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GENERALIZATIONS OF SCHWARZ LEMMA CHEN JIANBO The final year report of Math 2999 under the guidance of Professor Ngaiming Mok A. In Section 1, we introduce some background knowledge of complex geom- etry. In Section 2, classical Schwarz lemma and its interpretation is discussed. In Section 3, we study the Ahlfors-Schwarz’s lemma and its generalization to holomorphic maps between the unit disk and K¨ ahler manifolds with holomorphic sectional curva- ture bounded from above by a negative constant. In Section 4, we focus on the case when equality holds at a certain point is discussed for holomorphic maps between the unit disk and classical bounded symmetric domains of type I, II and III. In Section 5, two higher-dimensional generalizations of the Ahlfors-Schwarz lemma for holomorphic maps from a compact K¨ ahler manifold to another K¨ ahler manifold, both of which sat- isfy respective conditions on curvature, are studied. In Section 6, we investigate two applications of various versions of Ahlfors-Schwarz lemma. C 1. Preliminaries: fundamentals of Hermitian, K ¨ ahler and Bergman manifolds 2 1.1. Hermitian and K¨ ahler manifolds 2 1.2. The Hermitian connection and its curvature 2 1.3. The Bergman kernel and the Bergman metric on bounded domains 3 2. Schwarz lemma and its interpretation 4 3. Ahlfors-Schwarz’s Lemma and its generalization 5 4. Holomorphic maps between the unit disk and the classical bounded symmetric domains 7 4.1. Classical bounded symmetric domains of type I 7 4.2. Classical bounded symmetric domains of type III 11 4.3. Classical bounded symmetric domains of type II 12 5. Holomorphic maps between K¨ ahler manifolds 15 5.1. A Schwarz lemma for metrics 15 5.2. A Schwarz lemma for volume elements 16 6. Two applications 18 6.1. Kobayashi metric 18 6.2. Normal families 19 References 20 1

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Page 1: GENERALIZATIONS OF SCHWARZ LEMMAjianbo/schwarz.pdf · GENERALIZATIONS OF SCHWARZ LEMMA CHEN JIANBO The final year report of Math 2999 under the guidance of Professor Ngaiming Mok

GENERALIZATIONS OF SCHWARZ LEMMA

CHEN JIANBO

The final year report of Math 2999 under the guidance of Professor Ngaiming Mok

Abstract. In Section 1, we introduce some background knowledge of complex geom-etry. In Section 2, classical Schwarz lemma and its interpretation is discussed. InSection 3, we study the Ahlfors-Schwarz’s lemma and its generalization to holomorphicmaps between the unit disk and Kahler manifolds with holomorphic sectional curva-ture bounded from above by a negative constant. In Section 4, we focus on the casewhen equality holds at a certain point is discussed for holomorphic maps between theunit disk and classical bounded symmetric domains of type I, II and III. In Section 5,two higher-dimensional generalizations of the Ahlfors-Schwarz lemma for holomorphicmaps from a compact Kahler manifold to another Kahler manifold, both of which sat-isfy respective conditions on curvature, are studied. In Section 6, we investigate twoapplications of various versions of Ahlfors-Schwarz lemma.

Contents1. Preliminaries: fundamentals of Hermitian, Kahler and Bergman manifolds 21.1. Hermitian and Kahler manifolds 21.2. The Hermitian connection and its curvature 21.3. The Bergman kernel and the Bergman metric on bounded domains 32. Schwarz lemma and its interpretation 43. Ahlfors-Schwarz’s Lemma and its generalization 54. Holomorphic maps between the unit disk and the classical bounded

symmetric domains 74.1. Classical bounded symmetric domains of type I 74.2. Classical bounded symmetric domains of type III 114.3. Classical bounded symmetric domains of type II 125. Holomorphic maps between Kahler manifolds 155.1. A Schwarz lemma for metrics 155.2. A Schwarz lemma for volume elements 166. Two applications 186.1. Kobayashi metric 186.2. Normal families 19References 20

1

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GENERALIZATIONS OF SCHWARZ LEMMA 2

1. Preliminaries: fundamentals of Hermitian, Kahler and Bergman manifoldsThe purpose of this section is to review some well-known concepts and results. Most

of these results are taken from [1].1.1. Hermitian and Kahler manifolds.Definition 1.1. A Hermitian metric g on a complex manifold X is a J-invariant Rie-mannian metric on the underlying smooth manifold X, i.e., g satisfies g(Ju, Jv) =g(u, v) for real tangent vectors u and v, where J is the almost complex structure of TC

X ,the complexified tangent bundle of X.Definition 1.2. A Hermitian manifold (X, g) is said to be Kahler if and only if thetypes of complexified tangent vectors are preserved under parallel transport.Theorem 1.3. Let (X, g) be a Hermitian manifold such that g is given by 2Re(

∑gijdz

i⊗dzj) in local holomorphic coordinates (zj). Then, (X, g) is a Kahler manifold if and onlyif at every point P ∈ X, there exists complex geodesic coordinates (zj) in the sense thatthe Hermitian metric g is represented by the Hermitian matrix (gij) satisfying gij(P ) =δij and dgij(P ) = 0.1.2. The Hermitian connection and its curvature.Definition 1.4. Let X be a complex manifold and V be a holomorphic vector bundleover X with a Hermitian metric h on V . Let Γ(X) denote the set of smooth sectionsof V over open sets U . Let Λ(X) denote the set of smooth vector fields on X over U . Aconnection is a map D : Λ(X)× Γ(X)→ Γ(X) : (ξ, s)→ Dξs satisfying:

(1) D is complex linear in both ξ and s.(2) D satisfies the product rule Dξ(fs) = ξ(f)s + fDξs for smooth function f over

U .A connection D on V is said to be a complex connection if and only if for any local

holomorphic section σ and any tangent vector η of type (0, 1) in the domain of definitionof η, Dησ = 0. D is said to be a metric connection if and only if it is compatible withthe Hermitian metric h; i.e., for any open set U , any real tangent vector field v on U ,and for any two smooth sections s and t over U , v(s, t) = (Dvt, s) + (t,Dvs). We remarkthat the requirement that D be complex is consistent with the product rule since thetransition functions for V are holomorphic.

We shall define a complex metric connection on (V, h). Let U be a coordinate openset on X with holomorphic local coordinates (zi) such that V is holomorphically triv-ial over U . Let eα be a holomorphic basis of V |U and write s = sαeα. Let η = ηi ∂

∂z

be a smooth vector field of type (1, 0) over U . It suffices to define Dieα = Γγiαeγ whereDi = D∂/∂zi. The requirement that D be complex and metric determines (Γγiα) uniquelyby ∂i(eα, eβ) = (Dieα, eβ) + (eα, Dieβ) = (Dieα, eβ). Write (eα, eβ) = hαβ, giving

Γγiα = hγβ · ∂ihαβ,

where hγβ is the conjugate inverse of hαβ.Definition 1.5. The Hermitian connection of (V, h) is the unique complex metric con-nection D on (V, h).

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GENERALIZATIONS OF SCHWARZ LEMMA 3

Let (X, g) be a Hermitian manifold. The restriction of the Hermitian metric (., .) =g(., .) to T 1,0

X defines a Hermitian metric on T 1,0X . By conjugation the connectionD on T 1,0

X

extends to a connection on TCX = T 1,0

X ⊗ T0,1X . On the other hand, g, as a Riemannian

metric on the underlying smooth manifold X, gives a Riemannian connection ∇ on(X, g), which extends to the complexified tangent bundle TC

X .Theorem 1.6. The Hermitian connection D agrees with the Riemannian connection∇if and only if (X, g) is Kahler.

