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Holomorphic curvature of Finsler metrics and complex geodesics

arXiv:math/9207201v1 [math.CV] 10 Jul 1992Holomorphic curvature of Finsler metrics and complex geodesicsby Marco Abate and Giorgio Patrizio0. IntroductionIf D is a bounded convex domain in Cn , then the work of Lempert [L] and Royden-Wong[RW] (see also [A]) show that given any point p ∈D and any non-zero tangent vectorv ∈Cn at p, there exists a holomorphic map ϕ: U →D from the unit disk U ⊂C into Dpassing through p and tangent to v in p which is an isometry with respect to the hyperbolicdistance of U and the Kobayashi distance of D. Furthermore if D is smooth and stronglyconvex then given p and v this holomorphic disk is uniquely determined.For a general complex manifold it is hard to determine whether or not such complexcurves, called complex geodesics in [V], exist.

Therefore it is natural to investigate thespecial properties enjoyed by the Kobayashi metric of a strongly convex domain. In thiscase it is known that the Kobayashi metric is a strongly pseudoconvex smooth complexFinsler metric.

Furthermore, for a suitable notion of holomorphic curvature (see below),this metric in convex domain has negative constant holomorphic curvature (cf. [W], [S],[R]).In [AP] it was started a systematic differential geometrical study of complex geodesicsin the framework of complex Finsler metrics.

As in ordinary Riemannian geometry it isnatural to study geodesics as solutions of an extremal problem and not as globally length-minimizing curves, so in our case the natural notion turns out to be the one of geodesiccomplex curves, i.e., of holomorphic maps from the unit disk into the manifold sendinggeodesics for the hyperbolic metric into geodesics for the given Finsler metric. For instance,the annulus in C has no complex geodesics in the sense of [V], whereas the usual universalcovering map is a geodesic complex curve in the previous sense.It was shown in [AP] that geodesic complex curves for complex Finsler metrics satisfya system of partial differential equations and, under suitable hypotheses, it was given anuniqueness theorem.

Here we shall be concerned with the question of existence.The further ingredient needed to attack this problem is the notion of holomorphiccurvature of complex Finsler metrics. Given a complex manifold M and a complex Finslermetric F: T 1,0M →IR, i.e., a nonnegative upper semicontinuous function such thatF(p; λv) = |λ|F(p; v)for all (p; v) ∈T 1,0M and λ ∈C, the holomorphic curvature of F at p in the direction v isthe supremum of the Gaussian curvature at the origin of the (pseudo)hermitian metrics onthe unit disk obtained by pulling back F via holomorphic maps ϕ: U →M with ϕ(0) = pand ϕ′(0) = λv for some λ ∈C∗.

Here, following [H], we make the choice of computingthe Gaussian curvature using the weak laplacian rather then using Ahlfors’ notion ofsupporting metrics; this approach seems to be more natural for our applications and hasa better connection with the usual hermitian geometry. We remark that if F is the normassociated to a hermitian metric, then Wu in [Wu] showed that this definition yields the

2Marco Abate and Giorgio Patriziousual holomorphic sectional curvature in the direction v. We shall see in section 2 thatfor smooth strongly pseudoconvex (see below for definitions) Finsler metrics this definitionrecovers the holomorphic sectional curvature defined by Kobayashi in [K2].In section 1 we give a survey of the elementary implications of this notion of holo-morphic curvature of complex Finsler metrics gotten by application of Ahlfors’ lemmaand of its sharp form due to Heins [H]. For instance, as one could expect, it is easy toprove a hyperbolicity criterion in terms of negatively curved upper semicontinuous Finslermetric (cf.

Corollary 1.5). As for the investigation of geodesic complex curves and differ-ential geometric properties of intrinsic metrics, not much can be achieved without somesmoothness assumptions.

One of the few general facts obtained here is the explanation(Proposition 1.6) in terms of the holomorphic curvature of the known property of theCarath´eodory metric that a holomorphic disk which is an isometry at one point is infact an infinitesimal isometry at every point (i.e., an infinitesimal complex geodesic in theterminology introduced in [V]). Furthermore, in Proposition 1.7 it is given a very weakcharacterization of the Kobayashi metric which nevertheless is useful later in the paper.In order to prove more significant results it is necessary to consider the case of smoothFinsler metric, i.e., such that F ∈C∞T 1,0M \ {Zero section}, which are in additionstrongly pseudoconvex, that is such that for every p ∈M the indicatrixIp(M) = {v ∈T 1,0M | F(p; v) < 1}is strongly pseudoconvex.

Under these assumptions in section 2 we show how to computethe holomorphic curvature KF of F by means of a tensor explicitly defined in terms ofF and which agrees with the usual one in case F is the norm associated to a hermitianmetric. The expression we get has also been considered from a slightly different point ofview by Royden [R] and Kobayashi [K2].After this preparation we address the problem of the existence of geodesic complexcurves in section 3.

LetA(ζ) =−2¯ζ1 −|ζ|2for ζ ∈U, andΓα;i = Gα¯µG¯µ;i,where G = F 2, lower indices indicate derivatives (with respect to the coordinates of themanifold those after semicolon, with respect to the coordinates of the tangent space theothers), (Gα ¯β) = (Gα ¯β)−1and we are using the usual summation convention. From resultsof [AP] it follows that the geodesic complex curves are holomorphic maps ϕ: U →Msatisfying, for α = 1, .

. ., n,(ϕ′′)α + A · (ϕ′)α + Γα;i(ϕ; ϕ′)ϕi = 0,(0.1)and an additional set of equations which automatically hold if along the curve the metricF satisfies a K¨ahler condition introduced by Rund [Ru2], which reduces to the usual onefor hermitian metrics.The holomorphic solutions of (0.1) have nice properties on their own.

For instance,they realize the holomorphic curvature at every point for the direction tangent to the

Holomorphic curvature of Finsler metrics and complex geodesics3disk, and if they are isometry at the origin then they are infinitesimal complex geodesics(cf. Proposition 3.2).

Our first main result (Theorem 3.3) describes necessary and suffi-cient conditions for the holomorphic solvability of (0.1), and hence for the existence (anduniqueness) of complex geodesics through a given point and direction.The characterization given in Theorem 3.3 is completely expressed in terms of themetric, but rather technical; to give a clearer geometric characterization it is necessaryto bring the curvature into the picture.The previous list of properties of holomor-phic solutions of (0.1) shows that a natural necessary condition is that F has constantnegative holomorphic curvature. This is almost sufficient; to get the correct geometricconditions it is necessary to introduce a further tensor, defined on the sphere bundleS1,0M = {(p; ξ) ∈T 1,0M | F(p; ξ) = 1}.

Set for α = 1, . .

., n,Hα(v) = Hαi¯µ¯vµvivj =(Gτ ¯µΓτ;α¯)i −(Gτ ¯µΓτ;i¯)αvµvivj,where Γα;i¯= (Gα¯µG¯µ;i);¯.To understand the meaning of this tensor, let us considerthe case of a hermitian metric. Then Hαi¯µ¯= Riα¯µ¯−Rαi¯µ¯, where R is the Riemanniancurvature tensor of the Chern connection associated to the hermitian metric.

In particular,Hαi¯µ¯≡0 is equivalent to ∂T ≡0, where T is the torsion form of the Chern connection; Tis a T 1,0M-valued 2-form vanishing exactly when the given metric is K¨ahler. In conclusion,H may be interpreted as a torsion of the curvature; in fact, it can be shown that Hα ≡0is a simmetry condition — formally identical to (3.29) — on the curvature tensor of theChern connection induced by the Finsler metric on the vertical subbundle of the two-foldtangent bundle T 1,0T 1,0M.

We intend to pursue these matters elsewhere.Using the tensor H we may finally summarize our main results.We have (Theo-rem 3.6):Theorem 0.1: Let M be a complex manifold equipped with a strongly pseudoconvexsmooth complete Finsler metric F.Then there exists a unique holomorphic solutionϕ: U →M of (0.1) with ϕ(0) = p and ϕ′(0) = ξ for any (p; ξ) ∈S1,0M iffthe holo-morphic curvature KF ≡−4 and Hα ≡0 for all α.In other words, the existence and uniqueness of holomorphic solutions of (0.1) isequivalent to constant negative holomorphic curvature and a simmetry condition on acurvature tensor. Furthermore, it turns out the the K¨ahler condition along a holomorphicsolution of (0.1) holds iffit holds at one point, and thus (Corollary 3.9)Theorem 0.2: Let M be a complex manifold equipped with a strongly pseudoconvexsmooth complete Finsler metric F. Assume that the holomorphic curvature KF ≡−4 andthat Hα ≡0 for all α.

