and define geodesics as curves with locally length minimizing property with respect to F . We briefly review some concepts of the calculus of variations . For more details about the calculus of variations see [GF] and [S] . To define F, let ToM = {v E TM1v :~ 0}, and define : a bundle map xP :TóM~ToMby (2 .16) T (w) = 7r (X ,) for w ETó M . In coordinates, we have 7L (2 .17) xPx(w) _ F (w) F,, (w) 8 á .,, a= I . .1
14 6 M .-Y . PANG But from Lernma 3 .8, [X, JX] does not have a component in the direction of X and Xl, and thus [X, JX] , E L , . Again, from Lernma 3 .8, it follows that Sri (X) = 0, and we obtain (4 .3) JX = XII and [X, JX] = Si Z <, . From the structure equation (3 .6), it can be easily shown that the identity Ex Z a = X~ + ir' (X) Zp holds . This identity and the second identity of (4 .3) gives Ex [X, .JX] = Ex (Si Z a = (XSi ) Ztx + Si (,ex Za) _ {(X S-) Z,,+SpZr~(X)} Z < ,, +Sl X ' . But, Again, since we proved (Ex [X, JX]) , E RX , (D 1FI(Xl) , ® L,D, components of the vector Ex [X, JX] in the direction of X a for a =~ 0 has to vanish . Therefore, it follows that Si = 0 for a > 1 and [X, JX] _ Si Z, and the condition A2 follows . To prove the converse, suppose that [X, JX] = rZ holds for some r, E C°°(SrM) . We prove that the surface defined by C ,(t, s) = 7r(e tx e - S z iu) is a complex curve for all Zu E T*M by showing that J `~ -(-T, 0) is tangent to the curve C , for small T < 0 . Note that Gx £xZ = Ex [X, Z] = Ex (JX) = rZ, and let 771 = e-Tx u) . Using the definition of the Lic derivative, we compute 2 2 (e*txZ(etXv,)) tx{ .CXZ}~~tXw)) = .e*tx {GXGXZ}(etxm) = K ( e tx 7u ) {e* tx z ( ,IX ,) } . Theref'ore, if we lct W(t) = e* t xz( e tx,b) E T z (S*M), W(t) satisfies a second order ordinary difierential equation W"(t) = r(t)W(t) with initial conditions W(0) = Z,z and W'(0) = JX,T, . Consequently, we have W(t) E span{Z,b, JXú,} for small t, and in particular, W('r) _ e*Tx z ( ,rx,D) E sean{Z,7 JX,J . Substitutlng t7v = e-Tx7v, (4 .4) W (T) = e * ' x Z , E span{Zw, JX ,} . Using the identity 1 (T, 0) = 7r * X , and 7r * o J =J o 7r * , we obtain a a w (T, 0) = ás { 7 r(e 'x esz7 V) } s=o = 7r * e' x Z , = k 7r * (JX),í, = k J 7r * (X,D) = kJa a w (T, 0)
FINSLER METRICS AND TIiE KOBAYAS1i1 METItIC 147 for some k E IR, and hence J ~áe (T, 0) is tangent to the surface defined by C , . (ü) If [X, JX] = rZ holds for some rc E C' (S* M), then the involutivity of D follows from the identities (4 .5) [Z, X] = -JX, [Z, JX] = X and [X, JX] = KZ . The converse of this is an immediate conséquence of the the identity (3 .10) . 4 .6 Remark . (1) Note that, in the proof of Theorem 4 .2, the two equations Sri (X) _ 0 and SQ = 0 for /3 > 1 are equivalent to the single condition [X, JX] = rZ with rc = Si . The condition Sri (X) = 0 can be interpreted as a compatibility condition for F and the complex structure . For example, if F is the induced norm of a Kaehler manifold, this condition is satisfied . In this case, the condition Sp = 0 for ,(j > 1 puts restrictions on the curvature of the Kaehler metric . One special case of this is when M is a Kaehler manifold with constant holomorphic sectional curvature . In this case, it can be verified using results in [P] that Sá = cb í 3 for some constant e . (2) Conditions equivalent to Al-A3 of Theorem 4 .2 were introduced by Royden [R] and Faran [F] . The conditions Al-A3 in Theorem 4 .2 can be equivalently stated as conditions on Tó M . 4 .7 Theorem . The following conditions are equivalent : B1 . exp(U ) is a complex curve for all v E TOM . 