Consider now a Kahler manifold (X, g). Let R be the curvature tensor of the Rie-mannian manifold (X, g). The sectional curvature K(u, v) is given by

K(u, v) =R(u, v; v, u)

‖u ∧ v‖2.

With J denoting the almost complex structure of (X, g), we call K(u, Ju) the holomor-phic sectional curvature of the J-invariant real 2-plane generated by u. We extend thecurvature tensorR by complex linearity in the 4 variables to TC

X . Choosing the complexgeodesic coordinates adapted to g at P , we can express the curvature tensor in termsof the basis ∂

∂zi, ∂∂zi1≤i≤n of TC

X as:Rαβij = −∂i∂jgαβ,

where ∂i ≡ ∂∂zi

, ∂j ≡ ∂∂zj

.

We can identify T 1,0X with the real tangent bundle TR

X via ξ → 2Reξ. Writing ξ = ξi∂iand u = 2Reξ = ξ + ξ, we also call R(u,Ju;Ju,u)

‖u‖4 the holomorphic sectional curvature inthe direction of ξ. We have

R(u, Ju; Ju, u) = 4Rijklξiξjξkξl.

We define the notion of holomorphic bisectional curvature and the notion of the Riccicurvature form.Definition 1.7. Let (X, g) be a Kahler manifold, x ∈ X an arbitrary point and ξ, η ∈T 1,0X . Write u = 2Reξ and v = 2Reη. We define the holomorphic bisectional curvature

in the directions (ξ, η) to be R(u,Ju;Jv,v)‖u‖2‖v‖2 .

Definition 1.8. Let (X, g) be a Kahler manifold, x ∈ X an arbitrary point. The Riccicurvature form of (X, g) is Ric =

√−1Rijdz

i ∧ dzj, where Rij = gklRijkl.Remark 1.9. Let (X, g) be a Kahler manifold, x ∈ X an arbitrary point. The volumeelement of (X, g) is given by VM = in det(gαβ)dz1 ∧ dz1 ∧ · · · ∧ dzn ∧ dzn, and thereforethe Ricci curvature form has the expression

Rαβ = −∂α∂β log det(gαβ).

1.3. The Bergman kernel and the Bergman metric on bounded domains. Nowwe introduce a special family of Kahler manifolds. Let Ω b Cn be an arbitrary boundeddomain on a Euclidean space. Let L2(Ω) be the Hilbert space of square-integrablefunctions with respect to the Lebesgue measure dλ and H2(Ω) ⊂ L2(Ω) be the spaceof square-integrable holomorphic functions. By Montel’s Theorem, H2(Ω) is also a

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GENERALIZATIONS OF SCHWARZ LEMMA 4

Hilbert space. Let fi0≤i<∞ be an orthonormal basis ofH2(Ω). We define the Bergmankernel K(z, w) by

K(z, w) =∑

fi(z)fi(w).

Theorem 1.10. The infinite sumK(z, w) converges uniformly on compact sets to a func-tion jointly real-analytic in the variables (z, w), holomorphic in z and anti-holomorphicin w. K(z, w) is square-integrable in the variables (z, w) separately, is independent ofthe choice of the holomorphic basis fi0≤i<∞ and possesses a reproducing property forsquare-integrable holomorphic functions f given by

f(z) =

∫Ω

K(z, w)f(w)dλ(w).

We are going to define a Kahler metric from the Bergman kernel. Writing ω =√−1∂∂ logK(z, z), one can show ω(−iv ∧ v) > 0 for any non-zero tangent vector v of

type (1, 0). Therefore, we have

Definition 1.11. Let Ω b Cn be a bounded domain and K(z, w) be the BergmanKernel on Ω. Write φ(z) = logK(z, z). The Bergman metric is the Kahler metricg = 2Re

∑gαβdz

α ⊗ dzβ. In other words, it is the Kahler metric with Kahler form√−1∂∂φ.

Theorem 1.12. The Bergman metric is invariant under the automorphism group of Ω.

2. Schwarz lemma and its interpretationThe classical Schwarz lemma states that every holomorphic mapping from a unit

disk into itself is distance-decreasing with respect to the Poincare metric. We shallsee why it can be interpreted in this way.

Theorem 2.1 (Schwarz Lemma). Let f : D → D be a holomorphic function leaving theorigin fixed. Then |f(z)| ≤ |z| for all z ∈ D, and if equality holds for any nonzero point,then f(z) = cz for some c of modulus 1. Also we have |f ′(0)| ≤ 1 and if equality holds,then f(z) = cz for some c of modulus 1.

We can use Schwarz lemma to obtain

Aut(D) = f : D → D|f(z) = eiαz − a1− az

, where a is complex, |a| < 1,and 0 ≤ α ≤ 2π.

In fact, suppose f is an automorphism that fixes the origin. Then by applying Schwarzlemma to f and its inverse, we get that f(z)

zis of constant modulus and thus f(z) = cz

for some c of modulus 1. Suppose h is an arbitrary automorphism with h−1(0) = a.Because g(z) = (z − a)/(1 − az) is an automorphism, h g−1 is also an automorphismand it fixes the origin. Therefore h g−1 = eiα from the previous argument and henceh has the form h(z) = eiα z−a

1−az . The converse is obvious.

Aut(D) acts transitively onD via Mobius transformation Ψ(z) = eiα z−a1−az . We can there-

fore derive the Bergman metricKD(z, w) onD by the invariance of the Bergman metricunder Aut(D). In fact, normalizing the metric at the origin, for every point a ∈ D, we

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GENERALIZATIONS OF SCHWARZ LEMMA 5

have g(a) = 2Ref ∗(dz ⊗ dz) where f(z) = z−a1−az . A direct computation yields that

g = 2Redz ⊗ dz

(1− |z|2)2,

which we call the Poincare metric on the unit disk.

Remark 2.2. Consider the disk Dr of radius r. The metric gr = 2Re r2dz⊗dz(r2−|z|2)2

is thenormalized Bergman metric (the Poincare metric) on Dr.

The classical version of Schwarz lemma applies to a holomorphic map that fixes theorigin of the unit disk. Can we take away this condition and express the result in someinvariant form under the automorphism group? This consideration yields the follow-ing Schwarz-Pick Theorem, which is proved by transporting an arbitrary point z0 ∈ Dand its image f(z0) under f to the origin by two linear fractional transformations re-spectively and then applying the Schwarz lemma.

Theorem 2.3 (Schwarz-Pick Theorem). Let f : D → D be a holomorphic function.Then

|f ′(z)| ≤ 1− |f(z)|2

1− |z|2.

If f ∈ Aut(D), then the equality holds at any point, otherwise, there is strict inequalityfor all z ∈ D.

From a differential geometric viewpoint, we can restate Schwarz-Pick Theorem asfollows:

Theorem 2.4. Let D be the unit disk with the Poincare metric g. Then every holomor-phic mapping f : D → D is distance-decreasing. In other words, it satisfies f ∗g ≤ g,and if the equality holds at a single point, then f is an automorphism.

