Value Distribution Theory

VALUE DISTRIBUTION THEORY

THE UNIVERSITY SERIES IN HIGHER MATHEMATICS

THE UNIVERSITY SERIES IN HIGHER MATHEMATICS

Editorial Board

M. H. Stone, Chairman L. Nirenberg

S. S. Chern

HALMOS, PAUL R.-Measure Theory JACOBSON, NATHAN-Lectures in Abstract Algebra

Vol. I-Basic Concepts

Vol. II-Linear Algebra

Vol. III-Theory of Fields and Galois Theory KLEENE, S. C.-Introduction to Metamathematics LOOMIS, LYNN H.-An Introduction to Abstract Harmonic Analysis LOEVE, MICHEL-Probability Theory, 3rd Edition KELLEY, JOHN L.-General Topology ZARISKI, OSCAR, and SAMUEL, PIERRE-Commutative Algebra, Vols. I and II GILLMAN, LEONARD, and JERISON, MEYER-Rings of Continuous Functions RICKART, CHARLES E.-General Theory of Banach Algebras J. L. KELLEY, ISAAC NAMIOKA, and CO-AUTHoRs-Linear Topological Spaces SPITZER, FRANK-Principles of Random Walk NACHBIN, LEOPoLDo-The Haar Integral KEMENY, JOHN G., SNELL, J. LAURIE, and KNAPP, ANTHONY W.­ Denumerable Markov Chains SARIO, LEO, and NOSHIRO, KIyosHI-Value Distribution Theory

A series of advanced text and reference books in pure and applied mathematics. Additional titles will be listed and announced as pub­ lished.

f7alue Distribution Theory LEO SARlO Professor of J[ athematLcs L'niversity of California Los Angeles, California

AND

KIYOSHI NOSHIRO Professor of J1 athematics Nagoya Unil'ersity Nagoya, Japan

in collaboration with TADASHI KURODA

KIKUJI lVIATSUMOTO

~1ITSURU

X AKAI

D. VAN NOSTRAND COMPANY, INC. PRINCETON, NEW JERSEY TORONTO

NEW YORK

LONDON

D. VAN NOSTRAND COl\IPANY, INC. 120 Alexander St., Princeton, New Jersey (Principal office) 24 West 40 Street, New York 18, New York D. VAN NOSTRA~D COMPANY, LTD. 358, Kensington High Street, London, W.14, England D.

VA~ NOSTRAND COMPANY (Canada), LTD. 25 Hollinger Road, Toronto 16, Canada

COPYRIGHT

©

1966,

BY

D. VAN NOSTRAND COMPANY,

INC.

PublIshed simultaneously in Canada by D

VAN KOSTRAND COMPANY (Canada), LTD.

No reproductwn in any form of this book, in whole or in part (except for bnef quotation in critical artlcles or rcviews), may bc made wlthout wnttcn authonzatwn from the publlShers.

PRINTED IN THE UNITED STATES OF AMERICA

DEDICATED TO

The University of California which gave us the opportunity to discovel

CONTENTS

xi

ACKNOWLEDGMENTS

1

PREFACE

INTRODUCTION

1. 2. 3. 4. 5.

Historical New metric The fundamental A-, B-, and C-functions Method of areal proximity Summary

CHAPTER I

§1.

§2.

§3.

The proximity function s (C,a) The fundamental functions A, B, and C Euler characteristic Areal proximity Main theorem Nondegeneracy Exceptional points Ramification

MEROMORPHIC FUNCTIONS ON ARBITRARY RIEMANN S"CRFACES

9. 10. 11. 12.

Main theorems Sharpness of even bounds Sharpness of arbitrary bounds The class of Rp-surfaces

SURFACES

13. 14. 15. 16. 17. 18. 19.

7 8 10

MAPPINGS INTO CLOSED RIEl\IANN SURFACES

MAPPINGS OF ARBITRARY RIEMANN SURFACES

1. 2. 3. 4. 5. 6. 7. 8.

5 5 5

R.

AND CONFORMAL METRICS

Metric The fundamental functions Preliminary form of the second main theorem Evaluations Exceptional intervals Second main theorem Picard points vii

11 12 13 16 18 19 21 23 24 25 25 27 29 30 32 32 33 35 36 38 39 40

CONTENTS

Yill

CHAPTER II

MAPPINGS INTO OPEN RIEMANN SURFACES

§1. PRIKCIPAL FUNCTIONS 1. Preliminaries 2. Auxiliary functions 3. Linear operators 4. An integral equation §2. PROXr:YIITY FUNCTIONS ON ARBITRARY RIEMANN SURFACES 5. Boundedness of auxiliary functions 6. Uniform boundedness from below of s(',a) 7. Symmetry of s ("a) 8. Conformal metric

42 43 44 47 49 51 52 53 54 57

§3. AXALYTIC MAPPINGS 9. Main theorems 10. Affinity relation 11. Existence of mappings 12. Area of exceptional sets 13. Decomposition of s ("a) in subregions 14. Joint continuity of s ("a) 15. Consequences 16. Capacity of exceptional sets

59 60 62 63 65 67 68 70 71

CHAPTER III

FUNCTIONS OF BOUNDED CHARACTERISTIC

§1. DECG:\iPOSITION 1. Generalization of Jensen's formula 2. Decomposition theorem 3. Extremal decompositions 4. ConsequencGs §2. THE CLASS O~IB 5. Preliminaries 6. Characterization of O~IB 7. Decomposition by uniformization 8. Theorems of Heins, Parreau, and Rao CHAPTER IV

74 75 78 82 85 86 87 89 92 94

FUNCTIONS ON PARABOLIC RIEMANN SURFACES

§1. THE EVANS-SELBERG POTEKTIAL 1. The Cech compactification 2. Green's kernel on the Cech compactification 3. Transfinite diameter

98 99

101 105

CONTENTS

4. Energy integral 5. Construction §2. MEROM ORPHIC Ft:NCTIONS IN A BOUNDARY NEIGHBORHOOD 6. The af Hallstrom-Tsuji approach 7. Exccptional sets CHAPTER V

§2. FINITE PIC.\RD SETS Generalized Picard theorem AuxIliary results Proof of the generalized Picard theorem Classes of sets with the Picard property CHAPTER VI

120 120 123 125 125 127 129 132

RIEMANNIAN IMAGES

§1. MEAN SHEET NUMBERS 1. Base surface 2. Covering of subregions 3. Covering of curves §2. EULER CHARACTERISTIC 4. Preliminaries 5. Cross-cuts and regions 6. Main theorem on Euler characteristic 7. Extension to positive genus

§3. ISLANDS AND PENINSULAS 8. 9. 10. 11.

109

113 115 115 118

PICARD SETS

§1. INFINITE PICARD SETS 1. Sets of capacity zero 2. Scts of positive capacity 3. 4. 5. 6.

IX

Fundamental inequality Auxiliary estimates Proof of the fundamental inequality Defects and ramifications

§4. MEROMORPHIC FUNCTIONS 12. Regular exhaustibility 13. Application of the fundamental inequality 14. Role of the inverse function 15. Localized second main theorem 16. Localized Picard theorem §5. MAPPINGS OF ARBITRARY RIEMANN SURFACES 17. Conformal metrics

136 137 139 141 144 144 145 147 151 152 152 153 155 157 158 159 160 163 164 165 167 167

CONTENTS

x

18. 19. 20. 21. 22. 23.

Main theorem for arbitrary Riemann surfaces Integrated form Algebroids Sharpness of nonintegrated defect relation Direct estimate of M(p) Extension to arbitrary integers

ApPENDIX

I.

BASIC PROPERTIES OF RIEMANN SURFACES

II. GAUSSIAN MAPPING OF ARBITRARY Triple connectivity Arbitrary connectivity Arbitrary genus Arbitrary genus and connectivity Gaussian mapping Picard directions Islands and peninsulas Regular exhaustions Open questions

ApPENDIX

1. 2. 3. 4. 5. 6. 7. 8. 9.

MINIMAL SURFACES

169 171 173 175 176 178 179 194 194 195 196 196 197 198 199 199 200

BIBLIOGRAPHY

201

SUBJECT INDEX

231

A UTHOR INDEX

235

ACKNOWLEDGMENTS We are deeply grateful to the D. S. Army Research Office-Durham, in general, and to Drs. John W. Dawson and A. S. Galbraith in particular, for several D.C.L.A. grants during the five years 1961-1966 which the writing of the book has taken. Were it not for their patience with our everchanging plans, this work may never have been completed. Our sincere thanks are due to Professor S. S. Chern for the inclusion of our book in this distinguished series and for his continued stimulation. We are indebted to many colleagues who read the manuscript, in particular our collaborator 1\1. Nakai, who made substantial contributions to several parts of the theory and scrutinized the entire manuscript; our collaborator K. Matsumoto, who contributed his conclusive results on Picard sets; our collaborator T. Kuroda, who with Matsumoto and Nakai covered an early version of the manuscript in a seminar; our esteemed friend L. Ahlfors, whose council we had the advantage of obtaining on several occasions; K. V. R. Rao, who helped us with the second half of Chapter III; B. Rodin, who made valuable suggestions; 1\1. Glasner, who compiled the Indices and assisted us with the numerous tasks of preparing the manuscript for printing; P. Emig and S. Councilman, who with Glasner compiled the Bibliography. We were fortunate to have the typing of the several versions of the manuscript in the expert hands of Mrs. Elaine Barth and her efficient staff.