132 . [X, JX] = rZ onTó M for some te E C°°(TO M) . 133 . The distribution CX ® 0Z C T(TO M) is involutive . Moreover, these conditions are equivalent to conditions Al-A3 in Theorem 4 .2 . Proof : Note that the conditions Al and Bl are clearly equivalent . To prove the theorem, we show (i) that A2 implicas B2, and (ii) that A3 implies B3 . The converses of these are trivial to prove . (i) Suppose that [X, JX] =rc Z holds on SFM for some rc E C°° (SrM) . Extend K to Tó M by rc(tw) = tz K(W) for all t > 0 and w E SI*M . We claim that [X, JX] = rc Z on Tó M . To show this, we show that both W = [X, JX] and W = n Z on Tá M must satisfy the ordinary differential equation GyW =2W . Note that the integral curves of Y
14 8 M .-Y . PANG are the radial lirios in Tó M . Thus, if both [X, .IX] and rc Z satisfy the equation, they rnust coincide because the identity [X, JX] = r Z gives the same initial condition at points on SFM . To show that the vector field [X, JX] satisfies the differential equation, recall, frorrr Lemma 2 .11, that G y X =X and GyJX=JX . Using these identities and the fact that Gy is a, derivatiorr, we compute Gy [X, JX]=[GyX, IX] +[X, GyJX]=[X, JX]+[X, JX]=2[X, JX] . To show that the vector field r, Z satisfies the differential equation, recall that the vector field Y on Tó M is generated by the action of II3, by multiplication of e' . Using homogeneity of K, we obtain the following identity : For w E TO M, d ~~ dt, =o (Le . GyZ = [X, Y] = 0), we compute dt {e2r r (w) } = 2~ ;(v~) r .=o Using this identity and the fact that the vector fields Y and Z commute { .Cy (r Z)} = (Yr,) Z= 2(K Z) . (ii) The ; proof that the condition A3 irnplics the condition B3 imrnediatc ;ly follows frorrr thc ; identities in Lcrnrna 3 .8 . 4 .8 Totally Geodesic Complex Curves . We call a complex curve E totallly geodesic if, for any tangent vector v to the complex curve E and geodesic segnrcnt yv : (-e, e) - M such that ^c (0) = v, y, (t) is corrtairred in the complex curve for small t . The main result of this seetiorr is tlrat, under condition (1 .5), the geodesics of F can be uniquely extended to irnmersed complex curves that are totally geodesic submanifolds of M . In fact, tlrese curves are precisely thc ones defirred by the complex curves exp (U )'s in condition (1 .5) . Theorem 4 .9 . If the condition (1 .5) holds, the complex curve exp (U ) can be uniquely extended to a maximal totally geodesic complex curve f : E - M immersed in M . Moreover, there is a canonical (n-1)- dimensional holomorphic vector subbundle T'E of f*(TM) transversal to E . Proof :: Recall from Theorem 4 .2 that D= span{X, JX, Z} is an involutive distribution . By the Frobenius theorerrr, this irrrplies that S* M is foliated by 3-dirnerrsional maximal integral subrnanifolds of D . Let
(4 .10) FINSLGRMETRICS AND THC KOBAYASII1 MGTRIC 149 E be a leaf of this foliation .-D, then there is a well defined S I C e action on E since Z is tangent to E . Let us denote the quotient space E/S I by E . It is not difñcult to see that E is a complex curve with the complex structure induced from the complex structure of E C T* M, and that there is a holomorphic immersion f such that the following diagram commutes : inclusion E~Srm E J M Recall that the complex curve exp (U ) is the surface defined by (4 .11) (t,, s) - exp (tesw) = n (etXe- .yi4,(v)) . From this, it is clear that the complex curve f : E - M is locally defined by exp (U ) since e' x c -9z iv E E . From this, it