3. Ahlfors-Schwarz’s Lemma and its generalizationAhlfors [4] established a beautiful connection between curvature and holomorphic

maps. Before stating the result, we remark here that the Gaussian curvature k of ametric g = 2Rehdz ⊗ dz on a Riemann surface is given by

k = −1

h

∂2 log h

∂z∂z.

In particular, the Gaussian curvature of the Poincare metric g = 2Re dz⊗dz(1−|z|2)2

on theunit disk D is k = −2.

Theorem 3.1 (Ahlfors-Schwarz’s Lemma). Let f : D → N be a holomorphic mapping.If N is a Riemann surface equipped with a Kahler metric h with curvature boundedfrom above by a negative number −K, then

f ∗h2 ≤ 2

Kg,

where g is the Poincare metric of the unit disc D.

We can generalize Ahlfors-Schwarz’s lemma in the following form:

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GENERALIZATIONS OF SCHWARZ LEMMA 6

Theorem 3.2. Let f : D → N be a holomorphic mapping, where (D, g) be the Poincareunit disk, (N, h) a Kahler manifold of dimension n with holomorphic sectional curva-tures bounded above by −K(K > 0). Then

f ∗h ≤ 2

Kg.

To prove the theorem, we first introduce an inequality. To simplify notations, wedenote ∂

∂z

∣∣z=z0

by ∂∂z

here and henceforth when there is no confusion.

Lemma 3.3. For any z0 ∈ D, we have

∂z∂z log ‖df(∂

∂z)‖2h ≥ K‖df(

∂z)‖2h.

Proof. It suffices to prove the case when z0 = 0. Take the complex geodesic coordi-nates (wi) about f(0). The Kahler metric h of N is represented by the Hermitianmatrix (hij) satisfying hij(f(0)) = δij and dhij(f(0)) = 0. Define ϕ(z) = ‖df( ∂

∂z)‖2h =∑

hij(f(z))∂fi

∂z∂fj

∂z. We obtain

∂2ϕ

∂z∂z(0) =

∑ ∂2hij∂wl∂wk

(f(0))∂f i

∂z(0)

∂f j

∂z(0)

∂f l

∂z(0)

∂fk

∂z(0) +

∑ ∂2f i

∂z2(0)

∂2f i

∂z2(0),

1

ϕ(0)

∂2 logϕ

∂z∂z(0) =

1

ϕ(0)2

∂2ϕ

∂z∂z(0)− 1

ϕ(0)3

∂ϕ

∂z(0)

∂ϕ

∂z(0) ≤ 1

ϕ(0)2

∂2ϕ

∂z∂z(0),

which is the negative of the holomorphic sectional curvature in the direction of f∗( ∂∂z

)at z = 0.Hence we have 1

ϕ(0)∂2 logϕ∂z∂z

(0) ≥ K.

Now we shall prove Theorem 3.2.Proof. It suffices to prove

supz∈D

‖df( ∂∂z

)‖2h

‖ ∂∂z‖2g

≤ 2

K.

We first assume the supremum is obtained at some point z0 ∈ D. The fact that z0 is amaximum point implies we have

∂z∂z log‖df( ∂

∂z)‖2h

‖ ∂∂z‖2g

≤ 0 at z0.

Therefore it yields

K‖df(∂

∂z)‖2h

(lemma 3.3)≤ ∂z∂z‖

∂z‖2g ≤ ∂z∂z log ‖ ∂

∂z‖2g = 2‖ ∂

∂z‖2g,

where the last equality follows from the fact that the curvature of the Poincare unitdisk is −2.

In general, the supremum may not be obtained at any point z0 ∈ D. We proceed witha limiting argument with respect to the radius of the disks. In fact, for any z0 ∈ D,take r ∈ (|z0|, 1). Let Dr be the disk of radius r. Clearly, we have f |∗Drh = f ∗h|Dr . Letgr = 2Re r2

(r2−|z|2)2dz⊗ dz be the Poincare metric on Dr. Define ur(z) =

f∗h|DrgDr

. Then ur(z)

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GENERALIZATIONS OF SCHWARZ LEMMA 7

tends to 0 as |z| tends to r. Extending ur(z) continuously to Dr, ur(z) attains maxi-mum value at some z1 ∈ Dr by compactness of Dr. As in the previous discussion, wecan prove ur(z1) ≤ 2

K. It follows that

ur(z0) ≤ ur(z1) ≤ 2

K.

That is, ‖df( ∂∂z

)‖2h ≤ 2

Kr2

(r2−|z|2)2at z0. Letting r tend to 1, we get f ∗h ≤ 2

Kg at z0.

Remark 3.4. Kobayashi [9] gives the following generalization: Let M be a boundedsymmetric domain with an invariant Kahler metric g whose holomorphic sectionalcurvature is bounded below by −A(A > 0), (N, h) be a Kahler manifold with holomor-phic sectional curvatures bounded above by −B(B > 0). Then every holomorphic mapf : M → N satisfies f ∗(h) ≤ A

Bg. In fact, it suffices to consider the case where M is

a Poincare polydisk by the fact that for every tangent vector X of M , there exists atotally geodesic complex submanifold S such that X is tangent to S and S is isometricto a Poincare polydisk Dr. Now let X be a tangent vector of the polydisk Dr at theorigin. We can find j : D → Dr, where j(z) = (α1z, · · · , αnz) with

∑|αi|2 = 1 and D is

the unit disk with the Poincare metric multiplied by 2A

, such that j∗Y = X. For anyholomorphic map f : Dr → N , we have

‖f∗X‖2 = ‖f∗j∗Y ‖2 ≤ A

B‖Y ‖2 =

A

B‖X‖2,

where the inequality follows from Theorem 3.2(The constantA appears from the scaledPoincare metric.) and the last equality follows from the fact that j∗g equals to thescaled Poincare metric on D.

4. Holomorphic maps between the unit disk and the classical bounded symmetricdomains

The purpose of this section is to investigate the case when the equality in Theorem3.1 holds at a certain point when the codomain is a classical bounded symmetric do-main of type I, II or III. Before stating our results, we derive some properties of thebounded symmetric domain of each type, which was taken from Mok[1].

Let M(p, q;C) denote the space of matrices with complex entries of order p × q. Wedefine

DIp,q = Z ∈M(p, q;C) ∼= Cpq : Iq − ZtZ > 0, (p ≤ q)

DIIn = Z ∈ DI

n,n : Zt = −Z,

DIIIn = Z ∈ DI

n,n : Zt = Z.

4.1. Classical bounded symmetric domains of type I. Consider now DIp,q. To de-

scribe DIp,q we embed it as an open subset of the Grassmannian G(p, q) of complex q-

planes in Cp+q, with Euclidean coordinates (w1, · · · , wp+q). Identify Z ∈M(p, q,C) withthe (p+ q) by q matrix

[ZIq

], which represents the complex q-dimensional vector space

generated by its q C-linearly independent column vectors. Define SU(p, q) to be the spe-cial unitary group for the indefinite formHp,q =

∑1≤i≤p |wi|2−

∑p+1≤i≤p+q |wi|2. Clearly

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GENERALIZATIONS OF SCHWARZ LEMMA 8

SU(p, q) keeps DIp,q invariant. Let M ∈ SU(p, q) and write M

[ZIq

]=

[WE

]∼[WE−1

Iq

].