Take (p0; ξ0) ∈S1,0M. Then there is a (a fortiori unique) geodesiccomplex curve passing through p0 tangent to ξ0 iffthe K¨ahler condition holds at (p0; ξ0).From this result it is also possible to obtain a geometric characterization of theKobayashi metric.

The following corollary (Corollary 3.10) improves results of Pang [P]and it is closely related to those of Faran [F]:Corollary 0.3: Let F be a strongly pseudoconvex smooth complete Finsler metric withconstant holomorphic curvature KF ≡−4 and such that Hα ≡0 for all α. Then F is theKobayashi metric of M.

4Marco Abate and Giorgio PatrizioThe second named author thanks the Max-Planck-Institut f¨ur Mathematik of Bonnfor its hospitality and support while this paper was completed.1. Holomorphic curvature for semicontinuous metricsLet U denote the unit disk in the complex plane.

A pseudohermitian metric µg of scale gon U is the upper semicontinuous pseudometric on the tangent bundle of U defined byµg = g dζ ⊗d¯ζ,(1.1)where g: U →IR+ is a non-negative upper semicontinuous function such that Sg = g−1(0)is a discrete subset of U.If µg is a standard hermitian metric on U, i.e., g is a C2 positive function, the Gaussiancurvature of µg is defined byK(µg) = −12g ∆log g,(1.2)where ∆denotes the usual Laplacian∆u = 4 ∂2u∂ζ∂¯ζ . (1.3)The (lower) generalized Laplacian of an upper semicontinuous function u is defined by∆u(ζ) = 4 lim infr→01r2 12πZ 2π0u(ζ + reiθ) dθ −u(ζ).

(1.4)It is worthwhile to remark explicitely some features of this definition. First of all, when uis a function of class C2 in a neighbourhood of the point ζ0, (1.4) actually reduces to (1.3).In fact, for r small enough we can writeu(ζ0 + reiθ) = u(ζ0) + ∂u∂ζ (ζ0)reiθ + ∂u∂¯ζ (ζ0)re−iθ+ 12∂2u∂ζ2 (ζ0)r2e2iθ + ∂2u∂ζ∂ζ (ζ0)r2 + 12∂2u∂ζ2 (ζ0)r2e−2iθ + o(r2);hence12πZ 2π0u(ζ0 + reiθ) dθ −u(ζ) = r2 ∂2u∂ζ∂¯ζ (ζ0) + o(r2),and the claim follows.Second, if u is an upper semicontinuous function, it is easy to see that ∆u ≥0 isequivalent to the submean property; so ∆u ≥0 iffu is subharmonic.Finally, if ζ0 is a point of local maximum for u, then clearly ∆u(ζ0) ≤0.Now let µg be a pseudohermitian metric on U.

Then the Gaussian curvature K(µg)of µg is the function defined on U \ Sg by (1.2) — using the generalized Laplacian (1.4);

Holomorphic curvature of Finsler metrics and complex geodesics5clearly, if µg is a standard hermitian metric, K(µg) reduced to the usual Gaussian curva-ture.The idea behind the classical Ahlfors lemma is to compare a generic pseudohermitianmetric with an extremal one — usually the Poincar´e metric. For a > 0, let ga: U →IR+be defined byga(ζ) =1a(1 −|ζ|2)2 ;then µa = ga dζ ⊗d¯ζ is a hermitian metric of constant Gaussian curvature K(µa) = −4a.Of course, µ1 is the standard Poincar´e metric on U.The classical Ahlfors lemma is true in this more general situation:Proposition 1.1: Let µg = g dζ ⊗d¯ζ be a pseudohermitian metric on U such thatK(µg) ≤−4a on U \ Sg for some a > 0.

Then g ≤ga.Proof : The proof follows closely the classical one due to Ahlfors. For the sake of com-pleteness we report it here.For 0 < r < 1, define Ur = {ζ ∈C | |ζ| < r} and gra: Ur →IR+ bygra(ζ) =1a(1 −|ζ|2/r2)2 = ga(ζ/r),and set fr = g/gra: Ur →IR+.

Being upper semicontinuous, g is bounded on Ur; sincegra(ζ) →+∞as |ζ| →r, there is a point ζ0 ∈Ur of maximum for fr. Clearly, ζ0 /∈Sg;hence0 ≥∆log fr(ζ0) ≥∆log g(ζ0) −∆log gra(ζ0) = −2g(ζ0)K(µg)(ζ0) −8agra(ζ0).

(1.5)By assumption, K(µg)(ζ0) ≤−4a; therefore (1.5) yields g(ζ0) ≤gra(ζ0) and thus, by thechoice of ζ0,∀ζ ∈Urg(ζ) ≤ga(ζ/r).Letting r →1−we obtain the assertion.To complement this result, we recall a theorem due to Heins [H, Theorem 7.1], showingthat µg ̸= µa in the statement of Proposition 1.1 implies that g is strictly less than gaeverywhere:Theorem 1.2: (Heins) Let µg = g dζ ⊗d¯ζ be a pseudohermitian metric on U suchthat K(µg) ≤−4a on U \ Sg for some a > 0. Assume there is ζ0 ∈U \ Sg such thatg(ζ0) = ga(ζ0).

Then µg ≡µa.Now we start looking to the several variables situation. If M is a complex manifold, weshall denote by TM its real tangent bundle endowed with the almost-complex structure Jinduced by the complex structure of M; by T cM the complexification of TM and by T 1,0Mthe (1, 0)-part (i.e., the i-eigenspace of J) of T cM.

As well known, T 1,0M is naturallycomplex-isomorphic to TM. In this paper we shall mainly use T 1,0M as representative ofthe tangent bundle of M, except for an argument needed in section 3.

6Marco Abate and Giorgio PatrizioA complex Finsler metric F on a complex n-dimensional (n ≥1) manifold M is anupper semicontinuous map F: T 1,0M →IR+ satisfying(i) F(p; v) > 0 for all p ∈M and v ∈T 1,0pM with v ̸= 0;(ii) F(p; λv) = |λ|F(p; v) for all p ∈M, v ∈T 1,0pM and λ ∈C.We shall sistematically denote by G: T 1,0(M) →IR+ the function G = F 2. Note that,thanks to condition (ii), the definition of length of a smooth curve in a Riemannian manifoldmakes sense in this context too; so we may again associate to F a topological distance on M,and we shall say that F is complete if this distance is.

For the same reason, it makes senseto call (real) geodesics the extremals of the length functional. General introductions toreal Finsler geometry are [Ru1, B].Take p ∈M and v ∈T 1,0pM, v ̸= 0.

The holomorphic curvature KF (p; v) of F at (p; v)is given byKF (p; v) = sup{K(ϕ∗G)(0)},where the supremum is taken with respect to the family of all holomorphic maps ϕ: U →Mwith ϕ(0) = p and ϕ′(0) = λv for some λ ∈C∗, and K(ϕ∗G) is the Gaussian curvature dis-cussed so far of the pseudohermitian metric ϕ∗G on U. Clearly, the holomorphic curvaturedepends only on the complex line in T 1,0pM spanned by v, and not on v itself.The holomorphic curvature may also be defined (see e.g.

[S]) taking the supremumwith respect to the family of all holomorphic maps ϕ: Ur →M with ϕ(0) = p and ϕ′(0) = v,where Ur ⊂C is the disk of center the origin and radius r. We chose the given definitionto stress the similarities with the definitions of the Kobayashi and Carath´eodory metrics.If F is a complex Finsler metric on U — and so G = F 2 is a pseudohermitian metricG = g dζ ⊗d¯ζ on U —, a priori we have defined two curvatures for F: KF (ζ; 1) andK(G)(ζ). As anybody may guess, they actually coincide; this is a consequence ofLemma 1.3: Let µg = g dζ ⊗d¯ζ be a pseudohermitian metric on U, and ϕ: U →U aholomorphic self-map of U.