clearly follows that f : E - M is totally geodesic . To define the transversal holomorphic subbundle T - L E of f * (TM), note that 7rz E is a principal circle bundle . Define the fiber T,LE ofT l EatxEEbyT~E={vEf*(TM)Jw(f * v)=0 for al¡ wEE - ,,} . It is clear that T l E is a holomorphic vector bundle with dimension n-1 . To show that T~E is transversal to E, suppose v E TLE n TE . Note that xP(w) is tangent to E for all w E E y since : xP(rv) = 7r * X , and X is tangent to E . Moreover, 'Y(ui) :,¿ 0 because w (xP(w)) = w(7r * X ,) = FF a ua =F 2 = 1 z~ 0, where uJ =Ea -l , a dx a . Since v, f * Y' (w) E T, ; E, we have f * v = zkP(w) for some z E (E . But, recall that v E T L E, and therefore, v)(v) = 0 . This implies that v = 0 because 71) (V) =w (zXP(w)) = z {w (xP(w))} = z . 4 .12 The Holomorphic Sectional Curvature . If F satisfies the property (1 .5), there is a natural way to define holomorphic sectional curvature K of F . In this sectio'n, we show that K is determiried by the smooth function K in the condition A2 of Theorem 4 .2 . Holomorphic sectional curvature K of a complex Firisler rnetric F has been studied by Wong and Royden [W] [R] . To define K(v) for a unit vector v E T, M (Le . F(v) = 1), note that each complex curve U C M tangent to v has a canonical complex Finsler rnetric, defined by the restriction F,1 , u : TU -> IR . In fact, because U is of complex
15 0 M .-Y . PANG dimension one, it can be easily seen that the metric F1 h u is a norm induced by a Hermitian metric g on U : If F(v) = 1, F ((a + i/l)v) = l a + i,31 F(v) = a2 -+Q2 . In [W], Wong defined the holomorphic sectional curvatura K(v) as the supremum of the Gaussian curvatura of g at x E U, where supremum is taken over all complex curves tangent to v . In the special case when F is the norm induced by a Hermitian metric, this defines the usual holomorphic sectional curvatura of Hermitian metric . Observe that, if F satisfies the property (1 .5), then by Theorem 4 .2, there is a naturally defined totally geodesic complex curve of the form exp (U ) tangent to v . In [R], Royden showed that the Gaussian curvatura of the induced metric g at x on this complex curve attains the greatest value and, hence, it defines the holomorphic sectional curvatúre K(v) . Thus, the following theorem holds : 4 .13 Theorem . Let, N be a complex submanifold of M, and let K' be the holomorphic sectional curvatture of the induced metric FITN on N . Then the inequality K'(v) < K (v) holds every unitt vector v tangent to N . In particular, if U C M is a complex curve tangent to a unit vector v E TM, the Gaussian curvatura of the induced, metric g defined by FITU is bounded from above by K(v) . The function K can be regarded as a function on the quotient space S* MIS' . Recall from (2 .12) and(4 .5) the identities [Z, X] = -JX, [Z, JX] = X and KZ= [X, ,IX], and compute (ZK)Z =G Z(KZ) = £z [x, JX] - [L ZX, Jx] + [x, L Z (Jx)] = [-Jx, Jx] + [x, x] = 0 . From this ideritity, we conclude ZK= 0 . This implies that the function K is invariant under the circle action of unimodular complex numbers defined by multiplication on TO M, and hence the function K can be regarded as a function on S*M/S 1 . 4 .14 Theorem . If K is the holomorphic sectional curvatura of F, for every unit vector v E TM . K (v) = K o -D(v) Before we hegin the proof, note that by Theorem 4 .9 the complex curve U C M has a unique extension to a maximal totally geodesic complex curve f : E -+ M, and that there is a circle bundle rrr : É , E ovar E .