xi

PREFACE The purpose of this research monograph is to build up a modern value distribution theory for complex analytic mappings between abstract Riemann surfaces. All results presented herein are new in that, apart from the classical background material in the last chapter, there is no overlapping with any existing monograph on meromorphic functions. Broadly speaking the division of the book is as follows: The Introduction and Chapters I to III deal mainly with the theory of mappings of arbitrary Riemann surfaces as developed by the first named author; Chapter IV, due to Nakai, is devoted to meromorphic functions on parabolic surfaces; Chapter V contains Matsumoto's results on Picard sets; Chapter VI, predominantly due to the second named author, presents the so-called nonintegrated forms of the main theorems and includes some joint work by both authors. For a complete list of writers whose results have been discussed we refer to the Author Index. The value distribution theory had its inception with Picard's celebrated theorem, one of the most beautiful results in classical analysis. It was the starting point of the pioneering work of the French school: Borel, Hadamard, Valiron, and Julia. In 1924 Collingwood and Littlewood made a fundamental discovery: there can be infinitely many defectively covered points, and the defect sum (for entire functions) cannot exceed 1. This Collingwood-Littlewood defect relation is still the cornerstone of value distribution theory, and the primary object of the fundamental theorems. The relation was generalized by Nevanlinna, who also introduced the present terminology: "counting function" for the function initiated by Valiron to describe the coverage of a point, "characteristic function" for the Valiron function for fully covered points, and" proximity function" for the deviation of the former from the latter. The most effective methods currently in use, both for the integrated and nonintegrated forms, were devised by Ahlfors. To N evanlinna is due the development of value distribution theory into a beautiful unity, a masterpiece in the art of mathematics. Despite its elegance the classical theory suffers from the following restriction: a merom orphic function is a locally defined concept, and its potentialities are curtailed by confining it to a globally chosen special carrier such as the plane or the disk. Full richness of the theory can be expected only on the corresponding locally defined carrier, the most general one on which the concept of analytic function makes sense. 1

2

PREFACE

Building up a general theory of complex analytic mappings between Riemann surfaces thus appears to be of compelling importance. In the following sense we have reached this goal: we have established the integrated forms of the main theorems for analytic mappings into both closed and open Riemann surfaces, and the nonintegrated forms for mappings into closed surfaces. Moreover, we have obtained a bound for the number of Picard points for mappings into closed surfaces, and have shown that the capacity of the set of exceptional points vanishcs for mappings into arbitrary surfaces, closed or open. In contrast, we have only fragmentary results to report on the existence of mappings between given surfaces. In this direction the road is open for further research. Nowhere in the book have we made any attempt at completeness. We have been led mainly by our own interests and a desire for natural unity. The Introduction is intended to orient the reader with the techniques we have used, in particular, the new proximity function 8(~, a) and the method of areal proximity as compared with the classical curvilinear proximity. These tools permit us to obtain in Chapter I the main theorems for analytic mappings of arbitrary Riemann surfaces. In contrast with the classical theory our forms of the main theorems are valid for all subregions, with no exceptional ones omitted. As an extension of an elegant result by Chern we obtain the bound 7J - es for the number of Picard points. The class of Rp-surfaces is then introduced. It is characterized by the existence of capacity functions p with compact level lines. In Chapter I these surfaces provide us with the strictness of our bound for the number of Picard values and with a short proof of Nevanlinna's classical defect relation. Chapter II opens with the important theory of principal functions, indispensable in Chapters I to IV, and VI. In Chapter II these functions are used to establish the uniform boundedness from below of 8(~, a) in both variables for an arbitrary Riemann surface. This result leads to the main theorems for given nondegenerate mappings into arbitrary Riemann surfaces and to the affinity relation for such mappings into surfaces of finite Euler characteristic. It also gives, as shown by Nakai, the joint continuity of 8(~, a) and the vanishing ofthe capacity of the set of exceptional points under all mappings of arbitrary Riemann surfaces. Rodin's and Ozawa's results on the existence of analytic mappings are included. Chapter III starts with a decomposition theorem for meromorphic functions of bounded characteristic on arbitrary Riemann surfaces. The dass OMB of surfaces without such functions is thcn studied, and theorems of Heins, Parreau, and Rao on the dccomposition into quasi-bounded and singular parts are established. Chapter IV contains Nakai's penetrating proof, using the Cech compactification, of the existence of the Evans-Selberg potential on arbitrary

PREFACE

3

parabolic surfaces. This solution ofthe long open problem places parabolic surfaces in the class of Rp-surfaces. For functions on such surfaces it is shown that the set of exceptional values has vanishing capacity, an extension of the af HiHlstrom-Kametani-Nevanlinna theorem. That this theorem is sharp is the striking result of Matsumoto given in Chapter V: for cvery compact set K of vanishing capacity there exists a meromorphic function with a set E of essential singularities of vanishing capacity and with exactly K as the Picard set at every point of E. A necessary condition is then obtained for every merom orphic function in the complement of a given E to have a finite Picard set at each point of E. The theorem is a sharpening of a recent interesting result of Carleson. The longest chapter is VI. It starts with Ahlfors' elegant theory of covering surfaces, which is used to derive the nonintegrated forms of the main theorems on analytic mappings of arbitrary Riemann surfaces into closed surfaces. Corresponding theorems are also proved in a form localized to a transcendental singularity of the inverse function. At the end of Chapter VI we once more return to Rp-surfaces and show, generalizing an idea of Dinghas, that integrated forms of the main theorems can be derived from the nonintegrated forms. The general case of Rp-surfaces is then compared with the important special case of algebroids. To avoid interrupting the train of thought in value distribution theory proper, we have compiled in Appendix I some basic properties of Riemann surfaces that are referred to in Chapters I to VI. In Appendix II we first give an explicit construction of complete minimal surfaces of arbitrary connectivity and genus, smoothly immersed in E3. Although this construction, due to Klotz and the first named author, is somewhat isolated from the rest of the book, we believe that the value distribution theory of Gaussian mappings of these surfaces illuminates the general theory since thc fundamental quantities assume concrete meanings. Taking the proximity function into account, neglected thus far, should be fruitful in further research on Gaussian mappings. Beyond the above broad description of the book we have given a more detailed orientation on its plan and interconnections in the introductions to the chapters, sections, and appendices. The reader will do well to read them before starting a systematic study. The expert will note that several results not previously published are scattered through the book, and earlier ones or their proofs are improved. For example, Theorems I.2E and II.9A now appear without remainders and simplify the entire theory considerably. However, no changes have been made for the sake of changes. Where the authors had no improvements to report, their original presentations have been followed rather closely. The reader is not expected to have any previous knowledge of value distribution theory. For general prerequisites a standard Ph.D. curriculum

4

PREFACE

in complex analysis, real analysis, functional analysis, topology, differential geometry, and algebra should be sufficient. Outside of this we have in a few instances quoted well-known theorems if their proofs are easily obtainable from an established source; exact reference is then made. Bibliographical references concerning main results are placed in the introductions to chapters and sections, and again at the theorems. Sometimes" Remarks" are used for this purpose. Cross-references are self-explanatory: e.g., I for Chapters, 1. §1 for sections, 1.1 for numbers, I.IA for subnumbers, 1.(1) for formulas, and Theorem 1.2E for statements. For the convenience of the reader we have occasionally repeated some definitions and related preliminaries. For comprehensive studies of the main tools used in this book we refer to the forthcoming monographs Rodin-Sario, "Principal functions" (to appear in this series); Oikawa-Sario, "Capacity functions"; and NakaiSario, "Classification theory". In accordance with the plan of our book we have only lightly touched on the classical theory of meromorphic functions as presented in Nevanlinna's French monograph on the Picard-Borel theorem and later in his well-known German treatise. In this classical direction far-reaching further results have been obtained by Hayman and Edrei-Fuchs in their monographs on meromorphic functions and by Matsumoto in Chapter V of the present book. At the end of our book a rather comprehensive bibliography on value distribution theory, classical and modern, is published for the first time. (For reasons of space, literature on entire functions and other more restricted topics was generally not included.) It is our hope that the bibliography will be useful to workers in the field. It also reveals the profound influence Picard's theorem and the Collingwood-Littlewood defect relation have had on the evolution of complex analysis. Los Angeles, California April 1, 1966

LEO SARIO

KIYOSHI

N OSHIRO

INTRODUCTION The purpose of this Introduction is to give, without proofs, a general framework into which our method will be built in Chapters I and II. For a reader with some previous knowledge of value distribution theory and the theory of Riemann surfaces it also offers a comparative survey of the classical and the new approach. 1. Historical. During the nine decades that have elapsed since the publication of Picard's theorem, evolution has taken place toward greater generality: the Picard-Borel-Nevanlinna theory was first extended from the plane to more general plane regions by af Hallstrom [2] and Tsuji [7], then to various Riemann surfaces by Ahlfors [11], [13], Heins [1], Kunugui [3], Kuramochi [1], L. Myrberg [1], Noshiro [5], Ohtsuka [3], Parreau [2], Tamura [1], Tsuji [15], Tumura [4], and others. The most general result was obtained in 1960 by Chern [1], who considered as domain R a closed Riemann surface less a finite number of points, and as range S a closed Riemann surface. He showed that under a nondegenerate complex analytic mapping ~ = f(z) of R into S, z and ~ being the local complex parameters, the number P of Picard points, and more generally the defect sum, cannot exceed the negative of the Euler characteristic of S: (1)

This beautiful result of Chern's paves the way for the following question: Can generality be pushed further by allowing both Rand S to be arbitrary? A priori this did not seem likely. In fact, Heins [1] had exhibited an interesting Riemann surface of infinite genus, which carried meromorphic functions with infinitely many Picard values. A look in a somewhat different direction reveals, however, rather interesting new aspects. To this end let us consider current methods and see if we can introduce simplifications which permit greater generality. 2. New metric. The first tool we need is a function to describe the proximity of a generic point ~ E S to a given point a E S. The standard method is the following: one first forms a conformal metric with area element dw=)..2 dS, where dS is the Euclidean area element in the parametric disk, and ).. is covariant and strictly positive. Throughout our 5

6

INTRODUCTION

presentation let t (~, a, b) be a harmonic function of the variable ~ on S with a positive logarithmic pole at a and a negative logarithmic pole at b. One integrates t with respect to dw(b) over S. The resulting function (2)

q(~, a) =

Is t(~, a, b) dw(b)

is bounded from below and has a positive logarithmic singularity at a. It thus qualifies to describe the proximity of ~ to a. Moreover, !1q is simply the" density" of the metric: (3)

This makes it possible to use effectively the standard relations between line and area integrals. There are, however, two drawbacks to this approach. First, if S is open, it seems difficult, if not impossible, to establish the convergence of integral (2). Second, even when S is closed, a rather lengthy reasoning in partial differential equations is needed to show that!1q actually is ,\2. If S is open, there seems to be no way of putting the reasoning through. To overcome this difficulty we suggest the following reversal of the process: start with a function

(4) with given ~o, ~1 E S. The singularities together with a normalization ofthe additive constant uniquely determine to if S is closed. If S is bordered and compact, then we add the condition that the function be constant on the border. If S is open, we take for to the directed limit of the functions thus constructed on bordered subregions as the subregions exhaust S. The limiting function is a special case of the so-called principal function, and its existence is assured by the related linear operator method (Sario [1]). The function (5)

is bounded from below but continues to have a positive logarithmic pole at ~o. For any other point a take t(~, a, ~o) and add it to 80 The singularities at ~o cancel and the function

m.