Then on U \ [ϕ−1(Sg) ∪Sϕ′]K(ϕ∗µg) = K(µg) ◦ϕ.Proof : Let {gn} be a sequence of C2 functions such that gn ≥gn+1 and with gn(x) →g(x).Then on U \ [ϕ−1(Sgn) ∪Sϕ′] ⊃U \ [ϕ−1(Sg) ∪Sϕ′] we haveK(ϕ∗µgn) = −12|ϕ′|2(gn ◦ϕ)∆log(|ϕ′|2gn ◦ϕ)= −2|ϕ′|2(gn ◦ϕ)∂2 log(gn ◦ϕ)∂ζ∂¯ζ+ ∂2 log |ϕ′|2∂ζ∂¯ζ= −2|ϕ′|2(gn ◦ϕ)|ϕ′|2∂2 log gn∂ζ∂¯ζ◦ϕ= −12(gn ◦ϕ)∆log gn◦ϕ = K(µgn) ◦ϕ.Letting n →+∞and applying the dominated convergence theorem we get the assertion.

Holomorphic curvature of Finsler metrics and complex geodesics7The holomorphic curvature defined in this way is clearly invariant under holomorphicisometries. More generally, if f: M →N is holomorphic and F is a Finsler metric on N,we haveKFf(z); dfz(v)≥Kf ∗F (z; v),that is f ∗KF ≥Kf ∗F .When F is a honest smooth hermitian metric on M, KF (p; v) coincides with the usualholomorphic sectional curvature of F at (p; v) (see [Wu]).

The aim of this section is toextend a couple of results already known for hermitian metrics to this more general case.A piece of terminology: we say that a complex Finsler metric F has holomorphiccurvature bounded above (below) by a constant c ∈IR if KF (p; v) ≤c (respectively,KF (p; v) ≥c) for all (p; v) ∈T 1,0M with v ̸= 0.Our first result is the usual several variables version of Ahlfors’ lemma:Proposition 1.4: Let F be a complex Finsler metric on a complex manifold M. Assumethat the holomorphic curvature of F is bounded above by a negative constant −4a, forsome a > 0. Thenϕ∗F ≤µa(1.6)for all holomorphic maps ϕ: U →M.Proof : ϕ∗F is a pseudohermitian metric on U; by assumption (and by Lemma 1.3),K(ϕ∗F) ≤−4a.

Then the assertion follows from Proposition 1.1.As a consequence, we obtain a generalization of a well-known criterion of hyperbolicity:Corollary 1.5: Let M be a complex manifold admitting a (complete) complex Finslermetric F with holomorphic curvature bounded above by a negative constant. Then M is(complete) hyperbolic.Proof : Up to multiplying F by a suitable constant, we may assume KF ≤−4.

Let d denotethe distance induced by F on M, and ω the Poincar´e distance on U. Then Proposition 1.4yieldsdϕ(ζ1), ϕ(ζ2)≤ω(ζ1, ζ2),for all ζ1, ζ2 ∈U and holomorphic maps ϕ: U →M.But this immediately implies(cf.

[K1, Proposition IV.1.4]) that the Kobayashi distance kM of M is bounded below by d,and the assertion follows.In particular, then, a complex manifold admitting a complete complex Finsler metricwith holomorphic curvature bounded above by a negative constant is necessarily taut.The notion of holomorphic curvature for (non-smooth) Finsler metric has been intro-duced recently in connection with the Carath´eodory and Kobayashi metrics. In particular,Wong [W] and Suzuki [S] (see also [Bu]) have shown that the holomorphic curvature ofthe Carath´eodory metric is bounded above by −4, whereas the holomorphic curvature ofthe Kobayashi metric is bounded below by −4.An interesting immediate consequence of this is an interpretation in terms of curvatureof a well known property of the Carath´eodory metric:

8Marco Abate and Giorgio PatrizioProposition 1.6: Let F be a complex Finsler metric on a manifold M with holomorphiccurvature bounded above by −4. Let ϕ: U →M be a holomorphic map.

Then the followingare equivalent:(i) ϕ∗F(0; 1) = Fϕ(0); ϕ′(0)= 1, that is ϕ is an isometry at the origin between thePoincar´e metric on U and F;(ii) ϕ is an infinitesimal complex geodesic, that is ϕ∗F is the Poincar´e metric of U.Proof : By definition and Lemma 1.3, the Gaussian curvature of ϕ∗F is bounded aboveby −4. The assertion follows from Heins’ Theorem 1.2.Bounds on the holomorphic curvature allow to compare a complex Finsler metric tothe Kobayashi metric — and maybe to prove that a given Finsler metric actually is theKobayashi metric.

For instance, Pang [P] and Faran [F] gave conditions under which asmooth complex Finsler metric of constant negative holomorphic curvature coincides withthe Kobayashi metric. We shall discuss the smooth case in detail in the next two sections;here, to provide the right set-up to the problem, we examine a bit the general situation.We need an auxiliary notion to formulate our observation.

Let F be a complex Finslermetric on a manifold M, and take (p; v) ∈T 1,0M. We say that F is realizable at (p; v)if there is a holomorphic map ϕ: U →M such that ϕ(0) = p and λϕ′(0) = v with|λ| = F(p; v).

In other words, ϕ is an isometry at the origin between the Poincar´e metricof U and F.Obviously, the Kobayashi metric is realizable in any taut manifold; on the other hand,as a consequence of the next result, the Carath´eodory metric is realizable iffit coincideswith the Kobayashi metric.Proposition 1.7: Let F be a complex Finsler metric on a manifold M, and choose(p0; v0) ∈T 1,0M. Then:(i) If F is realizable at (p0; v0), then F(p0; v0) ≥κM(p0; v0);(ii) If KF ≤−4, then F ≤κM;(iii) If F is realizable at (p0; v0) and KF ≤−4, then F(p0; v0) = κM(p0; v0).Proof : (i) Let ϕ: U →M be a holomorphic map with ϕ(0) = p0 and v0 = λϕ′(0) suchthat F(p0; v0) = |λ|.

ThenF(p0; v0) = |λ| ≥κM(p0; v0). (ii) Take (p; v) ∈T 1,0M and let ϕ: U →M be a holomorphic map with ϕ(0) = pand v = λϕ′(0).

Then ϕ∗G is a pseudohermitian metric on U with Gaussian curvaturebounded above by −4 (by Lemma 1.3); it follows from Proposition 1.1 that ϕ∗G ≤µ1.ThusF(p; v) = Fϕ(0); λϕ′(0)= ϕ∗F(0; λ) ≤|λ|.Since this holds for all such ϕ, we get F ≤κM. (iii) Obvious, now.This is the best it can be done on the basis of Ahlfors’ lemma and Heins’ theoremonly.

In order to get deeper results it is necessary to use more tools — as we shall see inthe smooth case discussed in the rest of the paper.

Holomorphic curvature of Finsler metrics and complex geodesics92. Holomorphic curvature: the smooth caseIn this section we shall derive a tensor expression of the holomorphic curvature of a smoothcomplex Finsler metric.First of all, we need a few definitions, notations and general formulas.

Let F be acomplex Finsler metric on a complex manifold M, and set G = F 2, as usual. We shallassume that F is smooth, that is that F is of class Ck (k ≥4) out of the zero section ofT 1,0M.

By the way, (T 1,0M)0 will denote the complement in T 1,0M of the zero section.If (z1, . .

., zn) are local coordinates on M, a local section of T 1,0M will be written asnXj=1vj ∂∂zj ,and we shall use (z1, . .

., zn; v1, . .

., vn) as local coordinates on T 1,0M.We shall denote by indexes like α, ¯β and so on the derivatives with respect to thev-coordinates; for instance,Gα ¯β =∂2G∂vα∂vβ .On the other hand, the derivatives with respect to the z-coordinates will be denoted byindexes after a semicolon; for instance,G;ij =∂2G∂zi∂zjorGα;¯=∂2G∂zj∂vα .A smooth complex Finsler metric F will be said strongly pseudoconvex if the F-indicatrices are strongly pseudoconvexes, i.e., if the Levi matrix (Gα ¯β) is positive definiteon (T 1,0M)0. As usual in hermitian geometry, we shall denote by (Gα ¯β) the inverse matrixof (Gα ¯β), and we shall use it to raise indexes.

The usual summation convention will holdthroughout the rest of the paper.The main (actually, almost the unique) property of the function G is its (1,1)-homo-geneity: we haveG(z; λv) = λ¯λ G(z; v)(2.1)for all (z; v) ∈T 1,0M and λ ∈C. We now collect a number of formulas we shall use lateron which follows from (2.1).