FINSLP ;I1 METRICS ANll TIIG KOr3AYAS1ll METRIC 151 4 .15 Lemma . The f bers of the circle bundle 7rz : E - E defines a Legendre foliation . :L with respect to a natural contact, 1-form deffned by the pull-back of 77 to E . The structure equations of this Legendre foliation are the pull baks to E of the equations : (4 .16) dB' =- r7A~', drt=0 1 n t ;', d~' =K17A0 1 Proof . : The structure equations (4 .16) are obtained by pulling back the equation (3 .4) to É . Note that, since JX= XI and Z = ZI, we llave Ba (JX) = ~' (Z) = 0 for cx > 1 . Therefore, on E, we llave dB' = -77 A t ;' + G'110 1 A ~' dri=0 1 A~' d tl =SigAB 1 +Q 1 r10 1 A~ l . But, from thc ; identities (4 .5), it easily follows that G'r1 = Qr 11 = 0 arrd Si = r, . Hence, we obtain the structure equations (4 .16) : From the equations, it is clear that the pulí back to É of rl is a contact form since r) A drt = rt A B' A ' 0on E . Proof .. To prove tire theorern, choose v E T,,M and let U C M be a totally geodesic complex curve such that x E U arrd v E T,;U . Also, let f : E , M be the unique extension of U described in Theorem 4 .9, and let Éu denote the restriction of the circle bundle 7rr : É -~ E to U . Observe that U is also a submanifold of S M/S' since U C E= E/S' C SrM/S' . We claim that the Gaussian curvatura of g orr U is Klu E C_ (U), wlrcre K is regarded as a function orr S¡,M/S' . The theorern follows from the claim . To sea this, proveed as follows : Note that by commutativity of the diagram (4 .10), 7r = f o 7rz . Hence, if w E Eu, then xP(ur) = 7r,X w = f* o (7rr),X , . Therefore, the rrrap SrM , SpM sends Eu into the zenit tarrgerrt bundle S .U of g . Since T is a bundle map ovar U such that T(es'u» = e`T(w), kP maps tu dif eomorphically onto S .U, or equivalently, we llave a bundle rrrap 1 Is,u = <PIs,u : SU - Eu ovar U . Thus, since r, is corrstant along fibers of Eu arrd v E T,,U, we llave r, o <D(v) = K(x) . B,y thc ; clairrr, K(x) is the Gaussian curvatura of g at x E U, and the identity K(v) = Koq>(v) follows . To prove the clairn, we denote tire Gaussian curvatura of g on U by ic E C'(U), and show K = k . Recall that, f'rorrr Propositiorr 3 .5, thc ; Legendre foliation on Sg*U iras the structure equations (4 .17) dé' =-~At : ', di7=8'A~', d~'=k~AB',
15 2 M .-Y . PANC wherc : {H1, ~, `1} is the invariant coframe on S* U . The ; proof of the claim is done by est S ablishing the equivalente of the Legendre foliation on S . .U with ,Fz . This is proved by showing that there is a dif eomorphism 0 : Eu ~ S9 U such that (4 .18) 0*~=r7, and 7r o0=7rr . It follows that O*Hl = 0 1 and ?P*~ 1 = ~1, and in particular, the pull-back by 0 of the equations (4 .17) are the structure equations (4 .16) of .Fz . The identity k o 0= r, follows . We define O(w) for w E Eu C T*M as the by pull-back of w to the tangent space of U . In coordinates, we have (4 .19) l n+1 1 n n+1, . . ., 2ra ( :L ) 0, . .,O,X' ,0, . .,0¡U , . . .,26 ,4L 2L ~(x l xn+1 u1 ,9xn+1) where (x 1 , . . ., .xn, xn+1 ,. . ., x2,) is taken so that U C M is locally defined by .r,°' = 0 for a =~ 1, n + 1 . To complete the proof ; it remains to show t hat (1) rnaps Eu diffeornorpllically orto S .