(6)

is bounded from below and has a positive logarithmic singularity at a. We choose this function to describe the proximity of ~ to a: closer proximity gives greater values. The function exists on every Riemann surface S, open or closed, of finite or infinite genus. Having formed 8 we introduce a conformal metric with area element ,\2 dS by choosing the density ,\2 =!18 =!18 0 .1t is independent of a. Thus the

INTRODUCTION

7

problem of convergence of (2) and the proof of (3) are eliminated, and the metric is obtained on an arbitrary S. In this metric A has zeros which are those of grad to. But these zeros turn out to be helpful and, in fact, constitute a rather essential aspect of the theory. In passing we remark that the Gaussian curvature of our metric is constantly 1 and its total area is Isdw=47T. As a by-product we thus have a conformal metric (which has zeros of A) of constant curvature and finite total area on an arbitrary Riemann surface. 3. The fundamental A., B., and C·functions. We can now at once write down the first main theorem; it was earlier considered from different viewpoints by Heins [3], Kuramochi [1], L. Myrberg [1], and Parreau [2]. Here we give it in a form that directly serves the first purpose of this book: Picard's theorem on an arbitrary R. Remove from R a parametric disk Ro with boundary 130' and consider an adjacent regular region i1 c R with boundary 130 U f3n. On n form the harmonic function u with u = 0 on 130, u = k, a constant, on f3n,

such that the flux

J-

~o

du* = 1. For h c [0, k] consider the level line f3h =

u- 1 (h) and the region i1 h = U - 1 ((0, h)) between 130 and f3h. Given a point a c S let {Zj} be its inverse images under f and denote their number in Rh=Ro U i1 h by v(h, a). For the a-points we introduce the A-function

(7)

A(h, a) = 47T

f

v(h, a) dh.

It reflects the frequency of the a-points off on R. In particular, it vanishes for a Picard point a, i.e., a point which is not covered by f. For the f3-curves we take the B-function (8)

B(h, a) =

r

J~h-~O

s(f(z), a) du*.

Its geometric meaning is clear: it is the mean proximity to a of the image of f3h - 130 under f. Finally, the growth of the image area is characterized by the C-function (9)

C(h) =

lh JRrh

dw(f(z)) dh,

o

with C'(h) the area

JR hdw(f(z))

under f of Rh over S.

of the (multisheeted) Riemannian image

INTRODUOTION

8

Zj

A simple application of Stokes' formula to Oh less small disks about the that shrink to their centers gives the following

Theorem. For every regular region Q c R under an analytic mapping of an arbitrary Riemann surface R into another arbitrary Riemann surface S, (10)

A(k, a)+ B(k, a)

=

C(k).

Thus the elegant classical balance prevails: the (A + B)-affinity, so to speak, off is the same for all points a E S. In particular, for a Picard point a, A == 0 and we have a strong proximity of f(f3h -f3o) to a. 4. Method of areal proximity. We now come to the main question: How many Picard points aI' .. " aq can there exist? The answer is given by the second main theorem which we shall here give for mappings of an arbitrary R into a closed S. It is well known that in the classical second main theorem the remainder cannot be estimated for all values of the variable r. It is the integral of the integral of the remainder that can be given a dominating function. The remainder itself can behave arbitrarily wildly in certain intervals whose length can be estimated but which must be omitted in stating the second main theorem. When one then takes the defect relation, these exceptional intervals and the related changing of the coordinate system with varying 0 prevent the use of directed limits. But ordinary limits cannot be employed on an arbitrary Riemann surface R: there is no one parameter that would give an exhaustion of R. Thus the classical theory does not carryover to the general case. This difficulty can be overcome by the following simple device. We replace the proximity function B by the integral of its integral. Geometrically the first integration means that we replace the mean proximity of the image curve f(f3h) by what is just as natural if not more so, the mean proximity of the image regionf(Oh)' and then we take the integral of this. Analytically this means that, in some sense, we bring all quantities involved to the same level of integration. Then the remainder term in the second main theorem has an estimate for every subregion 0, directed limits can be employed, and the theory established on an arbitrary R. The actual derivation of the second main theorem consists of little more than another application of Stokes' formula. The proof is further facilitated by the presence of the zeros of ,\ to which we referred earlier. Their number is the Euler characteristic of the punctured (at ~o and ~I) Sand we obtain es without using the Gauss-Bonnet formula. Geometrically this makes it unnecessary to set up the tangent bundle, and we can dispense with borrowing from differential geometry. When the computations are carried out we obtain the following result:

INTRODUCTION

9

Theorem. For every 0 c R under an analytic mapping of an arbitrary R into a closed S, q

(11)

(q+e s )C2(k)
- 0, the essential condition reflecting the topology of R: the characteristic function must grow at least as rapidly as the Euler characteristic. For these mappings

10

INTRODUCTION

we have what we set out to find, a Picard theorem on an arbitrary R: the number of Picard points cannot exceed the excess of 7J over es , (17)

More accurately, P:5, 7J - es - L {3(a). In the case of the sphere S, i.e., for meromorphic functions on arbitrary Riemann surfaces, the bound 2 + 7J was shown to be sharp by an interesting example constructed by Rodin [2]. In the classical case of meromorphic functions in the plane we have an elementary proof of the defect relation, and a second main theorem without exceptional intervals. 5. Summary. The above is the approach our book starts with. To summarize, the advantages of the method are as follows:

(a) The most cumbersome part of the reasoning, that involving exceptional intervals, is eliminated. (b) The resulting "second main theorem" is valid for all subregions, with no "exceptional" ones omitted. (c) The degeneracy of mappings is characterized in a simple and uniform manner for all surfaces. (d) The results are obtained simultaneously in all cases, without the necessity of distinguishing between sup k = 00 and sup k < 00. In particular, meromorphic functions in the plane and in the disk are treated at once. (e) EVen the classical theorems in these special cases are more effectively derived as consequences of the general theorems. (f) These results are valid for merom orphic functions on arbitrary Riemann surfaces. (g) The results are not restricted to merom orphic functions but apply to mappings into arbitrary closed Riemann surfaces as well. (h) The method can be largely extended to the most general case of complex dimension 1: given mappings of arbitrary Riemann surfaces into arbitrary Riemann surfaces. Remark. The above Introduction is, in essence, an invited lecture "Complex analytic mappings" delivered before a meeting of the American Mathematical Society (Sario [12]).

CHAPTER I MAPPINGS INTO CLOSED RIEMANN SURFACES A Riemann surface R is a connected Hausdorff space with a conformal structure (cf. e.g., Ahlfors-Sario [1, p. 114]). We shall use the same symbol z for a generic point and its parametric image. Let S be another Riemann surface, and ~ its parameter. A mapping ~ = f(z) of R into S is by definition analytic if it is so in terms of the parameters. We are interested in the distribution of values off. Typically we ask: How many points of S canf omit? Using the time-honored terminology we shall call a Riemann surface closed or open according as it is compact or not. We shall proceed to full generality of the main theorems in two steps: in Chapter I we consider analytic mappings into closed surfaces; in Chapter II, those into arbitrary surfaces. This arrangement will permit us to clearly bring forth in Chapter I the essentials of our method, and the somewhat delicate reasoning on the proximity function in the general case can be postponed to the beginning of Chapter II. Proofs of peripheral intuitively clear steps in Chapter I (e.g., in IB, IC) can also be relegated to the corresponding passages of Chapter II (7D, 8C, resp.). The slight overlapping of the chapters will only facilitate the access to the main results in Chapter II. The general theory of the present chapter will be developed in §l. The above question on omitted points is answered in Theorem 7A, which is a consequence of our main result, Theorem 5C for arbitrary domain surfaces. In §2 the general theory is applied to the case of the sphere S, i.e., meromorphic functions on arbitrary Riemann surfaces R. For this case we give in §3 another proof along more classical lines, partly for completeness, partly for comparison. The proof involves exceptional intervals and we are restricted to a special class Rs of domain surfaces. In the Introduction we listed earlier literature, essential for later work. For the method developed in the present chapter for analytic mappings of arbitrary Riemann surfaces we refer to Sario [4], [5], [6], [9], and [12]. The presentation here will be self-contained.

§l. MAPPINGS OF ARBITRARY RIEMANN SURFACES In this section we shall first introduce the proximity function 8(~, a), the basic tool in our approach. We then define the fundamental A-, B-, and 11

I. MAPPINGS INTO CLOSED RIEMANN SURFACES

12

[IA

C-functions and derive the main theorems 2E and 5C governing them. As consequences we obtain the defect and ramification relations for analytic mappings of arbitrary open Riemann surfaces into arbitrary closed Riemann surfaces. Some terminology is taken from the theory of Riemann surfaces. A subregion of an open Riemann surface is called regular if its closure is compact and its boundary consists of a finite number of analytic Jordan curves. An axiomatic treatment of this and related concepts is to be found, e.g., in Ahlfors-Sario [1].

1. The proximity function

s(~,

a)

lAo Let f be an analytic mapping of an arbitrary Riemann surface R into a closed Riemann surface S. Value distribution theory deals with the affinity (to be specified) of f with respect to given points aI' ... , a q on S. First we shall construct a proximity function on S, i.e., a function to measure the nearness of a generic point ~ to a given point a. Choose ~o, ~l on S, different from the ai' i = 1, ... , q, and take arbitrary but then fixed disjoint parametric disks Do, DI about ~o, ~l' Let to be a harmonic function in S - ~o - ~l with singularities - 2 log 1~ - ~o 1 and 2 log 1~ - ~ll in Do and D I , respectively. The existence of such a function on a closed S is classical. (For the general case of a closed or open S the construction of to is carried out in II.6A.) We normalize the additive constant by the condition to(~) + 2 log 1~ - ~o 1 ---? 0 as ~ ---? ~o. The f'lllction (1)

continues to have a positive logarithmic pole at ~o and is nonnegative on S. It is our proximity function for ~o: greater proximity gives greater values.

lB. For any other point a i= ~o we could form the proximity function in the same manner. But we wish it to have the same Laplacian as 8 0 , so as to effectively use Stokes' formula. This we accomplish by adding to 80 the harmonic function t=t(~, a) with singularities -2 log I~-al and 2 log 1~ - ~o I· For normalization we choose (2)

as

~

---?

