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    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-Schwarzs 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-Schwarzs 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. 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 TCX ,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(


    idzj) 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 ) =ij 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 (X) denote the set of smooth sectionsof V over open sets U . Let (X) denote the set of smooth vector fields on X over U . Aconnection is a map D : (X) (X) (X) : (, s) Ds satisfying:

    (1) D is complex linear in both and s.(2) D satisfies the product rule D(fs) = (f)s + fDs 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 and any tangent vector of type (0, 1) in the domain of definitionof , D = 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 {e} be a holomorphic basis of V |U and write s = se. Let = i zbe a smooth vector field of type (1, 0) over U . It suffices to define Die = ie whereDi = D/zi. The requirement that D be complex and metric determines (

    i) uniquely

    by i(e, e) = (Die, e) + (e, Die) = (Die, e). Write (e, e) = h, givingi = h


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


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

    1,0X . By conjugation the connectionD on T


    extends to a connection on TCX = T 1,0X 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 TCX .Theorem 1.6. The Hermitian connection D agrees with the Riemannian connectionif 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 v2.

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

    zi, zi}1in of TCX as:

    Rij = ijg,

    where i zi , j zj


    We can identify T 1,0X with the real tangent bundle TRX via 2Re. Writing = iiand u = 2Re = + , we also call R(u,Ju;Ju,u)u4 the holomorphic sectional curvature inthe direction of . We have

    R(u, Ju; Ju, u) = 4Rijklijkl.

    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 , T 1,0X . Write u = 2Re and v = 2Re. We define the holomorphic bisectional curvaturein the directions (, ) to be R(u,Ju;Jv,v)u2v2 .

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

    1Rijdzi 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(g)dz1 dz1 dzn dzn, and thereforethe Ricci curvature form has the expression

    R = log det(g).

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


    Hilbert space. Let fi{0i


    have g(a) = 2Ref (dz dz) where f(z) = za1az . 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 r2dzdz

    (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-Schwarzs 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 = 1h

    2 log h


    In particular, the Gaussian curvature of the Poincare metric g = 2Re dzdz(1|z|2)2 on the

    unit disk D is k = 2.

    Theorem 3.1 (Ahlfors-Schwarzs 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 2Kg,

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

    We can generalize Ahlfors-Schwarzs lemma in the following form:


    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 2Kg.

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



    by z

    here and henceforth when th