*U, and (2) " ~ = 71 (1) : To show that 0 is a diffeornorphism onto Sg*U, recall that T maps Eu onto SU . Hence, if w E Eu, then the vectors xP(w) and T(Jw) _ -JT(w) f'orm an orthonormal frame of T, ;U for some x E U . The following computation shows that the covectors 0(7u) and 0(Jul) = ,Iz/1(w) forro the dual coframe of {xP(w),T(Jul)} : Using the iderltities (2 .10), compute ; {1h(1u)} (IP(w)) = w(7r * 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(1v)) = {-Jw}(7r*X ,) = uaJ," F F, = 0 . Hence z/>(Elx) C S .*U . Since V) is a bundle map preserving the circle action, it casily follows that 0 is a dif eorrlorpllisrn . (2) : To prove tlle identity z)*~ = TI, recall f'ronl (2 .2) that fl = ul dx 1 + un - " dx n+1 . Tllerefóre, fronl (4 .19), 't%1 * f = ul dx 1-}- u "+1 . dxn+l . On the other hand, the contact 1-forro 77 on E is defined by the pull-back of 71 _ 1 :2n .= to E . But, since dx° = 0 for a =,~ 1, n+ 1 on E, we have 77 = 7x 1 d x 1 + u n+1dxn+1, Hence the identity 0*~ = 17 follows . a The following corollary is a consequence of the proof of Theorem 4 .14 :
FINSI,I ;R MI . :TItICS AND 'CII1 ; KOBAYASIII MI TILIC 153 4 .20 Corollary . The Gaussian curvatura of the induced metric .q on E ás r,,É . 4 .21 Relation to the Kobayashi Metric . In this section, we prove a version of a theorem of Faran which states that vanishing of certain local invariants forres F to be the Kobayashi metric of M, provided that F is complete and satisfies the condition K = -4 (sea Introduction) . In the version of the theorem presented here, the condition of vanishing of invariants is replaced by the equivalent condition (1 .5) . Recall frorri Lempert's result described in the introduction that, if D C T' is a bounded strictly linearly convex domain with smooth boundary, then cvery extremal disk f : A -> D is an isornetric imbedding (Le . f *FK coincides with the Poincaré norm en ,), and that f (A) is a maximal totally geodesia cornplex curve ; in D . Since thc ; Poincaré nietrie has Gaussian curvatura -4, the holornorphic ; sectional curvaturc of thc ; Kobayashi rnctric FK is -4 . Ori the other hand, if F is a ny cornplex Finsler metric : on a cornplex manifold M, the condition of constant holornorphic ; sectional curvatura K =-4 imposes a restriction on the metric F . In faca, we show that, if F is any complete cornplex Finsler metric ; with the properties K = -4 and (1 .5), then F rnust coincide with the Kobayashi metric . To show this, we need thc ; following lerrima due to Ahlfors [A] [K] : 4 .22 Generalized Schwarz Lemma . Let (N, g) be a, 1-dimensional . Hermitian manifold such thatt the Gaussian cu,r'vatur'e is bounded, above by a negative consttantt -C . For' any holornorphic mal) f : A , N, the inequality (4 .23) Il .f*vll .~ <_ c Ilvll holds for allll v E TA, v)hcre II Ils is the norm on N induced by y and II II denotes the nor'rn defined by the Poincaré rnetric on A . We call a cornplex Finsler metric : F complete if the geodesic : vector field X is complete ; (or equivalently, if every geodesic can be ; extended to a geodesic define(¡ en all of IR) . 4 .24 Theorem [F] . Suppose F is a complete complex Finsler metric on a complex manifold M with constante holomorphic sectionall curvatura K = -4 satisfyinq the property (1 .5) . The dual metric F coincides with the Kobayashi metric FI< of M . Proof .- We ve ;rify the equality FK =F by verifying thc ; inequalities FK<FandF<FK .