~o.

The function

(3)

has a positive logarithmic pole at a as its only singularity. Thus qualifies as the proximity function for an arbitrary' on S.

8(~,

a)

2A]

§1. MAPPINGS OF ARBITRARY RIEMANN SURFACES

13

The proximity function is symmetric:

=

s(a, b)

(4)

s(b, a)

for a, bin S. This is immediately seen by applying the Green's formula to t(', a) and t(" b) over S less small disks about a, band

'0.

Ie. In terms of to we introduce on S

the metric

dw = ,\2 dS,

(5)

where (6)

and dS is the Euclidean area element in the parametric disk. It is easily seen that ,\ is finite everywhere and its only zeros are those of grad to. In passing we note that, by virtue of Igrad tol 2 dS = dto dtt, the total area of Sis 47T: (7)

w =

i

S

dw =

Ji oo

-

00

IJ x

(1

eta

+ et a )2 dtt dx

= 47T,

oon·

I

where f3x=g tom =X E (-00, The Gaussian curvature, defined at all points with Igrad tol #0, is constantly 1:

K - _~log~ - 1

(8)

-

,\2

-.

2. The fundamental functions A, B, and C 2A. We turn to the domain surface R, an arbitrary Riemann surface. Take a parametric disk Ro with boundary 130 such thatf(f3o) does not meet aI' ... , a q , ~l. Let Q be an adjacent regular region with boundary 130 U f3n. Denote by u the harmonic function in n with u f30 = 0, u f3n = k(a) = const. > such that f du* = 1. The Dirichlet problem here and

'0'

°

I

I

JlJa

throughout the book is solvable, e.g., by the Perron method (see AhlforsSario [1, p. 138]). For hE[O, k] consider the level line f3,,=u- 1(h) and the region Q,,= U-I((O, h)) bounded by f30 u f3". Without loss of generality we may aSStlme that f(f3o U f3,,) does not meet a. In fact, since the only singularity of s is logarithmic, the curvilinear integrals of s that we shall consider will be finite and continuous in h, and our formulas will extend. to the case a Ef(f3o U f3,,). Denote by Zj the a-points of f in a" and let Uj C 0" be disjoint clockWise oriented level lines of s(f(z), a) about Zj' encircling simply connected

I. MAPPINGS INTO CLOSED RIEMANN SURFACES

14

[2B

regions !1 j • The curves flh and flo are oriented so as to leave Ro to the left. An application of Green's formula to v(z) =h-u(z) and s(f(z), a) gives v ds* -s dv* = (

(

(9)

v!1 z s dR,

Jnh-u{j.1

)r.aj+l3h-SO

where dR is the Euclidean area element in the parametric disk. In view of

f

aj dv*

f dv* --+ 0 as shrinks to through level lines of s. fa, ds* tends to 41T times the multiplicity of the a-point whence

= 0 we have

Clearly

al S

aj

Zj

Zj,

(

J~al

v ds* -s dv* --+ 41T

2, v(Zj) = 41T (It (h-x) dv(x, a).

Jo

Here the sum is taken over all a-points in QIt, counted with their multiplicities, and vex, a) is the number of a-points in 110 u Qx, counted similarly. We integrate f~ by parts and obtain from (9) in the limit, (9)'

-41Thv(0,a)+41T (It v(h,a) dh+ (

Jo

sdu*-h (

JJJo

JJJh-JJO

=

ds*

1

v(z) dw(f(z)).

Oh

2B. On the other hand, we consider the equation

r

(10)

ds*

= (

Jpo+'EaOI

!1 z s dR,

JRO-UtlOI

where the !10j c Ro are small disks about the a-points of fin Ro. As the disks shrink to their centers, ds* gives again 41TV(0, a) and we obtain

f

41TV(0, a) +

(10)'

~aOI

(

JJJo

ds*

= (

dw(f(z)).

JRo

We multiply (10)' by h and add to (9)': (11)

47T (\(h, a) dh+ (

Jo

s du*

=

h ( dw+l v dw.

JJJh-JJO

JRo

Oh

This is the preliminary form of Theorem 2E. 2C. Here the right-hand side has a simple meaning. To see this set v = 1 in (9), 41T(V(h, a) -v(O, a)) +

( JJJh-PO

ds* =

1

which added to (10)' gives (12)

41TV(h, a) +

r ds* = JRhr dw.

J8h

Oh

dw,

2DJ

§1. MAPPINGS OF ARBITR4RY RIElvIANN SURFACES

1.5

The left-hand side here is the h-derivative of the left-hand side of (11) except for a finite number of values of h in [0, k] such that a Ef(f3h)' In fact, let 0(0) be the part of R bounded by f3( 0) = f3h + 6 - f3h, the constant 0>0 chosen so small that 0(0) nf-l(a)= 0. Then by Stokes' formula r

s du*

= r u ds* - r

J P(6)

J 0(6)

u!:l.zs dR,

J (2(6)

where ds* + h r

u ds* = 0 r

r Jp(6)

JPh+o

ds*

JO(1i)

!:l.zsdR+h r

=or ds*+or

JO h

J(2(6)

!:l.zsdR.

J(2(6)

It follows that

I ~o Jr

O(6)

sdu*- r dS*/

JPh

which tends to 0 with

o.

~

r J(2(6)

I!:l.zsl dR+ J(2(6) r Ih-:u/1!:l.zsl 0

dR

The same is true for 0 < 0, and we conclude that

the right-hand side of (Il) is the integral of

f

Rh

dw.

2D. We are ready to introduce our fundamental quantities. The Afunction (counting function) (13)

f

A(h, a) = 47T

v(h, a) dh

reflects the frequency of a-points. In particular, for a Picard point, i.e., a point not covered at all, A =0. The B-function (proximity function) (14)

B(h, a) = r

JPh -Po

s du*

is the mean proximity to a of the image of f3h (characteristic function) (15)

C(h)

=

lh o

130 under J.

The C-function

r dw dh JRh

is the integral of the area of the (multisheeted) Riemannian image of Rh = RovQ h overS. Our notations deviate from their counterparts in classical value distribution theory partly because of significantly different definitions, partly because our terminology conveys their meanings in a natural manner: the a-points are counted by the A-function, the boundary curves 13 are dealt with by the B-function, and the characteristic is the C-function.

16

1. MAPPINGS INTO CLOSED RIEMANN SURFACES

[2E

2E. We have arrived at a generalization of Nevanlinna's [22] celebrated first main theorem on meromorphic functions in the plane (see 14B). Equality (11) takes on the following exact meaning, with no remainders to estimate (Sario [9]): Theorem. For complex analytic mappings of arbitrary Riemann surfaces into closed Riemann surfaces

(16)

A(k, a) + B(k, a) = C(k).

This expresses a beautiful balance. If a point a is only lightly covered, then A is small but B must be correspondingly large; i.e., the image of f3h -f3o comes close to the point a. Conversely, if a point a is strongly covered, then the image of f3h -f3o stays at a great mean distance from a. In short, what a point a loses in its coverage, it gains in the proximity of the image of f3h - f3o. Moreover, the value of the sum A + B is always C. Note that the primary variable in (16) is not k but Q which determines k. We next ask: How many Picard points ai' i = 1, ... , q, can there exist? To study this problem we must find a lower bound for L: A(k, ail or, what amounts to the same, an upper bound for L: B(k, at).

3. Euler characteristic 3A. As preparation we evaluate the Euler characteristic of Qh. Without loss of generality we may suppose that f3h consists of a finite number of analytic Jordan curves. This can indeed always be achieved by a sufficiently small increment of h. Denote by vh(grad u) the number of zeros of grad u in Qh. Lemma. The Euler characteristic e(h) of Q h has the value (17)

vh(grad u)

=

e(h).

The geometric meaning of this is clear from the following observation (Sario [5]). If Q h is doubly connected, the level lines of the harmonic conjugate u* of u cover Q h smoothly without any branchings, and vh(grad u) = O. If, however, we have two contours constituting f3h' then some level line of u* must branch off to reach both contours and we have a zero of grad u. In general, if the connectivity of Q h is c, the number of zeros of grad u is c-2. Suppose then we have a "handle" in Qh' i.e., a torus-shaped part between f30 and f3h. Then some level line of u* must branch off before entering the tubes of the torus and again combine after completing its passage through the tubes. Thus we have two zeros of grad u, and if the genus is g we obtain 2g zeros by virtue of g handles.

30]

§1. MAPPINGS OF ARBITRARY RIEMANN SURFACES

17

In the general case of connectivity c and genus g we conclude that vh(grad u) = 2g+c-2. This is the Euler characteristic of Qh· The intuitive meaning of (17) is thus clear, and we proceed to give two proofs.

3B. The following demonstration making use of Riemann-Roch's theorem (see, e.g., Ahlfors-Sario [1, p. 324]) is due to Rodin [1]. We reflect 0. h about f30 U f3h so as to form its double Oh' a closed Riemann surface [loco cit., p. 119]. Let rp be the reflection of Oh' an indirectly conformal self-mapping leaving each point of f30 U f3h fixed. The genus g of Clio is g = 2g+c-l, where g is the genus and c the number of contours of Qh. Since u takes constant values on each component of f30 U f3h' the differential w defined by wl0. h = du+i du*, Wlrp(Qh)

=

-d(u

0

rp)-id(u

0

rp)*

is regular analytic on Oh. By Riemann-Roch's theorem w has 2g-2 zeros in Cl h , and consequently vh(grad u) = vh(du+i du*) = g-l = 2g+c-2 = e(h). 3C. A less function-theoretic but equally rapid proof (Sario [9]) can be given by making use of the well-known consequence of the Lefschetz fixed point theorem (see, e.g., Milnor [1, p. 37]): the sum of the indices of a differentiable vector field on a compact differentiable manifold is equal to its Euler characteristic. Let Qh and rp be as in 3B. The vector field X defined by X IQh

I

X rp(Q h)

= grad u, =

-grad u

0

rp

is differentiable since u is constant on each component of f30 U f3h. The sum of the indices of X is twice the number of zeros of grad u on Qh' counted with multiplicities. Therefore 2vh(grad u)

= e(h),

where e(h) is the Euler characteristic of the closed surface (\. Since e(h) = 2g-2 = 2(2g+c-2) = 2e(h),

We conclude that (17) holds.