First of all, differentiating with respect to vα and vβ we getGα(z; λv) = ¯λGα(z; v),Gα ¯β(z; λv) = Gα ¯β(z; v),Gαβ(z; λv) = (¯λ/λ)Gαβ(z; v).Thus differentiating with respect to λ or ¯λ and then setting λ = 1 we getGα ¯β vβ = Gα,Gαβ vβ = 0,(2.2)

10Marco Abate and Giorgio PatrizioandGαβγ vγ = −Gαβ,Gαβ¯γ vγ = Gαβ,Gα ¯βγ vγ = 0,(2.3)where everything is evaluated at (z; v).On the other hand, differentiating directly (2.1) with respect to λ or ¯λ and puttingeventually λ = 1 we getGα vα = G,Gαβ vαvβ = 0,Gα ¯β vαvβ = G.(2.4)It is clear that we may get other formulas applying any differential operator acting onlyon the z-coordinates, or just by conjugation. For instance, we getG¯α;i vα = G;i,(2.5)and so on.Assuming now F strongly pseudoconvex, we get another bunch of formulas we shallneed later on.

First of all, applying Gα ¯β to the first equation in (2.2) we getGα ¯βGα = vβ,(2.6)and thus, applying (2.5),G ¯β;iGα ¯βGα = G;i. (2.7)Recalling that (Gα ¯β) is the inverse matrix of (Gα ¯β), we may also compute derivativesof Gα ¯β:DGα ¯β = −Gα¯νGµ ¯β(DGµ¯ν),(2.8)where D denotes any first order linear differential operator.

As a consequence of (2.3)and (2.8) we getGα ¯β¯σ vσ = −Gα¯νGµ ¯βGµ¯ν¯σ vσ = 0,(2.9)and recalling also (2.6) we obtainG ¯βGα ¯βγ= −G ¯βGµ ¯βGα¯νGµ¯νγ = −Gα¯νGµ¯νγvµ = 0. (2.10)Now we may start to work.

Our first goal is to compute the holomorphic curvatureof our strongly pseudoconvex smooth complex Finsler metric F. SetS1,0M = {ξ ∈T 1,0M | F(ξ) = 1},and choose p ∈M and ξ ∈S1,0p M. To compute KF (p; ξ) we should write the Gaussiancurvature at the origin of ϕ∗G, where ϕ: U →M is any holomorphic map with ϕ(0) = pand v = ϕ′(0) = λξ, where |λ| = F(p; v) = [ϕ∗G(0; 1)]1/2.Writing ϕ∗G = g dζ ⊗d¯ζ, we haveg(ζ) = Gϕ(ζ); ϕ′(ζ),g(0) = |λ|2,

Holomorphic curvature of Finsler metrics and complex geodesics11andK(ϕ∗G)(0) = −12g(0)(∆log g)(0) = −2|λ|2∂2(log g)∂¯ζ∂ζ(0).The computation of the Laplacian yields∂2(log g)∂¯ζ∂ζ= −1G(ϕ; ϕ′)2G;i(ϕ; ϕ′)(ϕ′)i + Gα(ϕ; ϕ′)(ϕ′′)α2+1G(ϕ; ϕ′)nG;i¯(ϕ; ϕ′)(ϕ′)i(ϕ′)j + Gα ¯β(ϕ; ϕ′)(ϕ′′)α(ϕ′′)β+ 2 ReG¯α;i(ϕ; ϕ′)(ϕ′)i(ϕ′′)αo.Hence writing η = ϕ′′(0) we getK(ϕ∗G)(0) = −2G;i¯(p; ξ) −G;i(p; ξ)G;¯(p; ξ)ξiξj−2|λ|4Gα ¯β(p; ξ) −Gα(p; ξ)G ¯β(p; ξ)ηαηβ−4|λ|4 Reλ2G¯α;i(p; ξ) −G¯α(p; ξ)G;i(p; ξ)ξiηα. (2.11)We must compute the supremum (with respect to λ and η) of this formula.For themoment, let us consider λ fixed, and look for the infimum ofGα ¯β(p; ξ) −Gα(p; ξ)G ¯β(p; ξ)ηαηβ + 2 Reλ2G¯α;i(p; ξ) −G¯α(p; ξ)G;i(p; ξ)ξiηα,that is ofIλ(η) = Aα ¯β ηαηβ + 2 Reλ2B¯α;i ξiηα,(2.12)whereAα ¯β = Gα ¯β(p; ξ) −Gα(p; ξ)G ¯β(p; ξ),B¯α;i = G¯α;i(p; ξ) −G¯α(p; ξ)G;i(p; ξ).Let us study the hermitian form (Aα ¯β).

By assumption, the matrixGα ¯β(p; ξ)inducesa positive definite hermitian product on Cn; so we may decompose Cn accordingly as theorthogonal sum of Cξ and its orthogonal (Cξ)⊥. Since, by (2.4) and (2.5),Aα ¯β ηαξβ = Gα ¯β ηαξβ −GαG ¯β ηαξβ = Gαηα −Gαηα = 0,B¯α;i ξiξα = G¯α;i ξiξα −G¯αG;i ξiξα = G;iξi −G;iξi = 0,for every η ∈Cn, if we denote by˜η = η −Gα ¯β ηαξβξthe orthogonal projection of η into (Cξ)⊥, we get

12Marco Abate and Giorgio PatrizioLemma 2.1: Let F be a strongly pseudoconvex smooth complex Finsler metric on M,and take p ∈M and ξ ∈S1,0p M. Then Iλ ≡0 on Cξ and Iλ(˜η) = Iλ(η) for all η ∈Cn.So it suffices to study Iλ on (Cξ)⊥. Note that ˜η ∈(Cξ)⊥iff0 = Gα ¯β ˜ηαξβ = Gα˜ηα;therefore on (Cξ)⊥we haveAα ¯β ˜ηα = Gα ¯β ˜ηα.

(2.13)In particular (Aα ¯β) is positive definite on (Cξ)⊥.Thus Iλ is a quadratic polynomial on (Cξ)⊥with positive definite leading term; henceIλ attains a minimum at ˜η ∈(Cξ)⊥given byAα ¯β ˜ηα = −λ2B ¯β;i ξi,β = 1, . .

., n,that is, by (2.13),˜ηα = −λ2Gα ¯βB ¯β;i ξi,α = 1, . .

., n.(2.14)Putting (2.14) into (2.12) we find that the minimum of Iλ is−|λ|4Gα ¯βBα;¯ξjB ¯β;i ξi< 0,and thus (2.11) yieldsKF (p; ξ) = −2G;i¯−G;iG;¯−Gα ¯βBα;¯B ¯β;iξiξj.Now, using (2.4) and (2.6) we getGα ¯βBα;¯B ¯β;i = Gα ¯βGα;¯G ¯β;i −G;iG;¯;thereforeKF (p; ξ) = −2G;i¯−Gα ¯βGα;¯G ¯β;iξiξj. (2.15)It is easy to check (cf.

[Wu]) that when F is a standard hermitian metric on M, then (2.15)reduces to the usual holomorphic sectional curvature in the direction ξ. Furthermore, (2.15)exactly yields the holomorphic sectional curvature introduced by Kobayashi in [K2].There is a shorter way of writing KF .

SetΓα;i = Gα¯µG¯µ;i,and put Γα;i¯= (Γα;i);¯; then, by (2.8),Γα;i¯= Gα¯µG¯µ;i¯−Gα¯νGβ¯µGβ¯ν;¯G¯µ;i,and so, by (2.6) and (2.7),GαΓα;i¯= G;i¯−Gβ¯µGβ;¯G¯µ;i. (2.16)Summing up, we have proved the

Holomorphic curvature of Finsler metrics and complex geodesics13Proposition 2.2: Let F be a strongly pseudoconvex smooth complex Finsler metric onM, and take (p; ξ) ∈S1,0M. Then the holomorphic curvature of F in the direction of ξ isKF(p; ξ) = −2GαΓα;i¯ξiξj.