15 4 M .-Y . PANG (i) To show that the inequality FI< < F holds, let v E TM and recall frorn Tlicorenr 4 .9 that there is an irnrnersed totally geodesic corriplex curve f : E , M tangent to v . Sinee F is a complete metric with K = -4 and E is totally geodesic, the induced metric g on E defined by f*F is a complete Kaehler metric with Gaussian curvature -4 (see Theorem 4 .14 and Corollary 4 .20) . Therefore, there is a holornorphic covering rnap f2 : A - E which ¡s 'a local isometry between the Poincaré metric and 9 (see chapter IX of [KN]) . By composing f r and f2, we obtain a holornorphic rnap fro f 2 : A - M that is an isometric immersion with respect to the Poincaré metric and F (Le . {(f o f2) * F}(w) = 11wil) . Recall that the Kobayaslri metric FK(v) is defined as the infirnum of Ji .f * vil over all complex curve f : A - M tangent to v . Hence, the inequality follows : FK(v) < 11 (f l o f2)*vjj = F(v) for v E TM . (ü) To prove the inequality F(v) < FK(v), note that, for each complex curve : f : A -> M tangent to v E TM, there is a Hermitian metric metric gf on 0 defined by f*F . By Theorem 4 .13, tlre Gaussian curvature nf is bounded above by -4 . Applying the Gencralized Schwarz Lemrna 4 .22 to thc identity mal) id : -> (0, g f), we obtain the inequality {f*F}(v» <_ 11wII, or equivalently, F(f * w) <_ jjwjj for all uj E TA . In particular, this implies that F(v) < lif *v il for any complex curve tangent to v . Since FK (v) is the infirnum of 11 f * v 11 over all such f , the inequality F(v) < FK(v) follows . Referentes [A] L .V . AnLFORS, An extension of Schwarz's lemma, Trans . of Amer . Math . Soc . 43 (1938), 359-364 . [BD] J . BLAND, T . DUCHAMP, Moduli for Pointed Convex Dornains Invent . Math ., (to appear) . [F] J .J . FARAN, Hermitian Finsler Metrics and the Kobayashi Metric, Journal of Differential Geometry 31, no . 3 (1990), 601-625 . [GF] I .M . GEAYAND, S .V . FOMIN, "Calculas of Variations," Prentice-Hall, Inc ., Englewood Cliffs, 1963 . [K] S . KOl3AYAS11l, "Hyperbolic Manifolds and Holornorphic Mappings," Marcel Dekker, Inc ., New York, 1970 . [KN] S . KOBAYASM, K . NOMizu, "Foundations of Differential Geometry," Vol . 1 arrd 2, John Wiley & Sons, Inc ., New York, 1963 .
FINSLrR METRICS AND TI-Ir KOBAYASIII MI TRIC 155 [L1] L . LrmPrRr , , La metrique de Kobayashi et la representation des domains sur la boule, Bull . Soc . Math . France 109 (1981), 427-474 . [L2] L . LEMPERT, Intrinsic Distantes and Holomorphic Retracts, Complex Analysis and Applications 81 (1984), 341-364 . [L3] L . LEMPERT, Holomorphic invariants, normal forms, and the moduli space of convex domains, Annals of Math . 128 (1988), 43-78 . [P] M . PANG, The Structure of Legendre Foliations, Trans . of Amer . Math . Soc . 320, no . 2 (1990), 417-455 . [Pa] G . PATR1zIO, Disques extremaux de Kobayashi et equation de Monge-Ampere complexe, C . R . Atad . Sci . Pari s 305, Serie 1 (1987),721-724 . [R] H .L . ROYDGN, Complex Finsler Metrics, Contemporary Mathematics, Proceedings of Summer Research Conference, Aug . 12-18, Amer . Math . Soc ., 1984, pp . 119-124 . [S] S . STERNBERG, "Lectures on Differential Geometry," Prentice Hall, Englwood Cliffs, 1964 . [W] B . WONG, On the Holomorphic Curvature of Some Intrinsic ; Metrics, Proc . Amer . Math . Soc . 65, no . 1 (1977), 57-61 . Department of Mathematies Box 1146 Washington University St . Louis, MO 63130-4899 U .S .A . Rebut el 12 de Mare, de 1991