I. MAPPINGS INTO CLOSED RIEMANN SURFACES

18

[3D

3D. As a corollary of Lemma 3A we have for the closed range surface

S: Lemma. The number v(,\) of zeros of ,\ in the metric (5), (6) is

(18) Here es is the Euler characteristic of Sand gs is its genus. Let c be a positive number, so large that g Iitomi >c} consists of two disjoint "disks" about ~o and ~l' respectively. Let Do = g I tom> c}, DI = g I tom < - c} and v = a(c - to), where a is a positive constant with dv* = 1 along cycles of B=S - J\-DI separating Do and D I . On applying Lemma 3A with B and v in place of Ok and u we obtain

f

VB

(grad v) = e(B).

Clearly vB(grad v) = vB(grad to) = v(grad to) and, by (6), v(gradto) arrive at (18).

=

v(,\). On the other hand, e(B)=e s +2 and we

3E. We can again give an alternate proof using the reasoning in 3C. With B and to I B in place of Ok and u we conclude that the number of zeros of grad to is the Euler characteristic es +2 of B.

4. Areal proximity 4A. Our task of estimating L B(h, a i ) is facilitated if B is replaced by the integral of its integral (cf. Introduction). The first integration means that the proximity to a of the image of the curve 13k is replaced by that of the region Ok. The second integration is for expediency. For i > and for any integrable cP defined in [0, k] we set

°

(19)

cpi(h) =

!ok CPi-I(X) dx,

where CPo means cpo Our new proximity function is B2 and subindex 2 can be appended to each term in (16). Another simple device to shorten later reasoning is the following. Add to the points aI' ... , a q the 2g zeros a q + l , . • . , a q +2g of '\, where g noW stands for the genus of S, and for any function ljJ(h, a) set q+ 2g

(20)

ljJ(h) =

L: I

Then (21)

ljJ(h, a;).

5A]

§1. MAPPINGS OF ARBITRARY RIEMANN SURFACES

19

We are to find an upper estimate for B2(h).

4B. On S we distribute a mass dm=a dw heavily concentrated at the points a 1 , ••• , a q + 2g • Specifically, we set

a(~) = exp

q+ 2g

q+ 2g

1

1

[L: 8(~,ai)-210g (L: 8(~,ai)+const.)],

where the constant is chosen to satisfy L: 8(~, ai) + const. > 0 and the logarithmic term serves to make the total mass m= fsdm finite. In fact, for r= I~ -atl in a parametric disk we have a(~) = 0(r-2(log r) -2), and the mass over r < R, say, is O( (log R) - 1) = O( 1). If an a j , j > q, coincides with an at, i:5. q, then an obvious modification is needed. We can also choose ~o, ~l in lA so that this case does not occur. The density a,\2 of dm induces in the u+iu*-plane the density a/L2, where /L(z)

(22)

= '\(f(z))If'(z)llgrad U(Z)I-l.

f

By the convexity property JPh r log rp du* :5. log Ph rp du* of the logarithm for any nonnegative function rp on f3h we have B(h)
. k- 00 e

Therefore

A computation analogous to the one in lOC yields YJ 5, (2n-2)/N, and we conclude by (49) that

For N = 1, T=O and we again have the bound 2n. Clearly the number of Picard values of j is 2n/N. For any integer q~ 2 we can choose n=q and N =2, say, and obtain q Picard values. In this case (49) becomes P

5,

q+(I-T)

and since P and q are integers, P cannot exceed q. We have shown that the bound in (49) is sharp for all positive integers.

llB. We return to the question of the sharpness of (48). The following interesting modification of the surface of lOA was given by Rodin [2] and shows the sharpness for every integer YJ. Take n copies of the finite z-plane z=x+iy, each slit along the rays (55)

1m = {z I X

5,

0, y = 27Tim},

m=o, ± 1, ± 2,· ... For a fixed m the edges of 1m on all the sheets are identified so as to obtain a branch point of multiplicity n at 27Tim. This describes a covering surface R of the plane. We again choose p(z) = (27Tn)-110g Izl and define the region Rh as before.

I. MAPPINGS INTO CLOSED RIEMANN SURFACES

30

[12A

On this surface we consider the meromorphic function (56)

with poles at z=21Tim. We have e21lnh

v(h, (0) ~ 2 ~

and (57)

In view of the branch points of R over the points 21Tim we obtain e21lnh e(h) ~ 2(n-l) 21T' (58) (59)

E(h) ~ 2(n-l) e21lnh, 1Tn

· . f E2(k) 7J ~ 11m III A (k ) k-+oo

2

,00

1

= n- ,

and the bound is 2 + 7J ~ n + 1. On the other hand, the function f omits the values 0 and e21liu/n, fL= 0,···, n-1. Thus P=n+l, which establishes the assertion.

12. The class of Rp.surfaces 12A. In the examples of 11 we were able to replace the directed limits of the general theory by ordinary limits. This is possible whenever R is an Rp-surface characterized by the following condition: there exists a function p harmonic on R with a finite number of negative logarithmic poles and with compact level lines. Then Ro can be chosen as the region bounded by some level line f3o: P = c, and u on any Q bounded by f30 and another level line f3n is the restriction to Q of the same function p - c. The function (50) is a special case of a capacity function, constructed on an arbitrary Riemann surface as follows. (The construction is the same for one or several singularities.) For any set E let H(E) be the space of harmonic functions on E. Given a Riemann surface R choose a point Zo E R, a parametric disk D containing zo, and a regular subregion Q of R containing D. The capacity function Pn of Q with singularity at Zo is, by definition, the function Pn E H(D.-z o) with 1 (60) Pnl D = 21T log Iz-zol +h(z),

12C]

§2. MEROMORPHIC FUNCTIONS

31

where hE H(D), h(zo) =0, and Pn laO=kn=const. For 0 c 0' it is seen that kn ~ kn' (App. L§I), and the directed limit (61)

exists. Without using the monotonicity of kn we can also define kp as sup kn This gives the capacity (62)

of the ideal boundary f3 of R (Sario [3]). The surface is parabolic or hyperbolic according as cp =0 or cp > O. The class of parabolic surfaces is denoted by OG and can be shown to coincide with the class of Riemann surfaces on which there are no Green's functions (App. L§I).

12B. On an arbitrary R ¢ Oa the directed limit PB = limpn

(63)

n-R

exists [loco cit.] and is, by definition, the capacity function PB on R. However, there clearly are hyperbolic surfaces on which the level lines of P/J are not all compact. E.g., on the unit disk punctured at z = 1/2, PB = (I/27T) log Izl has a noncom pact level line Izl = 1/2. On a parabolic surface R the directed limit (63) does not always exist, but on every R there is a nested exhausting sequence {On} such that the corresponding capacity functions Pn converge [loco cit.]. The limit PB is again defined as a capacity function on R. In contrast with the Green's function there thus exists a capacity function on every Riemann surface, although for R E Oa it may not be unique. The usefulness of PB in this case is in that every R E Oa possesses a capacity function with compact level lines and is thus an Rp-surface. This is a recent result of Nakai [1], to be proved in Chapter IV. Other aspects of Rp-surfaces will be discussed in VLI9.

12C. For surfaces Rp that possess capacity functions with compact level lines we can replace A 2 , B 2 , C2 , E2 by A, B, C, E in defining <x, f3, 7J. In fact, if we set (64) (65)

l' . f B(k, a) a = l~;~ C(k)'

.,() o

&()

1·

. f A(k, f~)

a = le!~

C(k)

,

then by the reasoning leading to l'Hospital's rule, S~<x, and &:=:;,f3. Similarly we write 1· . fE(k) (66) K = l~!~ C(k)

1. MAPPINGS INTO CLOSED RIEMANN SURFACES

32

[13A

and obtain the defect and ramification relation in the general case of mappings into closed surfaces: (67)

For meromorphic functions on Riemann surfaces the bound is 2 + K. In the special case of merom orphic functions in the disk Izl < R5,oo this reduces to Nevanlinna's classical defect and raIllification relation

L S(a) + L &(a)

(68)

5,

2,

which we have thus obtained from the second main theorem without exceptional intervals, and sim'.lltaneously for the plane and the disk.

§3. SURFACES Rs AND CONFORMAL METRICS For the sake of completeness and comparison we shall also derive the defect and ramification relation using a more classical approach that involves exceptional intervals. Several aspects in our reasoning go back to the fundamental work of Ahlfors [10] on meromorphic functions in the plane. To develop a theory on certain Riemann surfaces Rs we make use of a metric to exhaust Rs. The metric we choose is suggested by ds = Igrad pplldzl, where pp is the cfl,pacity function with compact level lines, as exemplified in 11 and 12. We shall, however, consider a slightly more general situation, which offers interest in its own right. Our reasoning is self-contained and independent of the preceding sections.

13. Metric 13A. Let R be a Riemann surface endowed with a conformal metric (69)

ds

= '\(z)ldzl,

where ,\;:: 0 is defined in each parametric disk and ds is invariant under change of parameter. The length l(ex) of a rectifiable arc ex on R is well defined, and the distance d(zl' Z2) between two points is inf l(ex) for arcs ex from Zl to Z2. The distance d(El' E 2 ) between two subsets of R is defined as inf d(zv Z2) for Zl E Ev Z2 E E 2 • Let Ro be a regular subregion with boundary f3o. For a> 0 let (70)

f3"

= {z E R I d(z, Ro) = a}.

14A]

§3. SURFACES R, AND CONFORMAL METRICS

33

We impose the following requirements on our metric: (a) Log A is harmonic except for logarithmic singularities. (b) The length of the" level line " f3" is constantly

r ds = 1.

(71)

Jp a

(c) The level lines f3" are compact for every a> o. The last condition is understood to mean that f3" is either compact or void. We set ap=Sup a for nonvoid f3", and distinguish between ap=oo and ap e on R - Rao for some a, e and some O'o C' containing D" and with C n Xi = 0 .