(2.17)For future reference, we note here that more generally the holomorphic curvature of Fin the direction of a non-zero vector v ∈T 1,0pM — which coincides with the holomorphiccurvature in the direction of ξ = v/F(p; v) — is given by the formulaKF (p; v) = −2G(p; v)2 Gα(p; v)Γα;i¯(p; v) vivj.(2.18)3. Holomorphic curvature and geodesic complex curvesLet F be a (smooth) complex Finsler metric on a manifold M. Let Ur denote the disk{ζ ∈C | |ζ| < r} in C (with 0 < r ≤1), endowed with the restriction of the Poincar´emetric of U; note that Ur is a convex subset of U with respect to the Poincar´e metric.A holomorphic map ϕ: Ur →M is a segment of infinitesimal complex geodesic if ϕ∗F isthe Poincar´e metric on Ur, that is if ϕ is a local isometry from the Poincar´e metric to F.On the other hand, ϕ is said segment of geodesic complex curve if the image via ϕ ofany (real) geodesic in Ur is a (real) geodesic for F in M. In other words, ϕ is a localisometry and ϕ(Ur) is a totally geodesic complex curve in M. When r = 1, we shall talkof infinitesimal complex geodesics and geodesic complex curves tout-court.

In any case, if(p; v) =ϕ(0); ϕ′(0)we say that ϕ is tangent to (p; v).In [AP] we showed that ϕ is a segment of geodesic complex curve iffit is a holomorphicsolution of the system(ϕ′′)α + A(ϕ′)α = −Γα;i(ϕ; ϕ′)(ϕ′)i,α = 1, . .

., n,(3.1)Gαβ(ϕ; ϕ′)(ϕ′′ + Aϕ′)β =Gi;α(ϕ; ϕ′) −Gα;i(ϕ; ϕ′)(ϕ′)i,α = 1, . .

., n,(3.2)where the prime stands for ∂/∂ζ, and A: U →C is the functionA(ζ) = −2¯ζ1 −|ζ|2 .As we shall see later on, the main amount of informations is contained in equa-tion (3.1). For the moment, however, let us discuss equation (3.2) a bit.Let ϕ: Ur →M be a holomorphic solution of equation (3.1).

Then, putting (3.1)into (3.2), we find that ϕ is a segment of geodesic complex curve iffGαβ(ϕ; ϕ′)Γβ;i(ϕ; ϕ′)(ϕ′)i =Gα;i(ϕ; ϕ′) −G;α(ϕ; ϕ′)(ϕ′)i,that is iffalong the curve ϕ we haveGi;α −Gα;i + GαβΓβ;ivi = 0,α = 1, . .

., n.(3.3)

14Marco Abate and Giorgio PatrizioIn a more symmetric way, following [Ru2] we may introduce the torsion tensorTαi¯µ = (Gi¯µ;α −Gi¯µβΓβ;α) −(Gα¯µ;i −Gα¯µβΓβ;i);it is a (3,0)-tensor defined on (T 1,0M)0. Then (3.3) is equivalent toTαi¯µ vµvi = 0,α = 1, .

. ., n.If G(p; v) = gα ¯β(p) vαvβ is a standard hermitian metric, then Gαβ ≡0 and (3.3)reduces to∂gα¯µ∂zi = ∂gi¯µ∂zα ,that is to the usual K¨ahler condition.

For this reason, a strongly pseudoconvex smoothcomplex Finsler metric satisfying (3.3) will be said K¨ahler. Summing up, we have provedProposition 3.1: Let F be a strongly pseudoconvex smooth complex Finsler metric ona manifold M. Then a holomorphic solution ϕ of (3.1) is a segment of geodesic complexcurve iffF is K¨ahler along ϕ.It is possible to write (3.3) in still another way.

Set Γαβ;i = (Γα;i)β; then, using (2.3),(2.6) and (2.8),Γαβ;i = −Gγ ¯µGα¯νGγ¯νβG¯µ;i + Gα¯µGβ¯µ;i,(3.4)and soGαΓαβ;i = Gβ;i −GβγΓγ;i,Gα¯µΓαβ;i = Gβ¯µ;i −Gβ¯µγΓγ;i,GαΓαi;βvi = Gi;βvi.ThenTαi¯µ = Gβ¯µ(Γβi;α −Γβα;i),(3.5)[Gi;α −Gα;i + GαβΓβ;i]vi = Gβ(Γβi;α −Γβα;i)vi,(3.6)and (3.3) is equivalent to Gβ(Γβi;α −Γβα;i)vi = 0 for α = 1, . .

., n. We remark that when Gis a hermitian metric thenΓαβ;i = gα¯µ ∂gβ¯µ∂zi ,and so they are the coefficients of the Cartan-Chern connection associated to the hermitianmetric. In particular, then, (3.5) shows that in this case T actually coincides with thetorsion tensor of the connection.But let us now return to equation (3.1) and holomorphic curvature.

We shall say thata holomorphic curve ϕ: Ur →M realizes the holomorphic curvature at 0 ifK(ϕ∗G)(0) = KFϕ(0); ϕ′(0).More generally, ϕ realizes the holomorphic curvature at ζ0 ∈Ur if ϕ ◦γζ0 realizes it at 0,whereγζ0(ζ) = ζ + ζ01 + ζ0ζis the unique automorphism of U sending the origin to ζ0 with positive derivative at 0.

Holomorphic curvature of Finsler metrics and complex geodesics15Proposition 3.2: Let F be a strongly pseudoconvex smooth complex Finsler metric ona manifold M, and let ϕ: Ur →M be a holomorphic solution of (3.1). Then(i) ϕ realizes the holomorphic curvature at every point of Ur;(ii) ifϕ(0); ϕ′(0)∈S1,0M, then ϕ is a segment of infinitesimal complex geodesic for F.Proof : (i) By Lemma 2.1 and (2.14), a holomorphic ϕ: Ur →M realizes the holomorphiccurvature at 0 iffηα = −λ2Gα ¯β(p; ξ)B ¯β;i(p; ξ) ξi + c ξα,α = 1, .

. ., n,(3.7)where p = ϕ(0), v = ϕ′(0) = λξ with ξ ∈S1,0p M, η = ϕ′′(0) and c ∈C.

Since, by (2.6), wehaveGα ¯βB ¯β;i = Γα;i −G;iξα,it follows that (3.7) is equivalent toηα = −Γα;i(p; v)vi + c1vα,α = 1, . .

., n,(3.8)with a possibly different c1 ∈C. But (3.8) with c1 = 0 is just (3.1) evaluated in 0; so aholomorphic solution of (3.1) realizes the holomorphic curvature at the origin.Now take ζ0 ∈Ur, and set ψ = ϕ ◦γζ0.

Thenψ(0) = ϕ(ζ0);ψ′(0) = (1 −|ζ0|2)ϕ′(ζ0);ψ′′(0) = (1 −|ζ0|2)2ϕ′′(ζ0) + A(ζ0)ϕ′(ζ0).So ϕ realizes the holomorphic curvature at ζ0 iffϕ′′(ζ0) + A(ζ0)ϕ′(ζ0)α = −Γα;iϕ(ζ0); ϕ′(ζ0)ϕ′(ζ0)i + c2ϕ′(ζ0)α,and (i) follows. (ii) Assume (3.1) holds.

Then, recalling (2.6), we getGα(ϕ; ϕ′)(ϕ′′)α + AG(ϕ; ϕ′) = Gα(ϕ; ϕ′)(ϕ′′)α + A(ϕ′)α= −Gα(ϕ; ϕ′)Γα;i(ϕ; ϕ′)(ϕ′)i = −G;i(ϕ; ϕ′)(ϕ′)i.Therefore∂∂ζG(ϕ; ϕ′)= G;i(ϕ; ϕ′)(ϕ′)i + Gα(ϕ; ϕ′)(ϕ′′)α = −AG(ϕ; ϕ′).Now, along the curve t 7→eiθt we have∂∂ζ = 12e−iθ ddt;therefore t 7→Gϕ(eiθt); ϕ′(eiθt)is a solution of the Cauchy problemf ′(t) =4t1 −t2 f(t),f(0) = 1.But f(t) = (1 −t2)−2 is a solution of the same problem; thereforeGϕ(eiθt); ϕ′(eiθt)≡1(1 −t2)2 ,and we are done.

16Marco Abate and Giorgio PatrizioSo the main point now is to find when (3.1) has a holomorphic solution. Assumeϕ: Ur →M is such a solution, and apply ∂/∂¯ζ to (3.1).