Then Φ(Z) = WE−1 represents an automorphism of DIp,q ⊂ M(p, q,C). In coordinates,

we can write Φ =

[A BC D

], where A and D are square matrices of order p and q resp.

We have

Φ(Z) = (AZ +B)(CZ +D)−1.

Remark 4.1. The condition[A BC D

]∈ SU(p, q) is equivalent to the set of conditions

AtA− Ct

C = Ip;

BtB −Dt

D = −Iq;AtB − Ct

D = 0.

In particular, Φ =

[U 00 V

], with U and V unitary of order p and q resp., is an automor-

phism of DIp,q that fixes the origin.

Lemma 4.2. Given X ∈ M(p, q), p ≤ q there exist unitary matrices U and V of order pand q resp. such that

UXV =

α1

. . . 0αp

:= diagp,q(α1, · · · , αp).

Remark 4.3. By Lemma 4.2 and constructing a product Mobius transformation on aunit polydisk4n which extends to an automorphism ofDI

p,q, we can show that SU(p, q)

acts transitively on DIIIp,q .

Now we discuss the curvatures of (DIp,q, ω) where ω is the Kahler form of the Bergman

metric on DIp,q. Let K(Z,W ) be the Bergman kernel on DI

p,q. One can check thatK(Z,W ) = c det((I − W

tZ))−(p+q) is the Bergman kernel by invariance of det((I −

WtZ))−(p+q) under SU(p, q). Let Φ(Z) = log det((I − Z

tZ)) and ω =

√−1∂∂Φ so as

to get the normalization hijkl(0) = δij,kl, where we take h = 2Re(∑hijkldzij ⊗ dzkl). We

also have dgijkl(0) = 0 by the existence of the involution σ0(z) = −z. And the curvaturetensor is

Rij,kl,pq,rs = − ∂4Φ

∂zij∂zkl∂zpq∂zrs= −δikδprδjsδlq − δirδpkδjlδqs.

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GENERALIZATIONS OF SCHWARZ LEMMA 9

For the unit tangent vector of type (0, 1) X =∑X ij ∂

∂Zij, the holomorphic sectional

curvature in the direction of X at 0 is given byRij,kl,pq,rsX

ijXklXpqXrs

=−X ijX iqXpqXpj −X ijXpjXpqX iq

=−∑j,q

|∑i

X ijX iq‖2 −∑i,p

|∑j

X ijXpj‖2

=− ‖XX t‖2 − ‖X tX‖2

=− 2‖XX t‖2 ≤ −2

p.

Because Aut(DIp,q) acts transitively and the Bergman metric is preserved under the au-

tomorphism, the holomorphic sectional curvatures are bounded above by −2p

at everypoint of DI

p,q. Let f : D → DIp,q be a holomorphic map. We have f ∗h ≤ pg by Theorem

3.2, where g is the Poincare metric of D. Moreover, the constant p here is optimal byconsidering f(z) = diagp,q(z, · · · , z).

Now we discuss the case when equality is obtained at a certain point. Because theautomorphism group of DI

p,q acts transitively, it suffices to assume f(0) = 0. Letdf( ∂

∂z) = X =

∑X ij ∂

∂Zijat the origin 0. By Lemma 4.2, there exist unitary matrices U

and V of order p and q resp. such that UXV = diagp,q(α1, · · · , αp). Define Φ(Z) = UZV ,an automorphism of DI

p,q that fixes the origin. Then dΦ(0)(X) = diagp,q(α1, · · · , αp).Replacing f by Φ f , it suffices to assume df( ∂

∂z) = diagp,q(α1, · · · , αp) at the origin 0.

Theorem 4.4. Let f : (D, g) → (DIp,q, h) be a holomorphic map with f(0) = 0 and

df( ∂∂z

) = diagp,q(α1, · · · , αp) at the origin 0, where α1, · · · , αp ∈ C. If we have ‖df( ∂∂z

)‖2h =

p‖ ∂∂z‖2g at z = 0, then f must be of the form

f(z) = diagp,q(α1z, · · · , αpz),

where |α1|, · · · , |αp| = 1.

We first introduce two lemmas before we prove Theorem 4.4.Lemma 4.5. Let f : D → C be a holomorphic function with f(z)→ 0 as |z| → 1. Thenf must be identically 0.

Proof. Extend f to 0 on ∂D continuously. For any w ∈ D and any r ∈ ( |w|+12, 1), we have

f(w) =1

2πi

∫|w|=r

f(z)

z − wdz by Cauchy integral formula.

Because |f |, as a continuous function on D, is bounded on D, f(z)z−w is bounded on w :

|w| ∈ ( |w|+12, 1). By bounded convergence theorem, we have

f(w) = limr→1

1

2πi

∫|w|=r

f(z)

z − wdz =

1

2πi

∫|w|=1

f(z)

z − wdz = 0.

The result is proved as w is an arbitrary point on D.

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GENERALIZATIONS OF SCHWARZ LEMMA 10

Lemma 4.6. Let f : D → Bn be a holomorphic map, where (D, g) be the Poincare unitdisk, (Bn, h) the complex unit ball of dimension n with the Bergman metric h. Assumef(0) = 0 and df( ∂

∂z) = (∂f1

∂z, 0, · · · , 0) at z = 0. If we have |∂f1

∂z(0)| = 1, then f must be of

the formf(z) = (cz, 0, · · · , 0),

where c = ∂f1∂z

(0).

Proof. Because we have |f1(z)|2 ≤∑|fi(z)|2 < 1, f1 is a map from D to D with f1(0) = 0

and |∂f1∂z

(0)| = 1. By (classical) Schwarz lemma we get f1(z) = cz, where c = ∂f1∂z

(0) is ofunit norm. Therefore, for any i ∈ 2, · · · , n, we have

|fi(z)|2 ≤∑j≥2

|fj(z)|2 < 1− |z|2,

which implies fi(z) → 0 as |z| → 1. By Lemma 4.5, fi ≡ 0 for each i ≥ 2. Therefore,f(z) = (cz, 0, · · · , 0).

Now we proceed to prove the theorem.

Proof. Write f(z) in the matrix form as f(z) = (fij(z))1≤i≤p,1≤j≤q, with f(z)tf(z) < Iq.

Hence, we have

(1)∑i

|fij(z)|2 < 1, for each j = 1, 2, · · · , q,

which yields |fij(z)| < 1 for all i, j. Observing that fij(0) = 0 and ‖df( ∂∂z

)‖2g =

∑i |αi|2 =

p at z = 0, we get |∂fii∂z

(0)| = 1 and fii(z) = αiz with |αi| = 1 for each i by (classical)Schwarz lemma. (1) also yields ϕj(z) := (f1j(z), · · · , fpj(z)) is a map between D andBp with ϕj(0) = 0 and dϕj(

∂∂z

) = αjej at z = 0 where e1, · · · , ep are the standard basisof the vector space of column p dimensional vectors, with |αi| = 1. By Lemma 4.6, itfollows that f must be of the form

f(z) =

α1z. . . f

αpz

,where f = (fij)1≤i≤p,p+1≤j≤q.

We shall prove f ≡ 0 by induction on q. Without loss of generality, we can assumeα1, · · · , αp = 1. For p = q, we have nothing to prove. Assuming it is true for q − 1. Forq, write

f =[E F

],

where E and F are matrices of order p × (q − 1) and p × 1 respectively. We havef(z)

tf(z) < Iq the equation |z|2Ip zE zF

zEt

EtE E

tF

zFt

FtE F

tF

< Iq.