6. Uniform boundedness from below of s(~, a) 6A. We are now ready to carry out the construction of the proximity function according to our program in the introduction to §2. To form to take W=S-~O-~1 and W1=(Do-~o) u (D1-~d, where the parametric disks Do, D1 are centered at ~o, ~1. In Do-~o choose a= -2 log r, in D1 - ~1' a=2log r. We tacitly have also a neighborhood DfJ of the ideal boundary (3, but since we can choose a=O there we no longer write it down here or in later applications of Theorem 4C. For Wo take S - D~ - D~, the meaning of the primes being as in 5B. Since a I aW 0 = O( I) and the flux 'V'anishes, we infer that p-a=O(l) and pi Wo=O(l). The principal

54

1I. MAPPINGS INTO OPEN RIE"'dANN SURFACES

[613

function p is taken as to. Then to 1 Do = - 2 log r + 0(1) and it follows that So 1 Do =log (1 + r- 2e°(l») = - 2 log r +log (r2 + 0(1)). We have established the following estimate: Lemma. So 1 Do= - 2 log r+ 0(1) and 8 0 ;::: 0 on S. 6B. The construction of 8 = 8 0 + t will depend on the location of a. Let D and j) be disks disjoint from each other and from D~. Consider three cases: (I) aED~, (II) aES-D;-D, (III) aES-D~-D. The union of the three sets is S, and it suffices to establish a uniform lower bounel for 8 separately in each of the three cases. The third case can be dispensed \\'ith since it is the same as the second. Case I. aED~. Take TV=S-so-a, WI=Do-so-a, and Wo=S-Db. Set a=210g (ril-sai/is-al). Then a 1 awo is 0(1) and so are p-a and WOo The normalization is at So where p - 2 log r tends to the lImit -210g lal +c(a) as r ~ 0, with c(a) =0(1) uniformly for a E D~. By Lemma 6A this limit is 8 0 {a) + C I (a), where again c I (a) = 0(1). The function t = P - CI has the required normalization t(s,a)-210g Is-sol ~80(a) as s~so (this will entail the symmetry of 8=8 0 +t in 7D). Moreover, since 11-sal/ls-al > 1, we have t 1 Do> 2 log r+O(l) and t 1 S- Do=O(l). On combining this with Lemma 6A we obtain 81 Do> 0(1), 81 S - Do> 0(1), hence 8> O( 1) uniformly for a E D~. Case II. a E S- D~ - D. OnS" =S- D" we have -g(o 1 Do =210g r+O(l), -g(oIS"-Do=O(l). On applying Lemma 5C to E=aD'us o, 0= (D-D")uD~ we obtain gaIS";:::O, galaD'=O(l), gaISo=O(l). Consequently the restriction of ga - g(o to Do is > 2 log r + O( 1); to S" - Do, > 0(1); to aD', 0(1); and at So we have ga -g,o -2 log r ~ c(a) = 0(1) as S ~ So' uniformly in a. As the last application of Theorem 4C we take W =S - So - a, TV I = S"-So-a, Wo=D', and a=ga-g(o in WI' Then al aWo=O(l). The normalization is at So where p - 2 log r ~ cI{a) = 0(1). Take t = P + 80{a)cI(a). Since so(a)-cl(a»O(l), we conclude that tIDo>210gr+0(1), tiS" - Do > 0(1), and tiD" = 0(1). Adding t to 80 gives 81 Do> 0(1), sIS-Do>O(l). We have established the following result (Sario [10]): Theorem. The proximity function 8( S, a) i8 uniformly bounded from below for all S, a on an arbitrary Riemann 8urface S.

pi

7. Symmetry of s (~, a)

7A. We shall show that for a, b ES, 8(a, b)=s(b, a). We could use the same reasoning at; in I. lB. However, we also wish to prove that s(s, a) is the uniform limit of corresponding functions constructed on exhausting

55

§2. PROXIMITY PUNCTIONS

7BJ

subregions. This convergence is needed to show that the total area (see SB) in the metric we shall use is precisely 47T even for an open S. Suppose first that S is a bordered Riemann surface with compact border pand let Da, Do be parametric disks about a, ~o, with disjoint closures in S. Let V be the class of harmonic functions v on S - ~o -a with the same singularities and normalization as t: v

1

Da -

= - 2 log r + h,

a

vIDo-~o = 2 log r+k,

where h, k are harmonic in Da, Do, and k(~o)=so(a). Set c=47Th(a). In V single out the functions vO> VI determined by

~Vo 1,8 = on

0,

VI

1,8 =

const.,

and set for real A (47)

The quantities h, k, c corresponding to VA will be denoted by h", k A, c;!, For v, v'

E

V we write B(v) =

Ie vdv*,

B(v, v')

=

Ie v dv'*.

Lemma. The function VA minimizes B(v) + (2A-l)c in V: (48) Proof. The Dirichlet integral of V -VA over Sis D(v -VA) = B(v) + B(v;,l -B(v, VA) - B(v A, V). Let Ai(V), i

= 1,2, be the integral

I v dv* along 8D a,

oDo, respectively, and similarly for Ai(v, v'). In the same manner as in

2B we obtain 2

B(v A) = (1- A)A

L (Ai(V

O'

VI)

-Ai(V I , vol)·

I

Here the first summand is AI(h o, -2 log r)-AI(hl' -2log r)=cl-c O' Because of the normalization at ~o the second summand vanishes and we have Similarly B(v, vJ B(v A' v)

= =

AB(v, VI) = A(c I -c), (1- A)B(vo, v) = (1- A)(c - co).

Equality (48) follows.

7B. Now let S be an arbitrary open Riemann surface, and Q a regular subregion with border f3n.. Let VAn be the function (47) constructed on Q as above.

II. MAPPINGS INTO OPEN RIEMANN SURFACES

56

[7C

Lemma. The directed limit

VA = lim VAn

(49)

n~s

exists and the convergence is uniform in compact subsets of S - a - ~o. Proof. The train of thought is, in essence, the same as in 2D. Let n c Q' and indicate by primes the quantities corresponding to Q' and (30:. We apply (4S) to V=V~, VA =vo and obtain (50)

Analogously (51) and (52)

From these equations and from the relations B(v;):o:; B'(v;) =0, B(v l ) =0, D?:.O, we infer that CI increases while Co decreases with increasing Q, and CI :0:; Co for every Q. A fortiori the limit ci=lim Cm exists, with the obvious meaning of Cm' This implies (53)

In view of the normalization v m ,( ~o) - vm( ~o) = 0 the asserted convergence follows from Lemma lA. For later reference (SB) we let Q' --+ S while keeping Q fixed in (51), and obtain Bo(vI)+C I

For Q (54)

--+

=

cIn+Do(VI-VIn)'

S this gives lim Bn(v l ) = O. n~s

7C. Although both Vo and VI were needed in the convergence proof, we shall only make use of VI in the sequel. Lemma. The function t constructed in 6B and the limiting function VI in (49) are identical. Proof. By definition, t = LIt, and by virtue of the uniqueness of principal functions it suffices to show that VI = Liv i . Let WI = (Da-a) u

(Do-~o)

u Dp

and let .Q contain S- WI' Denote by LIn the Ll"0perator acting on functions on al = 0 WI and providing us with harmonic functions on

§2. PROXIMITY FUNCTIONS

sA]

57

n

n Wl· Then vln=LHlVul and we are to prove that lim LlnVHl=LlVl· On al we have Vln ---+ Vl and consequently Lwv ln - LlnV l = Lldv ln - Vl) ---+ 0,

i.e., lim LlnV ln = lim LlnV l · By the definition of Ll the latter expression is indeed Llvl · The above reasoning for t can also be applied to show that (55)

where ton is the to-function constructed on Q.

7D. It is now easy to see that s=so+t is symmetric (Sario [14]): Lemma. For any a, b =I- ~o

sea, b)

(56)

=

s(b, a).

Proof. Again first suppose S is bordered compact with border (3. Let ab' ao be the peripheries of parametric disks abouL a, b, ~o and set ta=t(~, a), tb=t(~, b) with ta I{3=const., tb I(3=const. Then CX a ,

Here

fo = 0

and, in the same fashion as in 7A, we obtain Lo =

4-n-(so(a)-so(b)). Analogous computations give

faa = 41Ttb(a)

and

to =

-41Tta(b). We infer that so(a)+t(a, b) = so(b)+t(b, a).

This is (56). If S is noncompact the statement follows from the above and the uniform convergence of the approximating functions formed on the QcS.

8. Conformal metric

8A. As in I.IC we shall now form a conformal metric in terms of to and show that even for open surfaces S the total area is 41T. Let the area element be dw=;"'2dS, where (57)

;"'2 = ~s = ~s = etolgradtol2. o (1 + eto )2

For x E ( -00, 00) denote by a(x, to) the level line to = x on S - ~o - ~1. When x is near -00 or 00, then a(x, to) is compact and encircles ~1 or ~o, respectively. On an open S, a is noncompact for some values x.

58

II. MAPPINGS INTO OPEN RIE1vIANN SURFACES

[813

Given a regular region Q c S containing ~o, ~l set an(x, to) = a(x, to) n Lemma. On a noncornpact S we have for x E ( -00, 00) lim

(58)

O---f.S

J

dt'!; ::; 417

n.

a.e.

an(X. to)

Proof. If the statement is false, then there exists a constant 10 > 0, a regular region Qo, and a value Xo with ao = ao o (xo, to), such that

r

. . ao

dt * o>

417+310, grad to I ao#O. and ao is not tangent to eQ o. Moreover, there is a

I

0>0 with grad to G#O and

L[dt'!;[ =(10), where

Y = G n aQ o consists of disjoint closed arcs Yl' Y2 joined by ao, and (e) stands for a quantity in the interval (- 10, e). Because of the uniform convergence too -+ to there exists a regular region Q => Q o such that

J

dt'!;o

=

ao

J

dt'!;+<e),

ao

and

For any arc a c G from YI to Y2 we have

J

dt'!;o

J

=

a

dt'6n+- ~'

On letting e --l>- 0 we conclude that

14B. We have seen in 13A that (b,a)--l>-u(b,a)=s(b,a)-2go.(b,a)= 8(b, a) is finitely continuous on flo. x Q and the same is true of the coefficients of dg~(b, ~) as functions of (b, ~) for some fixed local parameters

on flc. Thus Hn.{~, a) is finitely continuous on Q x Q. We have reached the following result (Nakai [6]):

70

II. MAPPINGS INTO OPEN RIEMANN SURFACES

[IGA.

