0 Preliminaries.- I Basic Properties.- II Hyperbolic Imbeddings.- III Brody's Theorem.- IV Negative Curvature on Line Bundles.- V Curvature on Vector Bundles.- VI Nevanlinna Theory.- VII Applications to Holomorphic Curves in Pn.- VIII Normal Families of the Disc in Pn Minus Hyperplanes.

This paper surveys some surprising applications of a lemma characterizing normal families of meromorphic functions on plane domains. These include short and efficient proofs of generalizations of (i) the Picard Theorems, (ii) Gol’dberg’s Theorem (a meromorphic function on C which is the solution of a first-order algebraic differential equation has finite order), and (iii) the Fatou-Julia Theorem (the Julia set of a rational function of degree d ≥ 2 is the closure of the repelling periodic points). We also discuss Bloch’s Principle and provide simple solutions to some problems of Hayman connected with this principle. Over twenty years ago, on the way to a partial explication of the phenomenon known as Bloch’s Principle, I proved a little lemma characterizing normal families of holomorphic and meromorphic functions on plane domains [68]. Over the years, the lemma has grown and, in dextrous hands, proved amazingly versatile, with applications to a wide variety of topics in function theory and related areas. With the renewed interest in normal families (arising largely from the important role they play in complex dynamics), it seems sensible to survey some of the most striking of these applications to the one-variable theory, with the aim of making this technique available to as broad an audience as possible. That is the purpose of this report. One pleasant aspect of the theory is that judicious application of the lemma often leads to proofs which seem almost magical in their brevity. In such cases, we have made no effort to resist the temptation to write out complete proofs. Hardly anything beyond a basic knowledge of function theory is required to understand what follows, so the reader is urged to take courage and plough on through. And now we turn to our tale. 1. Let D be a domain in the complex plane C. We shall be concerned with analytic maps (i.e., meromorphic functions) f : (D, | |R2) → (Ĉ, χ) Received by the editors October 15, 1997, and, in revised form, May 26, 1998. 1991 Mathematics Subject Classification. Primary 30D45; Secondary 30D35, 34A20, 58F23.

background

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Introduction to Complex Hyperbolic Spaces

Normal families: New perspectives

The field of complex dynamics in higher dimension was initiated in the 1920’s by Fa-tou. It was motivated by studies in Newton’s method, celestial mechanics and functional equations. Recently, new methods from pluripotential theory have been introduced to the subject. These techniques have produced many new interesting results. We give an introduction to this subject and a summary of the most relevant developments.

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Complex dynamics in higher dimensions

A holomorphic motion of E C C over the unit disc D is a map f: DxC-?C such that f(O, w) =w, w E E, the function f(z, w) = fz(w) is holomorphic in z, and fz: EC is an injection for all z E D. Answering a question posed by Sullivan and Thurston [13], we show that every such f can be extended to a holomorphic motion F: D x C C. As a main step a "holomorphic axiom of choice" is obtained (concerning selections from the sets C\fz(E) , z E D ). The proof uses earlier results on the existence of analytic 2 discs in the polynomial hulls of some subsets of C

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Holomorphic motions and polynomial hulls

Characteristics of the behavior of a meromorphic function and the first fundamental theorem Meromorphic functions of finite order The second fundamental theorem Deficient values Asymptotic properties of meromorphic functions and deficiencies Value distribution with respect to the arguments Applications of Riemann surfaces to value distribution On the magnitude of an entire function Notes A survey of some results after 1970 Bibliography References added to the English edition Author index Subject index Notation index.

Value Distribution of Meromorphic Functions

These notes are an expanded version of lectures delivered at the AMS Summer School on Algebraic Geometry, held at Santa Cruz in July 1995 The main goal of the notes is to study complex varieties (mostly compact or projective algebraic ones), through a few geometric questions related to hyperbolicity in the sense of Kobayashi A convenient framework for this is the category of “directed manifolds”, that is, the category of pairs (X, V ) whereX is a complex manifold and V a holomorphic subbundle of TX  If X is compact, the pair (X, V ) is hyperbolic if and only if there are no nonconstant entire holomorphic curves f : C → X tangent to V (Brody’s criterion) We describe a construction of projectivized kjet bundles PkV , which generalizes a construction made by Semple in 1954 and allows to analyze hyperbolicity in terms of negativity properties of the curvature More precisely, πk : PkV → X is a tower of projective bundles over X and carries a canonical line bundle OPkV (1) ; the hyperbolicity of X is then conjecturally equivalent to the existence of suitable singular hermitian metrics of negative curvature on OPkV (−1) for k large enough The direct images (πk)⋆OPkV (m) can be viewed as bundles of algebraic differential operators of order k and degree m, acting on germs of curves and invariant under reparametrization Following an approach initiated by Green and Griffiths, we establish a basic Ahlfors-Schwarz lemma in the situation when OPkV (−1) has a (possibly singular) metric of negative curvature, and we infer that every nonconstant entire curve f : C → V tangent to V must be contained in the base locus of the metric This basic result is then used to obtain a proof of the Bloch theorem, according to which the Zariski closure of an entire curve in a complex torus is a translate of a subtorus Our hope, supported by explicit Riemann-Roch calculations and other geometric considerations, is that the Semple bundle construction should be an efficient tool to calculate the base locus Necessary or sufficient algebraic criteria for hyperbolicity are then obtained in terms of inequalities relating genera of algebraic curves drawn on the variety, and singularities of such curves We finally describe some techniques introduced by Siu in value distribution theory, based on a use of meromorphic connections These techniques have been developped later by Nadel to produce elegant examples of hyperbolic surfaces of low degree in projective 3-space; thanks to a suitable concept of “partial projective projection” and the associated Wronskian operators, substantial improvements on Nadel’s degree estimate will be achieved here 2 J-P Demailly, Kobayashi hyperbolic projective varieties and jet differentials

Introduction to Complex Hyperbolic Spaces

Citations

Normal families: New perspectives

Complex dynamics in higher dimensions

Holomorphic motions and polynomial hulls

Value Distribution of Meromorphic Functions

Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials

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