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GENERALIZATIONS OF SCHWARZ LEMMA 11

By the fact that every principal submatrix of a positive definite matrix is positivedefinite, this implies

(2)[|z|2Ip zF

zEt

EtE

]< Iq−1,

(3)[|z|2 zfiq(z)

zfiq(z)∑

i |fiq(z)|2]< I2 for each i.

(2) enables us to define a function ψ : D → DIp,q−1 by ψ(z) =

[zIp E

]and by induction

hypothesis, we get E ≡ 0. (3) implies that

det

[1− |z|2 −zfiq(z)

−zfiq(z) 1−∑

i |fiq(z)|2]> 0,

⇔|fiq(z)|2 < 1

|z|2(1− |z|2)(1−

∑i

|fiq(z)|2) <1

|z|2− 1,

⇒|fiq(z)| → 0 as |z| → 1.

By Lemma 4.5, fiq(z) ≡ 0 for each i. Hence f ≡ 0.

4.2. Classical bounded symmetric domains of type III. To consider DIIIn we con-

sider a bilinear form J which is represented by the complex symmetric matrix Jn =[0 In−In 0

]. Define GIII(n, n) ⊂ G(n, n) to be the set of complex n-planes V in C2n such

that J |V ≡ 0. On the affine piece of G(n, n) consisting of Z ∈ M(n, n;C), the condi-tion J |V ≡ 0 is equivalent to Zt = Z, so that DIII

n can be identified as a subspaceof MS(n,C), the vector space of symmetric marices of order n × n. Define the groupSp(n,C) = M ∈ M(n, n;C) : M tJnM = Jn. GIII(n, n) is invariant under Sp(n,C).Therefore, GO = Sp(n,C)

⋂SU(n, n) keeps DIII

n invariant. Writing M ∈ GO =

[A BC D

]for square matrices A,B,C,D of order n×n. We haveM tJnM = Jn andM t

In,nM = In,n,which yields

(4) M−1 = J−1n M tJn =

[0 −InIn 0

] [At Bt

Ct Dt

] [0 In−In 0

]=

[Dt −Bt

−Ct At

],

(5) M−1 = I−1n,nM

tIn,n =

[In 00 −In

] [AtCt

BtDt

][In 00 −In

]=

[At −Ct

−BtDt

].

It follows that GO = M =

[A BB A

]: M ∈ SU(n, n). And the isotropy subgroup at the

origin is K = M =

[U 00 U

]: U−1

= U t. We have

Lemma 4.7. Let X be an arbitrary complex symmetric matrix of order n × n. Then,there exist nonnegative real numbers α1, · · · , αp and a unitary matrix U such thatUXU t = diag(α1, · · · , αp).

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GENERALIZATIONS OF SCHWARZ LEMMA 12

Remark 4.8. By Lemma 4.7 and constructing a product Mobius transformation on aunit polydisk4n which extends to an automorphism ofDIII

n , we can show thatGO actstransitively on DIII

n .

Now we consider the Schwarz lemma for DIIIn . Because the inclusion DIII

n → DIn,n

realizes DIIIn , equipped with their canonical metrics (up to normalizing constants), as

a totally geodesic submanifold of DIn,n, the curvature tensor on DIII

n is simply the re-striction of the curvature tensor of DI

n,n. Therefore, let f : D → DIIIn be a holomorphic

map. We have f ∗h ≤ ng. Moreover, the constant n here is optimal by consideringf(z) = diagn,n(z, · · · , z).

We shall discuss the case when equality is obtained at a certain point. Because Aut(DIIIn )

acts transitively and any tangent vector of type (1, 0) at the origin can be transferredto a matrix of the form diagn,n(α1, · · · , αn) by composing f with an automorphismfixing the origin by Lemma 4.7, it suffices to consider the case when f(0) = 0 anddf( ∂

∂z) = diagn,n(α1, · · · , αn) at the origin 0, where α1, · · · , αn are nonnegative real num-

bers. The result below follows directly from 4.4.

Theorem 4.9. Let f : (D, g) → (DIIIn , h) be a holomorphic map with f(0) = 0 and

df( ∂∂z

) = diagn,n(α1, · · · , αn) at the origin 0, where α1, · · · , αn ∈ C and h is the normalizedBergman metric. If we have ‖df( ∂

∂z)‖2h = n‖∂f1

∂z‖2g at z = 0, then f must be of the form

f(z) = diagn,n(α1z, · · · , αnz),

where |α1|, · · · , |αn| = 1.

4.3. Classical bounded symmetric domains of type II. To consider DIIn we con-

sider a bilinear form Σ which is represented by the complex symmetric matrix Sn =[0 InIn 0

]. Define GII(n, n) ⊂ G(n, n) to be the set of complex n-planes V in C2n such

that Σ|V ≡ 0. On the affine piece of G(n, n) consisting of Z ∈ M(n, n;C), the condi-tion Σ|V ≡ 0 is equivalent to Zt = −Z, so that DII

n can be identified as a subspace ofMa(n,C), the vector space of skew-symmetric marices of order n × n. Define the com-plex orthogonal group O(n,C) = M ∈ M(n, n;C) : M tSnM = Sn. GII(n, n) is invari-ant under O(n,C). Therefore, GO = O(n,C)

⋂SU(n, n) keeps DII

n invariant. WritingM ∈ GO =

[A BC D

]for square matrices A,B,C,D of order n×n. We have M tSnM = Sn

and M tIn,nM = In,n, which yields

(6) M−1 = S−1n M tSn =

[0 InIn 0

] [At Bt

Ct Dt

] [0 InIn 0

]=

[Dt Bt

Ct At

],

(7) M−1 = I−1n,nM

tIn,n =

[In 00 −In

] [AtCt

BtDt

][In 00 −In

]=

[At −Ct

−BtDt

].

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GENERALIZATIONS OF SCHWARZ LEMMA 13

It follows that GO = M =

[A B−B A

]: M ∈ SU(n, n). We can show that GO acts

transitively on DIIn . The isotropy subgroup at the origin is K = M =

[U 00 U

]: U−1

=

U. Now we let J1 =

[0 1−1 0

], and define

µ(α1, · · · , αp) =

α1J1

. . .αpJ1

if n = 2p

α1J1

. . .αpJ1

0

if n = 2p+ 1

Lemma 4.10. Let X be an arbitrary complex skew-symmetric matrix of order n × n.Then, there exist nonnegative real numbers α1, · · · , αp and a unitary matrix U suchthat UXU t = µ(α1, · · · , αp), where n = 2p or 2p+ 1.

Remark 4.11. By Lemma 4.12 and constructing a product Mobius transformation on aunit polydisk4n which extends to an automorphism of DII

n , we can show that GO actstransitively on DII

n .

For the same reason as that of DIIIn , the curvature tensor on DII

n is simply the re-striction of the curvature tensor of DI

n,n. Therefore, let f : D → DIIn be a holomorphic

map. We have f ∗h ≤ ng. But n is not always the optimal constant for DIIn . Below we

discuss in two cases.