Theorem. The function s(~, a) is continuous on S x S. Specifically, for every regular region 0 of S there exists a symmetric finitely continuou8 function vd~, a) on 0 x 0 such that (79)

15. Consequences

15A. Let jL be a regular Borel measure with compact support Sil in 8. The s-potential Sll of the measure jL is defined by Sllm

Since c=inf(C a).

S x

s8(~,

=

a) -1> 8(~,a)

f8(~,

a) djL(a).

00,

=

s(~,a)-c

is strictly positive on S x S. It is more convenient to use 8 than kernel of potentials. Accordingly we consider the potential

8

as the

which is strictly positive if jL ¢ 0, finitely continuous and sub harmonic in S-SIl' and lower semicontinuous on S. Let 0 be a regular region containing Sil and let gn be the Green's function of O. As the first application of Theorem 14B we have (80)

where g/5. is the Green's potential

f

onO, andvnm = (vd~,a) -c) djL(a) isa finitely continuous function onO. An immediate consequence of (80) is that the potentials 811 satisfy the continuity principle: If 811 ISil is finitely continuous on SIl' then 811 is finitely continuous on S. More precisely, for ~' E Sil we have as ~ ---*

r

(81)

lim sup 811 (.s-s.

m::; lim sup 8 m. (.s. 11

We also deduce from (80): A set of 8-capacity ( = s-capacity) zero is identical with a set of gn-capacity zero and hence of logarithmic capacity zero (App. 1.7).

16AJ

§3. ANALYTIC AIAPPINGS

n

15B. The most important consequence of (80) is the following, known as the fundamental existence theorem in potcntial theory. For the proof see, e.g., Ninomiya [IJ, Kishi [IJ, or Nakai [4J. Lemma. Let K be a compact subset of 8 and let u( 0 be a strictly positive finite upper semicontinuous function on K. There exists a measure jJ., with compact support 8~ in K such that

on K except for a set of capacity zero and with the further property

everywhere on 8 ~ . In particular 8"m =um on 8" except for a set of capacity zero.

15C. We turn to the maximum principle for our 8-potentials. The behavior of 8" near 8 1( is regulated by (81), but the behavior at the ideal boundary f3 of 8 is unknown. We can only state that for any compact set K (82)

M(K) = sup lim sup 8(~, a) < aE

K

00.

(-/3

som

In fact, we have 8(~, a)=som+t(~, a)-c, where is bounded on 8 outside of a neighborhood of ~o and (~, a) ~ t (~, a) is continuous. Moreover, ~~t(~,a) is harmonic on 8-a-~o and has L1-behavior at the ideal boundary of 8. From this, inequality (81), and the maximum principle for subharmonic functions we deduce the following form of the maximum principle for 8~: Lemma. If 8":::; M on 8~l then

If 8 E OGl then the ideal boundary is negligible and we have the usual maximum principle: Corollary. If 8 E OG and 8":::; M on 8 11 , then 8":::; M on 8.

16. Capacity of exceptional sets 16A. Let f be an arbitrary analytic mapping of R into 8. As in I2B we set (cf. App.I.22): (83)

II. MAPPINGS INTO OPEN RIEMANN SURFACES

72

[l6B

where 13k is the boundary of Rk=Ro u Q. Then 13k(a) is essentially the same as Bh(a) of I2B with h=k:

c=

(84)

infs(~,

a)-I.

Sxs

We set

E = {a Ia

E

S, lim inf 13k(a) = oo}. R,,-R

Assume E has positive capacity. By Lemmas I5B and I5C we can find a nonzero measure f1- such that S" c E and

on S. Then by Fubini's theorem

f

Bk(a) df1-(a) =

Is" {

ik

8(f(Z), a) du* } df1-(a) =

ik

8" (f(z)) du*

~

m.

By Fatou's lemma (cf. App.I.2I) 00

=

f(lim inf 13/c(a)) df1-(a) Rk-R

~

lim infJ13/c(a) df1-(a) Rk-R

~

m.

This contradiction shows that E must be of capacity zero. In view of (84) we have obtained the following result (Nakai [6]) anticipated at the end of I2C: Theorem. The set E of points of S for which lim inf B/c(a) is infinite has vanishing capacity.

16B. Again let f be an analytic mapping of R into S and set

En

=

{a Ia

E

S, v(k, a) ~ n for all k(Q)}

with n=O, I, ... and

In particular, Eo is the set of all Picard points. The following statement is due to Nakai [6]. Theorem. For arbitrary analytic mappings f with (85)

the set Eoo has capacity zero. In other words, f covers S infinitely often except for a set of S of capacity zero.

§3. ANALYTIC MAPPINGS

16B]

73

Proof. We have only to show that En has capacity zero. Suppose this were not so. As in 16A there would then exist a nonzero measure f.L such thatSJJ,cEn , on S, and we would have (86) for all R k. We replace B(k, a) by J'Jk(a) in (60) and obtain

=

A(k, a)+ J'Jk(a)

(87)

C(k) + O(k).

By (86) integration of both sides with respect to df.L(a) yields

f

(88)

A(k, a) df.L(a)

=

C(k)f.L(S)+O(k).

By Fubini's theorem

f

A(k, a) df.L(a)

= 47T

1: {Is

v(h, a) df.L(a)} dh ::;; 47Tf.L(S)nk.

This together with (88) gives C(k) = O(k), contradicting (85).

CHAPTER III FUNCTIONS OF BOUNDED CHARACTERISTIC A major part of the classical value distribution theory is based on the Poisson-Jensen formula. For the general theory of meromorphic functions f on Riemann surfaces R this is not a preferred method as even the first main theorem appears with a remainder that causes unnecessary complications. However, the approach has the advantage that log If I on a regular subregion .Q can be decomposed. In the limit .Q -+ R valuable information is thus obtained provided f has bounded characteristic. It is this class M B of merom orphic functions of bounded characteristic that will occupy us in the present chapter. The chapter constitutes a natural complementation of Chapters I and II, where the results are meaningful only for mappings of unbounded characteristic. In §1 we derive the decomposition theorem on Riemann surfaces; the proof is "direct" in that no use is made here of uniformization. In §2 we give a shorter proof employing uniformization and taking the corresponding classical theorem in the disk for granted. (For fundamentals on uniformization see, e.g., Ahlfors-Sario [1, p. 181].) Wc also relate t,he class OMB of Riemann surfaces without non constant functions of bounded characteristic to other properties of function-theoretic significance. It is possible to generalize Poisson's formula also to mappings between Riemann surfaces and to consider such mappings of bounded characteristic (Fuller [1]). Here we shall, however, always have the extended plane as the range surface. The results in §l arc due to Sario [8]. [11]; those in §2 to Hcins [3], Parreau [2], and Rao [2].

§l. DECOMPOSITION A merom orphic function of boundcd characteristic in a disk is the quotient of two bounded analytic functions. This classical theorem can be extended to open Riemann surfaces R as follows. Consider the class M B of merom orphic functions f of bounded characteristic on R, defined in terms of capacity functions on subregions. Let L be the class of harmonic functions on R, regular except for logarithmic singularities with integral 74

10]

§1. DECOMPOSITION

75

coefficients. Then f EM B if and only if log If I is the difference of two positive functions in L. In this section we shall construct these functions on the surface R itself rather than on the disk of uniformization. Iflog If I is regular at the singularity of the capacity functions, then the classical reasoning (e.g., Nevanlinna [22]) can be generalized without difficulty. In the general case we introduce the extended class Me of locally meromorphic functions eU + iU' , U E L, with single-valued moduli. This class seems to be of interest in its own right. 1. Generalization of Jensen's formula

lAo Let n be a compact bordered Riemann surface with border f3n, and let p denote the capacity function in n with pole at a given point ZoEQ. By definition, p(z)-log Iz-zol ~O as Z~Zo, and p(z)=kn= const. on f3n. (Here we use a slightly different normalization than in I.l2A.) Given a continuous real-valued function f on f3n, the solution v of the Dirichlet problem can be expressed in the form v(zo) = (217) -1

JDo f dp*. To

see this let a be a level line p=c near zo, oriented to leave Zo to its left. Then by Green's formula

J

Bo-a

v dp* - P dv* = 0, and the statement

follows on letting c ~ - 00. There is a simple relation between p with singularity at Zo and the Green's function with singularity at t E Q: g(zo, t)=kn-p(t). For the proof let abe a level line g(z, t) = C1 near t, encircling t counterclockwise. Then

JDo-a- 6 g dp* -p dg* =0, and the statement is obtained in the limit

as c ~ - 00, c1 ~ 00. 1;:; is also a direct consequence of the well-known symmetry g(zo, t) = g(t, zo).

lB. Let R be an arbitrary open Riemann surface. We consider the class L= L(R) of functions U on R, harmonic except for logarithmic singularities ~log Iz-zil at Zi' i=l, 2,···, with integral coefficients "i. By definition the class Me = M e( R) consists of locally meromorphic functions (1)

f = eu + fu*

with

U E

L.

The conjugate function U* has periods around Zi and along some cycles in R. Every branch of f is meromorphic, the branches differing by multiplicative constants d with Idl = 1. The modulus If I is single-valued throughout R. The class Me contains the class M of (single-valued) meromorphic fUnctions on R.

Ie. Given Zo E R let Q be a regular subregion containing Zo and with boundary f3n. Denote by au, bv the zeros and poles of a given! E Me on R.

III. FUNCTIONS OF BOUNDED CHARACTERISTIC

76

We first assume that f(zo) #0,

n the function

v(z)

=

[ID

and that no au' b. is on f3n. Consider on

00,

L g(z, au) - L g(z, b.),

log If(z) 1+

alJEO

b"en

where each au' b. is taken as many times as indicated by its multiplicity. Clearly v is harmonic on n, and

= 2~

(2) log If(zo)1

r

log If(z)1 dp*-

JPo

L (kn-p(alL)}+ L (kn-p(b.)). b"eO

alJEn

If an aIL or b. is on f3n, we first apply (2) to a slightly smaller region Okn _e C 0 bounded by the level line p = kn - e, and then let e --+ O. Since all terms in the equation are continuous in e, (2) remains valid for 0. ID. Suppose now f(zo)=O or Laurent expansion

fez)

=

00.