We get2(1 −|ζ|2)2 (ϕ′)α = Γα;i¯(ϕ; ϕ′)(ϕ′)i(ϕ′)j + Γα¯β;i(ϕ; ϕ′)(ϕ′)i(ϕ′′)β,(3.9)where, as before, Γα;i¯= (Γα;i);¯and Γα¯β;i = (Γα;i) ¯β.Now, (2.2) and (2.9) yieldΓα¯β;i vβ = Gα¯µ¯β G¯µ;i vβ + Gα¯µG ¯β¯µ;i vβ = 0. (3.10)So we can replace ϕ′′ by ϕ′′ + Aϕ′ in (3.9) and, calling in (3.1) again, we obtain2(1 −|ζ|2)2 (ϕ′)α = Γα;i¯(ϕ; ϕ′)(ϕ′)i(ϕ′)j −Γα¯β;i(ϕ; ϕ′)Γ¯β;¯(ϕ; ϕ′)(ϕ′)i(ϕ′)j.

(3.11)Now, ifϕ(0); ϕ′(0)∈S1,0M, then Proposition 3.2. (ii) yieldsGϕ(ζ); ϕ′(ζ)=1(1 −|ζ|2)2 ;therefore — setting v = ϕ′(ζ) — (3.11) becomesΓα;i¯−Γα¯β;iΓ¯β;¯vivj = 2G vα,α = 1, .

. ., n.(3.12)So (3.12) is a necessary condition for (3.1) to have a holomorphic solution.

The interestingfact is that it is sufficient too:Theorem 3.3: Let F be a strongly pseudoconvex smooth complex Finsler metric on amanifold M. Then the Cauchy problem (ϕ′′)α + A(ϕ′)α = −Γα;i(ϕ; ϕ′)(ϕ′)ifor α = 1, . .

., n,ϕ(0) = p,ϕ′(0) = v0,(3.13)admits a holomorphic solution for all (p; v0) ∈S1,0M iff(3.12) holds. Furthermore, thesolution, if exists, is unique.Proof : We have already proved one direction; so assume (3.12) holds.For any eiθ ∈S1, let consider the Cauchy problem( ¨g(t)α = −A(t)˙g(t)α −Γα;ig(t); ˙g(t)˙g(t)i,for α = 1, .

. ., n,g(0) = p,˙g(0) = eiθv0.

(3.14)The standard ODE theory provides us with an ε > 0 and uniquely determined mapsgeiθ: (−ε, ε) →M solving (3.14). Define ϕ: Uε →M byϕ(ζ) = gζ/|ζ|(|ζ|),(3.15)

Holomorphic curvature of Finsler metrics and complex geodesics17and assume for a moment that ϕ is holomorphic. Since, writing ζ = teiθ, we have∂∂ζ = −ie−iθ2t ∂∂θ + it ∂∂tand∂∂¯ζ = ieiθ2t ∂∂θ −it ∂∂t,(3.16)it follows that∂ϕ∂ζ (ζ) =¯ζ|ζ| ˙gζ/|ζ|(|ζ|),and thus ϕ is a holomorphic solution of (3.13).

In conclusion, we must prove that, assuming(3.12), the map ϕ defined by (3.15) is holomorphic. Note that, since a holomorphic map isuniquely determined by its restriction to the real axis, the uniqueness statement for (3.14)implies that ϕ is the unique possible holomorphic solution of (3.13).First of all, set f0(t) = tanh t andσθ(t) = geiθ(tanh t).

(3.17)Then˙σθ = (f ′0)(˙geiθ ◦f0)and¨σθ = (f ′0)2[(¨geiθ + A˙geiθ) ◦f0];so σθ satisfies ¨σαθ = −Γα;i(σθ; ˙σθ) ˙σiθ,for α = 1, . .

., n,σθ(0) = p,˙σθ(0) = eiθv0.Set h = G(σθ; ˙σθ). Thenh′ = G;i(σθ; ˙σθ) ˙σiθ + Gα(σθ; ˙σθ) ¨σαθ= G;i(σθ; ˙σθ) ˙σiθ + Gα ¯β(σθ; ˙σθ) ¨σαθ ˙σβθ= G;i(σθ; ˙σθ) ˙σiθ −G¯µ;i(σθ; ˙σθ) ˙σiθ ˙σµθ = 0.So h(t) ≡h(0) = 1, and the curve σθ lifts to a curve ˜σθ = dσθ = (σθ; ˙σθ) in S1,0M.Now, we define a global vector field X ∈ΓT 1,0(S1,0M)by settingX˜v = vi ∂∂zi −Γα;i vi ∂∂vα ,(3.18)where (z1, .

. ., zn; v1, .

. ., vn) are the local coordinates of ˜v ∈(T 1,0M)0.

It is not difficultto check that X is globally defined, and that for ˜v ∈S1,0M it is actually true thatX˜v ∈T 1,0˜v(S1,0M), as claimed.To proceed, we need to recall a basic fact of complex differential geometry. Let N be acomplex manifold, of complex dimension m. If we consider N with its real structure, thenTN is a (4m)-dimensional real vector bundle on N endowed with a complex structure J.If we denote by T cN its complexification, then T 1,0N is the i-eigenspace of J, and thecanonical isomorphism T 1,0N →TN is given byY 7→Y o = Y + Y ,

18Marco Abate and Giorgio Patriziowhere Y is the complex conjugate of Y in T cN. In particular, then,JY o = i(Y −Y ).The aim of this observation is that, by construction, ˜σθ is the integral curve in S1,0Mof the vector field Xo starting at (p; eiθv0).

If we denote by etXo the local one-parametergroup of diffeomorphisms induced by Xo on S1,0M, we may then write˜σθ(t) = etXo(eiθ˜v0)andσθ(t) = π(etXoeiθ˜v0),(3.19)where π: S1,0M →M is the canonical projection, and ˜v0 = (p; v0) ∈S1,0M.We need another vector field on S1,0M. The mapeiθ, (p; v)7→(p; eiθv) is a one-parameter group of diffeomorphisms of S1,0M; therefore it is induced by a vector field Z,namelyZ = ivα ∂∂vα ∈ΓT 1,0(S1,0M);note that π∗(Z) = 0.

Then (3.19) becomes˜σθ(t) = etXoeθZo ˜v0andσθ(t) = π(etXoeθZo ˜v0). (3.20)Now, we need to compute[Xo, JXo] = i[X + X, X −X] = −2i[X, X],and[Xo, Zo] = [X + X, Z + Z] = [X, Z]o + [X, Z]o.Using local coordinates we find[X, Z] = −iΓα;j vj ∂∂vα −ivβ ∂∂zβ −Γαβ;j vj ∂∂vα −Γα;β∂∂vα= −ivβ ∂∂zβ −Γα;j vj ∂∂vα= −iX,(3.21)because, by (2.2) and (2.9), Γαβ;j vβ = Γα;j.

It is clear by the definitions and (3.10) that[X, Z] = 0; finally,[X, X] = −hΓ¯α;¯hj −Γβ;jΓ¯αβ;¯hivjvh ∂∂vα +hΓα;h¯−Γ¯β;¯Γα¯β;hivjvh ∂∂vα= 2vα ∂∂vα −vα ∂∂vα= −2iZo,where we used (3.12) on S1,0M. So we get[Xo, JXo] = −4Zoand[Xo, Zo] = −JXo.

Holomorphic curvature of Finsler metrics and complex geodesics19Now fix τ > 0, and set ˜vτ = eτXo ˜v0. Putu(t) = etXo∗Zoe−tXo ˜vτ ∈T˜vτ (S1,0M).Thendudt (t) = ddtetXo∗Zoe−tXo ˜vτ= −etXo∗{LXoZo}e−tXo ˜vτ = etXo∗(JXo)e−tXo ˜vτ ,where LXo is the Lie derivative, andd2udt2 (t) = ddtetXo∗(JXo)e−tXo ˜vτ= −etXo∗{LXo(JXo)}e−tXo ˜vτ = 4etXo∗Zoe−tXo ˜vτ .In other words, u(t) is a solution of the Cauchy problem ¨u = 4u,u(0) = Zo˜vτ ,˙u(0) = (JXo)˜vτ .Thereforeu(t) = 14e2t2Zo˜vτ + (JXo)˜vτ+ 14e−2t2Zo˜vτ −(JXo)˜vτ,and, in particular,π∗eτXo∗Zo˜v0 = π∗u(τ) = e2τ −e−2τ4π∗(JXo)˜vτ = e2τ −e−2τ4Jπ∗Xo˜vτ .