When n = 2p, f(z) = µ(z, · · · , z) obtains n as the optimal constant. Now we dis-cuss the case when equality is obtained at a certain point. Because Aut(DII

n ) actstransitively and any tangent vector of type (1, 0) at the origin can be transferred toa matrix of the form µ(α1J1, · · · , αnJ1) by composing f with an automorphism fix-ing the origin by Lemma 4.12, it suffices to consider the case when f(0) = 0 anddf( ∂

∂z) = µ(α1J1, · · · , αnJ1)) at the origin 0, where α1, · · · , αn are nonnegative real num-

bers.

Theorem 4.12. Let f : (D, g) → (DII2p, h) be a holomorphic map with f(0) = 0 and

df( ∂∂z

) = µ(α1J1, · · · , αnJ1) at the origin, where α1, · · · , αn ∈ R+⋃0 and h is the nor-

malized Bergman metric. If we have ‖df( ∂∂z

)‖2h = 2p‖ ∂

∂z‖2g at z = 0, then f must be of the

form

f(z) = µ(z, · · · , z) =

zJ1

· · ·zJ1

,and α1, · · · , αn = 1.

The proof of Theorem 4.12 entirely mimics that of Theorem 4.4.

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GENERALIZATIONS OF SCHWARZ LEMMA 14

When n = 2p + 1, the optimal constant is 2p. In fact, recall that the holomorphicsectional curvature of DII

n at 0 in the direction of X is

B(X,X) =−2‖XX t‖2

‖X‖4.

Because X is skew symmetric, there exists a unitary matrix U such that UXU t =µ(α1, · · · , αp) by Lemma 4.12. From the fact that unitary transformations preservethe entrywise norm, we have

B(X,X) =−2‖UXU tUXU t‖2

‖UXU t‖4

=−2‖µ(α1, · · · , αp)µ(α1, · · · , αp)‖2

‖µ(α1, · · · , αp)‖4

= −∑|αi|4

(∑|αi|2)2

≤ −1

p

(8)

By Theorem 3.2, we have f ∗h ≤ 2pg for any holomorphic map f : D → DIIn . And the

optimal constant 2p is obtained by

f(z) = µ(z, · · · , z) =

zJ1

· · ·zJ1

0

.Now we discuss the case when equality is obtained at a certain point. By the samereasoning as in the case n = 2p, it suffices to consider the case when f(0) = 0 anddf( ∂

∂z) = µ(α1J1, · · · , αnJ1) at the origin 0, where α1, · · · , αn are nonnegative real num-

bers.Theorem 4.13. Let f : (D, g) → (DII

2p+1, h) be a holomorphic map with f(0) = 0 anddf( ∂

∂z) = µ(α1J1, · · · , αnJ1) at the origin, where α1, · · · , αn ∈ R+

⋃0 and h is the nor-

malized Bergman metric. If we have ‖df( ∂∂z

)‖2h = 2p‖∂f1

∂z‖2g at z = 0, then f must be of

the form

f(z) = µ(z, · · · , z) =

zJ1

· · ·zJ1

0

,and α1, · · · , αn = 1.

Proof. By employing exactly the same method used for Theorem 4.4, we can claim thatf must be of the form

f =

zJ1

zJ1

· · · fzJ1

0 0 · · · 0

,

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GENERALIZATIONS OF SCHWARZ LEMMA 15

where f = (f12p+1, · · · , f2p+12p+1)t,and α1, · · · , αn = 1. Because f(z) = −f(z)t, we havef(z) = µ(z, · · · , z).

5. Holomorphic maps between Kahler manifoldsIn this section, we shall prove two higher-dimensional generalizations of the Ahlfors-

Schwarz lemma for holomorphic maps from a compact Kahler manifold to anotherKahler manifold, both of which satisfy respective conditions on curvature.Remark 5.1. Below we assume the domain is a compact Kahler manifold. This as-sumption assures the existence of a maximal point of a certain function. In fact, byapplying a method developed in [7], it suffices to only assume the domain is complete.This more general case was studied by Yau in [6].5.1. A Schwarz lemma for metrics.Theorem 5.2. Let (M, g) be a compact Kahler manifold with Ricci curvature boundedfrom below by a constant K1. Let (N, h) be another Kahler manifold with holomorphicbisectional curvature bounded from above by a negative constant K2. If there is a non-constant holomorphic mapping f from M into N , we have

f ∗h ≤ K1

K2

g.

Proof. Denote f ∗h by s. Define u = Tracegs =∑gαβsαβ. First, we show u is globally

defined. Let (U, z) and (V,w) be two coordinate neighborhoods of M with nonemptyintersection. It suffices to prove u is invariant under the change of coordinates. Pickx ∈ U

⋂V . At x, let H = (hij) be the local expression of h, S1 = (s1

αβ) and S2 = (s2

αβ) be

the local expression of f ∗hwith respect to z and w respectively,5zf = (∂fi

∂zj) and5wf =

( ∂fi

∂wj) be the local expression of the gradient of f with respect to z and w respectively,

G1 = (g1ij) and G2 = (g2

ij) be the local expression of the metric g with respect to z and wrespectively and T = ( ∂zi

∂wj) and T−1 = (∂wi

∂zj).

It suffices to show TraceG−12 S2 = TraceG−1

1 S1. In fact,S2 = 5wf

tH5wf = T t5z fH5zfT = T tST ,

and G2 = T tG1T . Therefore,TraceG−1

2 S2 = TraceT−1G−1

1 S1T = TraceG−11 S1.

Now we show u ≤ K1

K2. For any x ∈ M , take complex geodesic coordinates (zi) and (wi)

in a neighborhood of x and in a neighborhood of f(x) respectively. At x, we have

u(x) =∑

sαα =∑

f ∗h(∂/∂zα, ∂/∂zα) =∑α,i

|∂fi

∂zα|2.

We also have

(9)∑k

∂k∂k log u(x) =∑

∂s∂sgαβ ∂f

i

∂zα

∂f i

∂zβ+∑‖∂s∂αf i‖2+

∑∂wp∂wrhij

∂f p

∂zs

∂f r

∂zs

∂f i

∂zα

∂f j

∂zα.

At x, the Ricci tensor is Rij = −∑

k ∂k∂kgij. From gijgkj = δik, we can derive Rij =∑

k ∂k∂kgji. Denote the Ricci curvature in the direction of (u/‖u‖, v/‖v‖) byRic(u/‖u‖, v/‖v‖),

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GENERALIZATIONS OF SCHWARZ LEMMA 16

which is bounded below by K1 as assumed. Denote the holomorphic bisectional cur-vature in the direction of (u/‖u‖, v/‖v‖) by B(u, v), which is bounded from above by K2

as assumed. We have ∑∂s∂sg

αβ ∂fi

∂zα

∂f i

∂zβ

≥∑

Ric(∂f i

∂z/‖∂f

i

∂z‖g,

∂f i

∂z/‖∂f

i

∂z‖g)‖

∂f i

∂z‖2g

≥K1

∑i,α

∣∣∂f i∂zα

∣∣2 = K1u.

(10)

∑∂wp∂wrhij

∂f p

∂zs

∂f r

∂zs

∂f i

∂zα

∂f j

∂zα

=−∑

B(∂f

∂zs,∂f

∂zα)‖ ∂f∂zs‖2h‖∂f

∂zα)‖2h

≥−K2

∑s,α

∑i

∣∣∂f i∂zs

∣∣2∑j

∣∣∂f j∂zα

∣∣2 ≥ −K2u2

(11)

It follows from (9), (10) and (11) that∑

k ∂k∂k log u(x) ≥ K1u(x) − K2u(x)2. Because uis defined on a compact space, u attains maximum at a certain point x0, and to provethe theorem, it is sufficient to prove u(x0) ≤ K1

K2. Because x0 is a maximum point, we

have∑

k ∂k∂k log u(x0) ≤ 0 in local coordinates. This yields 0 ≥ K1u(x0)−K2u(x0)2, andhence u(x0) ≤ K1

K2.

5.2. A Schwarz lemma for volume elements. A Schwarz lemma for volume ele-ments can also be established by considering another elementary function of the ten-sor f ∗h.

Theorem 5.3. Let (M, g) be a compact Kahler manifold with scalar curvature boundedfrom below by K1. Let (N, h) be another Kahler manifold with Ricci curvature boundedfrom above by a negative constant K2. Suppose dim M =dim N = n. If there is anon-constant holomorphic mapping f from M into N , we have

f ∗dVN ≤ (K1

nK2

)ndVM

where dVM , dVN are volume elements of M and N respectively.

Proof. Let u = f∗dVNdVM

, which is a globally defined function. We can take complex ge-odesic coordinates (zi) and (wi) in a neighborhood of x and in a neighborhood of f(x)respectively. We can associate the Ricci tensor with the volume element through

Rαβ = −∂α∂β log det(gij(0)),

andΘαβ = −∂wα∂wβ log det(hij(w))|w=f(0)

respectively, where Rαβ and Θαβ are the Ricci curvature tensors of M and N in lo-cal coordinates respectively. The derivatives of determinants are calculated through

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GENERALIZATIONS OF SCHWARZ LEMMA 17

Jacobi’s formula, the details of which we omit here. We also have

u(x) =det(hij(f(0)))| det(∂f i/∂zj)|2

det(gij(0))= | det(∂f i/∂zj)|2,

and

∑k

∂k∂k log u =∑k

∂k∂k log det(hij(f(0))) +∑k

∂k∂k log | det(∂f i/∂zj)|2 −∑k

∂k∂k log det(gij(0))

=∑k

∂k∂k log det(hij(f(0)))−∑k

∂k∂k log det(gij(0))

=∑k,i,j

∂wi∂wj log det(hij(w))|w=f(0)∂f i

∂zk(0)

∂f j

∂zk(0)−

∑k

∂k∂k log det(gij(0))

= −∑k,i,j

Θij

∂f i

∂zk(0)

∂f j

∂zk(0) +

∑k

Rkk

= −∑k

RicN(∂f

∂zk(0)/‖ ∂f

∂zk(0)‖h,

∂f

∂zk(0)/‖ ∂f

∂zk(0)‖h)‖

∂f

∂zk(0)‖2

h +∑k

Rkk

≥ −K2

∑k

‖ ∂f∂zk

(0)‖2h +K1.

(12)

Furthermore, we have

(13)∑k

‖ ∂f∂zk

(0)‖2h ≥ n(

∏k

‖ ∂f∂zk

(0)‖2h)

1n ≥ n| det(∂f i/∂zj)|

2n = nu

1n ,

where the first inequality follows from the arithmetic-geometric mean inequality andthe second inequality follows from the Hadamard’s inequality. This leads to the fol-lowing:

(14)∑k

∂k∂k log u ≥ −K2nu1n +K1.

Because M is compact, u attains maximum value at a certain point of M , say x0. Be-cause x0 is a maximum point,

∑k ∂k∂k log u(x0) ≤ 0. Therefore, we have u(x0) ≤ ( K1

nK2)n.

Because u attains maximum at x0, f ∗dVN ≤ ( K1

nK2)ndVM at each point of M .

Remark 5.4. A Kahler manifold (M, g) is Kahler-Einstein if there exists a constant λsuch that Ric(g) = λg. Theorem 5.3 has a direct consequence that the Kahler-Einsteinmetric of a compact Kahler manifold with negative Ricci curvature −A(for any A > 0fixed) is unique. In fact, suppose g and g′ are two Kahler-Einstein metrics on M . ByTheorem 5.3, the identity map i : (M, g) → (M, g′) and its inverse are both volumedecreasing, which implies i preserves the volume. In coordinates, it means i preservesthe determinant of the metric tensor and thus the Ricci curvature tensor. As themanifold is Einstein, it preserves the metric.

Remark 5.5. Mok and Yau [2] gave a stronger version of Schwarz lemma for volumeforms: Let (M, g) be a complete Hermitian manifold. Let (M, g) be a compact Kahler

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GENERALIZATIONS OF SCHWARZ LEMMA 18

manifold with scalar curvature bounded from below by K1. Let (N, h) be another com-plex manifold with a volume form dVN such that the Ricci form is negative definiteand (∂∂ log dVN)nK2 ≥ dVN . Suppose dim M =dim N=n and there is a non-constantholomorphic mapping f from M into N . Then we have

f ∗dVN ≤Kn

1

nnK2

dVM ,

where dVM is the volume element of (M, g).

6. Two applications6.1. Kobayashi metric. The Poincare metric on the unit disc D may be used to givean intrinsic pseudometric dM , the Kobayashi metric, on any complex manifold M ,which was first defined by Kobayashi [10]. To define d, we let p, q be two points on M,and we shall call a chain a sequence of holomorphic mappings fj : D → M(j = 1, ·, N)together with points xj, yj ∈ D such that

f1(x1) = p, fj(yj) = fj+1(xj+1), fN(yN) = q,

and then define

dM(p, q) = infN∑j=1

ρ(xj, yj),

where the infimum is taken over all chains and ρ is the Poincare distance on D. Itis clear that a holomorphic mapping f : M → N is distance decreasing with respectto their Kobayashi metrics. In other words, we have dN(f(p), f(q)) ≤ dM(p, q) for anyp, q ∈M .An infinitesimal version of the Kobayashi metric is given by Royden [12]:

Definition 6.1. Let M be a complex manifold. For any p ∈M and ξ ∈ T 1,0p , define

FM(p, ξ) = infa−1 : ϕ : Da →M,ϕ(0) = p, ϕ′(0) = ξ,

where Da is the disk of radius a centered at 0 and the infimum is taken over all holo-morphic maps ϕ which map some Da into M .

Theorem 6.2. The function FM(p, ξ) : T 1,0C → R is upper semi-continuous and for any

p, q ∈M ,

dM(p, q) = infγ

∫ 1

0

FM(γ(t), γ′(t))dt,

where the infimum is taken over all possible C1 curve γ : [0, 1]→M with γ(0) = p, γ(1) =q. In other words, the Kobayashi metric is the integrated form of FM .

Definition 6.3. A complex manifoldM is called Kobayashi hyperbolic if the Kobayashimetric is non-degenerate.

Then Theorem 3.2 gives a criterion of a Kahler manifold being Kobayashi hyperbolic,which we shall explain below.

Theorem 6.4. A complex manifoldM is hyperbolic if it carries a Kahler metric hwhoseholomorphic sectional curvatures are bounded above by some negative constant −K.

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GENERALIZATIONS OF SCHWARZ LEMMA 19

Proof. For any a ≥ 0, p ∈ M and ξ ∈ T 1,0p , let f : Da → M be a holomorphic map with

f(0) = p and df( ∂∂z

) = ξ. Theorem 3.2 states that f ∗h ≤ 2Kgwhere g = 2Re a2

(a2−|z|2)2dz⊗dz

is the Poincare metric on Da. It follows that

‖ξ‖2h ≤

2

K

1

a2.