(3.22)We are almost done. Recalling (3.15), (3.16), (3.17) and (3.20), it is clear that weshould prove that∂∂θπe(atanh t)XoeθZo ˜v0θ=0= tJ ∂∂tπe(atanh t)XoeθZo ˜v0θ=0,where we may take θ = 0 because ˜v0 is generic.

Let us compute; using (3.22) we get∂∂θπe(atanh t)XoeθZo ˜v0θ=0= π∗e(atanh t)Xo∗Zo˜v0 =t1 −t2 Jπ∗Xo˜vatanh t,whereas∂∂tπe(atanh t)XoeθZo ˜v0θ=0= ∂∂tπe(atanh t)Xo ˜v0=11 −t2 π∗Xo˜vatanh t,and the proof is complete.So we have found a necessary and sufficient condition for the existence of segments ofgeodesic complex curves:

20Marco Abate and Giorgio PatrizioCorollary 3.4: Let F be a strongly pseudoconvex smooth complex Finsler metric on amanifold M. Then:(i) if (3.12) holds, then for any (p; ξ) ∈S1,0M there is a segment of infinitesimal complexgeodesic tangent to (p; ξ);(ii) there exists a (unique) segment of geodesic complex curve tangent to (p; ξ) for any(p; ξ) ∈S1,0M iffF is K¨ahler and (3.12) holds.Proof : (i) Theorem 3.3 and Proposition 3.2.(ii). (ii) In [AP] it is shown that a segment of geodesic complex curve is a holomorphicsolution of the system (3.1)–(3.2).The assertion then follows from Theorem 3.3 andProposition 3.1.A natural question now is whether the completeness of the metric F — together withK¨ahler and (3.12) — would imply the existence of geodesic complex curves defined onthe whole unit disk U.

The answer is positive, but for the proof we beforehand need adiscussion of the geometrical meaning of (3.12).Thanks to Proposition 3.2. (i), we know that a holomorphic solution of (3.1) realizes theholomorphic curvature and it is an isometry from the Poincar´e metric to F; in particular,thus, the holomorphic curvature along the curve should be −4.

This suggests to look fora connection between (3.12) and the holomorphic curvature; and indeed the next resultshows that the connection is provided by a sort of simmetry condition on the curvature.Analogously to the tensor Tαi¯µ previously introduced, setHαi¯µ¯= Gτ ¯µ(Γτi;α −Γτα;i);¯+ Gτ ¯µiΓτ;α¯−Gτ ¯µαΓτ;i¯= (Gτ ¯µΓτ;α¯)i −(Gτ ¯µΓτ;i¯)α;it is a (4,0)-tensor on (T 1,0M)0. Note thatHαi¯µ¯vµvivj =Gτ(Γτi;α −Γτα;i);¯−GταΓτ;i¯vivj.

(3.23)ThenTheorem 3.5: Let F be a strongly pseudoconvex smooth complex Finsler metric on amanifold M. Then (3.12) holds iffKF ≡−4 andHαi¯µ¯vµvivj = 0,α = 1, . .

., n.(3.24)Proof : We start by showing that (3.12) implies KF ≡−4. Indeed, take (p; ξ) ∈S1,0M.Then, recalling (2.2), (2.6), and (2.10), we getGαΓα¯β;i = GαGα¯µ¯β G¯µ;i + GαGα¯µG¯µ ¯β;i = 0.Therefore (3.12) yieldsKF(p; ξ) = −2GαΓα;i¯ξiξj = −4Gαξα −GαΓα¯β;iΓ¯β¯ξiξj = −4Gαξα = −4.

Holomorphic curvature of Finsler metrics and complex geodesics21From now on we shall assume KF ≡−4; we ought to prove that in this case (3.12) isequivalent to (3.24), that is, by (3.23), toGτ(Γτi;α −Γτα;i);¯vivj = GταΓτ;i¯vivj,α = 1, . .

., n.(3.25)By (2.18), KF ≡−4 is equivalent toGβΓβ;i¯vivj = 2G2.Differentiating with respect to vν we get4GG¯ν =Gβ¯νΓβ;i¯+ GβΓβ¯ν;i¯vivj + GβΓβ;i¯ν vi;multiplying by Gα¯ν, and recalling (2.6), we obtain4G vα =Γα;i¯+ Gα¯νGβΓβ¯ν;i¯vivj + Gα¯νGβΓβ;i¯νvi. (3.26)Now,Γβ;i¯= Gβ¯µG¯µ;i¯−Gβ¯τGσ¯τ;¯Γσ;i;(3.27)Γβ¯ν;i¯= Gβ¯µ¯ν G¯µ;i¯+ Gβ¯µG¯µ¯ν;i¯−Gβ¯τ¯ν Gσ¯τ;¯Γσ;i −Gβ¯τGσ¯τ ¯ν;¯Γσ;i −Gβ¯τGσ¯τ;¯Γσ¯ν;i.Therefore, using (2.2), (2.3), (2.4), (2.6), (2.8) and (2.10), we getGα¯νGβΓβ¯ν;i¯= −Gα¯νGσ;¯Γσ¯ν;i = −Gα¯νGσ;¯(Gσ¯µ¯ν G¯µ;i + Gσ¯µG¯µ¯ν;i)= −Γ¯µ;¯Gα¯νG¯µ¯ν;i + Γ¯γ;¯Gα¯νGδ¯µGδ¯γ¯νG¯µ;i= −Γ¯µ;¯(Gα¯νG¯µ¯ν;i + Gα¯ν¯µ G¯ν;i) = −Γα¯µ;iΓ¯µ¯.So (3.26) becomes4G vα =Γα;i¯−Γα¯µ;iΓ¯µ;¯vivj + Gα¯νGβΓβ;i¯ν vi.

(3.28)Now, in (2.16) we showed thatGα¯νGβΓβ;i¯ν vi = Gα¯νG;i¯ν −Γ¯τ;¯G¯τ;ivi.Since, by (2.4) and (2.9),Γ¯τ¯;¯ν vj = Gσ¯τGσ¯;¯ν vj = Γ¯τ;¯ν,we getGα¯νGβΓβ;i¯ν vi = Gα¯νG¯;i¯ν −G¯τ;iΓ¯τ¯;¯νvivj= Gα¯νG¯ν;i¯−G¯τ;iΓ¯τ¯ν;¯vivj + Gα¯ν(G¯;¯ν −G¯ν;¯);i −G¯τ;i(Γ¯τ¯;¯ν −Γ¯τ¯ν;¯)vivj.

22Marco Abate and Giorgio PatrizioNow (2.8) yieldsGα¯νG¯τ;iΓ¯τ¯ν;¯= Gα¯νGσ¯ν;¯Γσ;i −Gα¯νGδ¯µ¯νΓ¯µ;¯Γδ;i= Gα¯νGσ¯ν;¯Γσ;i + Γα¯µ;iΓ¯µ;¯−Gα¯νG¯µ¯ν;iΓ¯µ;¯;so, by (3.27),Gα¯ν(G¯ν;i¯−G¯τ;iΓ¯τ¯ν;¯) = (Γα;i¯−Γα¯µ;iΓ¯µ;¯) + Gα¯νG¯µ¯ν;iΓ¯µ;¯.Summing up, we have foundGα¯νGβΓβ;i¯νvi =Γα;i¯−Γα¯µ;iΓ¯µ;¯vivj+ Gα¯νG¯τ ¯ν;iΓ¯τ;¯+ (G¯;¯ν −G¯ν;¯);i −G¯τ;i(Γ¯τ¯;¯ν −Γ¯τ¯ν;¯)vivj.For the moment, setT¯ν = [G¯;¯ν −G¯ν;¯+ G¯τ ¯νΓ¯τ;¯vj = T¯ν¯µ vµvj;recall that (3.6) says thatT¯ν = G¯τ(Γ¯τ¯;¯ν −Γ¯τ¯ν;¯)vj.Therefore(G¯;¯ν −G¯ν;¯);i vj = (T¯ν);i −(G¯τ ¯νΓ¯τ;¯);i vj = (T¯ν);i −G¯τ ¯ν;iΓ¯τ;¯+ G¯τ ¯νΓ¯τ;¯ivj;G¯τ;i(Γ¯τ¯;¯ν −Γ¯τ¯ν;¯)vj = (T¯ν);i −G¯τ(Γ¯τ¯;¯ν −Γ¯τ¯ν;¯);i vj.In conclusion, we have shown thatGα¯νGβΓβ;i¯νvi =Γα;i¯−Γα¯µ;iΓ¯µ;¯vivj + Gα¯νG¯τ(Γ¯τ¯;¯ν −Γ¯τ¯ν;¯);i −G¯ν¯τΓ¯τ;¯ivivj.Recalling (3.28), we have obtained4G vα = 2Γα;i¯−Γα¯µ;iΓ¯µ;¯vivj + Gα¯νG¯τ(Γ¯τ¯;¯ν −Γ¯τ¯ν;¯);i −G¯ν¯τΓ¯τ;¯ivivj,and the assertion follows.If G(p; v) = gµ¯ν(p) vµvν is an hermitian metric, then the tensor Hαi¯µ¯becomesHαi¯µ¯= gτ ¯µ∂∂zj T τiα = Riα¯µ¯−Rαi¯µ¯,where T τiα is the torsion of the Chern connection associated to the hermitian metric, andRαi¯µ¯is the Riemannian curvature tensor of the connection. So (3.24) is equivalent toRiα¯µ¯vivµvj = Rαi¯µ¯vivµvj(3.29)for all v ∈T 1,0M.