Therefore FM(p, ξ)2 ≥ K2‖ξ‖2

h by the definition of FM , implying the Kobayashi metric isnon-degenerate.

Remark 6.5. Essentially, we only need a distance decreasing property to derive theresult. Hence, given any Hermitian manifold (M,h) satisfying the inequality f ∗h ≤Cga for every a > 0, every holomorphic map f : Da → M and for some constant Cindependent of a and f , M is Kobayashi hyperbolic.6.2. Normal families. The notion of a normal family has played a central role in thedevelopment of complex function theory, weaving a line of thought through Picard’stheorems, the Riemann mapping Theorem and various modern results in the subject.Montel’s Theorem states that a locally bounded family of holomorphic functions de-fined on an open subset of C is a normal family. Here we employ our result to give avariant of Montel’s Theorem for Kahler Manifolds. A detailed study of normal familiesin the general setting of complex manifolds was carried out in [13] by Wu.

Given two metric spaces M and N , we denote by C(M,N) the space of all continu-ous maps f : X → Y equipped with the topology of uniform convergence on compactsets. Let M and N be two complex manifolds. Then we denote by Hol(M,N) the spaceof all holomorphic maps f : M → N .Definition 6.6. A sequence fn ∈ C(M,N) is called compactly divergent if and onlyif given any compact K ∈ M , and compact K ′ ∈ N , there exists an n0 such thatfi(K)

⋂K ′ = ∅ for any n ≥ n0.

Definition 6.7. A subset F of C(M,N) is normal if every sequence fn ⊂ F either hasa convergent subsequence in C(M,N) or is compactly divergent.Definition 6.8. Let the distance function of N be dN . Then a family F ∈ C(M,N) iscalled an equicontinuous family if and only if given any ε > 0 and any x ∈ M , thereexists a neighborhood U of x such that y ∈ U implies dN(f(x), f(y)) < ε for all f ∈ F .

Now we state and prove our main theorem. In the proof we will apply two well-knowntheorems. The first is Ascoli Theorem, which states that F ∈ C(M,N) is compact ifand only if it is equicontinuous, pointwise relatively compact and closed. The secondresult is Hopf–Rinow theorem, which states that a closed and bounded subset of aRiemannian manifold is compact.Theorem 6.9. Let (M, g) be a compact connected Kahler Manifold whose Ricci curva-ture is bounded from below and (N, h) be a connected Kahler Manifold whose holomor-phic sectional curvature is bounded from above by a negative constant. Then a familyF ∈ Hol(M,N) is equicontinuous. If N is complete, then F is normal.Proof. Let dM and dN be the distances on M and N respectively. By Theorem 5.2,we have that for any f ∈ F and any x, y ∈ M , dN(f(x), f(y)) ≤ CdM(x, y) for some

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GENERALIZATIONS OF SCHWARZ LEMMA 20

constant C > 0 independent of f and x, y. Therefore, for any ε > 0 and x ∈ M ,take U = y ∈ M : dM(x, y) < ε

C. Then y ∈ U implies dN(f(x), f(y)) < ε for all

f ∈ F ⊂ Hol(M,N). Hence, F is equicontinuous.

Now we prove the second statement. For K ⊂ M and F ⊂ C(M,N), define F (K) =f(x) : f ∈ F, x ∈ K. Let F be a sequence in F . If F is not compactly divergent, thenthere exist compact sets K0 ∈ M and K1 ∈ N such that gi(K0)

⋂K1 6= ∅ for infinite

numbers of gi ∈ F . Let G = gi. We would like to show that G(K) is relatively com-pact for each compact K ⊂M .

Since M is connected, it suffices to prove the claim for all connected compact K con-taining K0. Take such a K and we know that gi(K)

⋂K1 6= ∅. For any ε > 0 and each

x ∈ K, there exists a neighborhood U such that the diameter of gi(U) is less than ε forall gi by the equicontinuity ofG. The compact setK admits a finite cover by such neigh-borhood, so the diameter of gi(K) is less than 2lε for all i, where l is the number of setsin the cover. Let the diameter of compact K1 be η(η < ∞). Take a fixed point y ∈ K1.Then gi(K) ⊂ B(y, 2lε+η) for all i because gi(K)

⋂K1 6= ∅, where B(x, r) is an open ball

of radius r centered at x. That is, G(K) ⊂ B(y, 2lε + η). Because G(K) is a boundedset in the Kahler Manifold N , G(K) is relatively compact by Hopf–Rinow theorem.Consider G in C(M,N). Since G is equicontinuous, we can show G is equicontinuousand pointwisely relative compact. Therefore G is a relatively compact sequence. Thisproves F is normal since F is an arbitrary sequence in F .

Remark 6.10. Essentially, we only need a distance decreasing property to derive theresult. Hence, given any pair of connected Hermitian manifolds (M, g) and (N, h) sat-isfying the inequality f ∗h ≤ Cg for all f ∈ Hol(M,N) and for some constant C inde-pendent of f , the above theorem holds for M and N .

References[1] Mok, N., Metric Rigidity Theorems on Hermitian Locally Symmetric Manifolds, World Scientific

(1989)[2] Mok, N. and Yau, S. T., Completeness of the Kahler-Einstein metric on bounded Riemann domains

and the characterization of domains of holomorphy by curvature conditions, Proc. Sympos. PureMath., vol. 39, part 1, Amer. Math. Soc, Providence, RI, pp. 41-60, (1983)

[3] Fritzsche K., Grauert H., From Holomorphic Functions to Complex Manifolds, Springer-VerlagNew York, (2002)

[4] Ahlfors, L. V., An extension of Schwarz’s lemma, Tmns Amer. Math Soc. 43, 359-364 (1958)[5] Ahlfors, L. V., Conformal Invariants: Topics in geometric function theory, McGraw-Hill, New York,

(1973)[6] Yau, S.T., A General Schwarz Lemma for Kahler Manifolds, Amer. J. Math. 100, no. 1, 197-203

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28, pp. 201-228 (1975)[8] Kobayashi, S. and Nomizu K., Foundations of Differential Geometry, Vol. I (1963), Vol. II (1968),

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4, 481-485. (1967)[10] Kobayashi, S., Intrinsic Metrics on Complex Manifolds. Bull. Amer. Math. Soc. 73 , no. 3, 347–349.

(1967)

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GENERALIZATIONS OF SCHWARZ LEMMA 21

[11] Kobayashi, S., Hyperbolic Manifolds and Holomorphic Mappings: An Introduction, 2th edition, AWorld Scientific, Singapore, (2005)

[12] Royden, H., Remarks on the Kobayashi metric. Proc. Maryland Conference on Several ComplexVariables, Lecture Notes Math. 185, 368-383 (1971)

[13] Wu, H., Normal families of holomorphic mappings, Acta Math. 119 , 193–233. (1967)[14] Schiff, J.L., Normal Families, Springer-Verlag, New York, (1993)[15] Boas, H.P., Julius and Julia: Mastering the art of the Schwarz lemma. Amer. Math. Monthly 117,

770-785, (2010)[16] Gamelin, T., Complex Analysis, Springer-Verlag New York, (2001)