So (3.24) may be interpreted as a simmetry condition on a curvaturetensor; more precisely, as anticipated in the introduction, a simmetry condition on theChern connection induced by the Finsler metric on the vertical subbundle of T 1,0T 1,0M.Finally, we also remark that — at least in the hermitian case — (3.24) in particular holdswhen ∂T ≡0.Now that we have an idea of the geometrical meaning of (3.12), we may return to thestudy of geodesic complex curves. As anticipated, we are now able to prove the following

Holomorphic curvature of Finsler metrics and complex geodesics23Theorem 3.6: Let F be a strongly pseudoconvex smooth complete complex Finsler metricon a manifold M. Assume that the holomorphic curvature of F is identically −4 andthat (3.24) holds. Then for every (p; ξ) ∈S1,0M there is a unique holomorphic solutionϕ: U →M of (3.1) defined on the whole unit disk U such that ϕ(0) = p and ϕ′(0) = ξ.Proof : First of all, we remark that the distribution CXo ⊕CZo ⊂T(S1,0M) is involutive.Indeed, Theorem 3.5 shows that (3.12) holds, and in the proof of Theorem 3.3 we havealready computed [Xo, JXo] = −4Zo and [Xo, Zo] = −JXo.

For the remaining brackets,using (3.21) we get[Xo, JZo] = Xo,[JXo, Zo] = Xo,[JXo, JZo] = JXo,[Zo, JZo] = 0.Let eN denote the integral leaf of this distribution passing through (p; ξ). From the proofof Theorem 3.3 it follows that N = π( eN) ⊂M is a Riemann surface locally parametrizedby the holomorphic solutions of (3.1).In particular, F restricted to N is a completehermitian metric of constant Gaussian curvature −4, because of Proposition 3.2.

Thusthere is a unique holomorphic covering map ψ: U →N which is an isometry between thePoincar´e metric on U and F restricted to N and such that ψ(0) = p and ψ′(0) = ξ. But ifϕ: Uε →N is the holomorphic solution of (3.1) with ϕ(0) = p and ϕ′(0) = ξ, then ϕ too isan isometry between the Poincar´e metric restricted to Uε and F restricted to N; it followsthat ϕ = ψ|Uε, and ψ is the extension of ϕ to the whole U we were looking for.Corollary 3.7: Let F be a strongly pseudoconvex smooth complete complex Finsler met-ric on a manifold M. Assume that the holomorphic curvature of F is identically −4 andthat (3.24) holds.

Then:(i) for any (p; ξ) ∈S1,0M there is an infinitesimal complex geodesic tangent to (p; ξ);(ii) if moreover F is K¨ahler, then for any (p; ξ) ∈S1,0M there is a unique geodesic complexcurve tangent to (p; ξ).Proof : Theorems 3.5, 3.6 and Propositions 3.1 and 3.2.Actually, we can even get a sort of punctual version of the latter result. As usual, weneed a computation, which by the way clarifies the relationship among (3.12) and the twotorsion tensors we introduced, T and H. For the sake of simplicity, setΣα;i¯= Γα;i¯−Γα¯µ;iΓ¯µ;¯.In particular, (3.12) becomes Σα;i¯vivj = 2G vα.Proposition 3.8: Let F be a strongly pseudoconvex smooth complex Finsler metric ona manifold M. ThenHαi¯µ¯vµ vj = XTαi¯µ vµ+GiσΣσ;α¯−GασΣσ;i¯vj,for all i, α = 1, .

. ., n, where X is the complex conjugate of the vector field defined in (3.18).In particular,Hαi¯µ¯vµvivj = X(Tαi¯µ vµvi) −GασΣσ;i¯vivj,α = 1, .

. ., n.

24Marco Abate and Giorgio PatrizioProof : By (3.5) and (3.18)Tαi¯µ vµ = Gβ(Γβi;α −Γβα;i),X = vj ∂∂zj −Γ¯γ;¯vj ∂∂vγ .ThusX(Tαi¯µ vµ) =Gβ(Γβi;α −Γβα;i);¯−Γ¯γ;¯(Γβi;α −Γβα;i)¯γvj+ [Gβ;¯−Gβ¯γΓ¯γ;¯](Γβi;α −Γβα;i)vj. (3.30)Now, Gβ¯γΓ¯γ;¯= Gβ;¯, and so the second addendum in (3.30) vanishes.

Next, recalling (3.4)and the usual formulas,Gβ(Γβi;α −Γβα;i)¯γ = −[GiσΓσ¯γ;α −GασΓσ¯γ;i].ThereforeX(Tαi¯µ vµ) = Gβ(Γβi;α −Γβα;i);¯vj + [GiσΓσ¯γ;αΓ¯γ;¯−GασΓσ¯γ;iΓ¯γ;¯]vj= Gβ(Γβi;α −Γβα;i);¯vj −[GiσΣσ;α¯−GασΣσ;i¯]vj + (GiσΓσ;α¯−GασΓσ;i¯)vj= Hαi¯µ¯vµ vj −[GiσΣσ;α¯−GασΣσ;i¯]vj,and the assertion follows.As a corollary we haveCorollary 3.9: Let F be a strongly pseudoconvex smooth complex Finsler metric on amanifold M. Assume that the holomorphic curvature of F is identically −4 and that (3.24)holds. Take (p0; ξ0) ∈S1,0M.

Then there is a segment of geodesic complex curve tangentto (p0; ξ0) iffF is K¨ahler at (p0; ξ0), that is iffTαi¯µ(p0; ξ0) ξi0ξµ0 = 0,α = 1, . .

., n.(3.31)The segment, if exists, is unique.Furthermore, if F is complete then the segment ofgeodesic complex curve actually extends to a whole geodesic complex curve.Proof : One direction is known. Conversely, assume (3.31) holds.

By Theorem 3.5,GασΣσ;i¯vivj = 0;hence, by Proposition 3.8, XTαi¯µ vivµ= 0. So Tαi¯µ vivµ is constant (and thus zero) alongthe solution of (3.1) tangent to (p0; ξ0); the assertion then follows from Proposition 3.1.As already discussed in the introduction, one of the motivations behind this workwas to find a differential description of the properties of the Kobayashi metric in stronglyconvex domains.

We conclude then with the following:Corollary 3.10: Let F be a strongly pseudoconvex smooth complete complex Finslermetric on a manifold M. Assume that (3.12) holds or, equivalently, that KF ≡−4 and(3.24) holds. Then F is the Kobayashi metric of M.Proof : By Theorem 3.5, in both cases the holomorphic curvature of F is −4.

Furthermore,by Theorem 3.6 F is realizable. The assertion then follows from Proposition 1.7.

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22 (1973),1103–1108.Marco AbateDipartimento di MatematicaSeconda Universit`a di Roma00133 Roma, ItalyGiorgio PatrizioDipartimento di MatematicaSeconda Universit`a di Roma00133 Roma, ItalyJune 1992

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