A branch of f near Zo then has the

c,t(z-zO),t+CH1 (Z-ZO)Hl+ ... ,

and the other branches are obtained on multiplying by constants eir , with r real. The same is true of the branches of the function in

n and of ,p(z) =f(z)·cp(z)-,t = c,t+e(z-zo)EM e ,

where e(z-zo) log IC,t1

--+

= 21 7T

the sums

If

0 as z --+ zo0 On applying (2) to

r

JOn

,p we obtain

log If I dP*-Akn- If (kn-p(alL)}+ If (kn-p(b.)),

being extended over points in

n - Z00

IE. For - 00 < h ::; kn consider the region On C 0 bounded by the level line p=h. Let n(h, a) be the number of a-points, a=O or 00, of fin n", counted with their multiplicities. It is understood that n( -00, a) is the multiplicity (;::: 0) of the a-point at Z00 Then

If (kn-p(j-l(a)))

J

(kn-h) d(n(h, a)-n(-oo, a))

f:

(n(h, a) -n( -00, a)) dh,

kn

=

=

-<Xl

and A=n( -00, O)-n( -00, (0). We set (3)

N(h, a) =

r

<Xl

(n(h, a) -n( -00, a)) dh+n( -00, a)h

77

§1. DECOMPOSITION

IG]

and also use the notations N(O, a) =N(kn' a), N(O,j) =N(O, (0). We have obtained the following auxiliary result (Sario [11]): Jensen's formula on Riemann surfaces. For a locally meromorphic junction f E Me with single-valued modulus on an arbitrary open Riemann 81.trface R,

(4)

log Ic,d =

LIon

log If I dp*+N(O,j)-N(O,

j).

+

IF. Using log If I =max (0, log If!> we set I m(O,j) = 2"

(5)

7r

i

+ If I dp* log

Pn

and see that

2~ Ln log If I dp*

= m(O,j)-m(O,

j).

The counterpart of Nevanlinna's characteristic function is

T(O) = T(O,j) = m(O,fHN(O,j),

(6)

and Jensen's formula (4) takes the form

(7)

log Ic,d = T(O,f)-T(O,

j)

for allfE Me.

IG. We shall now consider differences f - a, and we therefore restrict our attention to the class Me Me of single-valued merom orphic functions f on

R. For a#oo we define the counterpart of Nevanlinna's proximity function

as (8)

m(O, a) = m(O, f~a) =

2~ Ion l;g If~al dp*,

and the counterpart of the counting function as (9) N(O, a)

= N( 0, f~a) =

f:

(n(h, a) -n( -00, a» dh+n( -00, a)kn ·

For a=oo the definitions were given by (5) and (3). We apply Jensen's formula (7) to f-a. Clearly N(O,f-a)=N(O,j), While the inequality log + (PI + P2) ::; log + PI + log + P2 + log 2 for arbitrary numbers PI' P2 > 0 yields +

m(O,f-a) ::;m(O,jHlog

lal + log 2.

78

III. FUNCTIONS OF BOUNDED CHARACTERISTIC

f2A.

We can now state the following extension to Riemann surfaces of Nevan. linna's form of the first main theorem: Theorem. For a mel'Omorphic function on an arbitrary Riemann surface R m(Q, a)

(10)

+ N(Q, a) =

T(Q)

+ cp(a),

+

where Icp(a)1 :::;log lal +log 2+ Ilog Icll, c being the first Laurent coefficient of f-a at zoo The theorem simply states that in the Poisson· Jensen decomposition of their log If(z)-al into positive and negative harmonic functions on values at Zo must add up to log If(zo) -al. The value of the positive com. ponent of log If(z) I at Zo is the characteristic T(Q).

n

2. Decomposition theorem

2A. We introduce the class llfB (or MeB) of functions f in M (or l11'e) with bounded characteristics T(Q)

(11)

=

0(1).

Explicitly, one requires the existence of a bound M < 00 independent of Q such that T(Q) < M for all Q c R. That (11) is independent of Zo will be

a consequence of a decomposition theorem which we proceed to establish. We continue considering arbitrary open Riemann surfaces R. Let LP be the subclass consisting of all positive functions in the class L defined in IB and the constant function O. We are ready to state (Sario [8]): Theorem. A necessary and sufficient condition for f E MeB on R is that (12)

log If I

=

u-v,

where u, v E LP. The proof will be given in 2B to 2L. As a corollary we observe that a function f EM is in M B on R if and only if (12) holds.

2B. First we shall discuss in 2B to 2E the casef(zo) =0 or 00. Suppose f E MeB. We begin by showing that R ¢ Oa. If f(zo) =00, then T(Q) ~ N(Q,j) ~ n(-oo, oo)kn, ~ kn,.

If R E Oa, then kn, --+ 00 as Q --+ R (App. 1.1) and consequently T(Q)00, a contradiction. If f(zo) =0, then from Jensen's formula T(Q,j)

we have T(Q,

=

T(Q.j)+O(I)

j) ~ N(Q, j) ~ n(-oo, O)kn, ~ ko.

and arrive at the same conclusion R ¢ Oa.

§1. DECOMPOSITION

2D]

79

On the other hand, if equality (12) is true, the existence of nonnegative 8uperharmonic functions u, v, at least one of which is non constant, implies R ¢ Oa (App. 1.4). Thus either condition of the thcorem gives the hyperbolicity of R, and we may henceforth assume the cxistence of the Green's function g(z, zo) on R if f(zo) =0 or 00.

2C. Let no, n", be the multiplicities of the zero or pole of fat Zo; then >t==no-n",. The functions f1 (z)

=

f(z)rp(z)

belong to Me> and we can reword the problem of characterizing MeB: Lemma. A necessary and sufficient condition for f E MeB is thatf1 E MeB. Proof. By definition, T(Q, rp)

=

N(Q, rp)+m(Q, rp).

For >t>o we have trivially N(Q, rp-1) =0, m(Q, rp-1) =0, whence T(Q, rp-1) sO, and it follows from Jensen's formula that. T(Q, rp) = 0(1). If A< 0, then N(Q, rp)=m(Q, rp)=O, T(Q, rp)=O, and consequently T(Q, rp-1)=0(1). In both cases T(Q, rp-1) = 0(1). T(Q, rp) = 0(1), The inequalities T(Q,j) :=:; T(Q,j1)

+

T(Q,j1) :=:; T(Q,j)

T(Q, rp-1)

=

T(Q,f1)+0(1),

+

=

T(Q,j)+O(I)

T(Q, rp)

yield (13)

T(Q, j)

=

T(Q, f1)

+ 0(1),

and the lemma follows. 2D. Reformulation of condition (12) is equally elementary: Lemma. A necessary and sufficient condition for (14) 'With u, v

log E

If I =

u-v

LP is that

(15) with u l , V l E LP. Proof. We know that

log

Ifll

=

log

If I + Ag

=

log

If I + (no-noo)g.

III. FUNCTIONS OF BOUNDED CHARACTERISTIC

80

[2E

If (14) is true, then

log Ifll

=

+ nog)-(v+nCX)g)

(u

and (15) follows. The converse is seen similarly. 2E. We conclude that Theorem 2A will be proved for f withf(zo)==o or 00 if we establish it for fl. Explicitly, we are to show that fl E MeB if and only if log Ifll =U l -Vl' U v Vl E LP. 2F. Let Pnz be the capacity function in

monic function h on

n with pole at z.

For a har.

n, h(z) = (27T) IBn h dpt,.z. -1

Denote by au' by the zeros and poles of fin R. Those in R -Zo are the zeros and poles of fl in R. Suppose first there is no au' by on f3n. Then the function h(z)

L

= log Ifl(z)1 +

L

gg(z, a u )-

allEQ-z o

gn(z, by)

b v EO:-2'O

is harmonic on n. Throughout this section the zeros and poles are counted with their multiplicities. We set (16) (17)

Yn(Z,fl)

=

L

gn(z, by),

bvEn-Zo

and (18)

Then (19)

Since all terms are continuous in n, the equation remains valid if there are zeros or poles of f on f3n. We observe that xn(ZO,fl)=m(n,fl) and Yn(zo,fl)=N(n,fl). Here we shall only make use of the consequence (20)

2G. We shall show next: Lemma. For

nocn

(21)

Uno (z, fl) :::; un(z, fl)'

(21)'

uno(z,fll) :::; un(z,fl l ).

81

§1. DECOMPOSITION +

proof. By (19), log If1 (Z) 15, Un(Z,fl) for every Q. It follows that

Xno(Z,fl) 5, 21 7r

=

21 7r

r

Un(t,fl)

J13no

dp~oz

Jorno (Un(t,fl)-Yno(t,fl)) dp~oz

= un(z,fl)-Yno(z,fl),

because this difference is regular harmonic in Q o. We have reached statement (21), and inequality (21)' follows in the same fashion.

2H. From (20) and (21) we infer that T(Q,fl) increases with Q. We can set T(R,fl) = lim T(Q,fl)

(22)

n~R

and use alternatively the notations T(Q)=O(I) and T(R) 1, then log 1~ - g1 is harmonic for 1~ 1 ::; 1, [md we infer from the Gauss mean value theorem that -u(g) =log 10 - gl = log Igl, or equivalently +

(55)

-u(g) = log

Itl·

§2. THE CLASS

6:8]

89

OMB

On the other hand, if Itl < 1 we observe that log lei & - tl = log 11- te-i&l. Since log 11 - ~t I is harmonic for I~ I :::; 1, we find as before that +

-u(t)

= log 11 - 0 . tl = log

In

Since (55) is thus valid for Itl ;;: 1 it holds by continuity for Itl = 1. On substituting in (54) we obtain the generalization to Riemann surfaces of Oartan's formula: + 1 2" (56) T(D., j) = log If(zo) 1+ 277 J° N(D., ei &) d{}.

r

This again shows that T(D., j) increases with D.. In terms of this evaluation one can also estimate (57)

and q

T(D.,fl + ...

(58)

+fq)

:::;

L T(D., fj)+ log q. 1

SE. We note that the functions in M B constitute a field. In fact, if

/1'/2 E M B, then by (57), (58) we have fl +f2' fd2 E M B, and Jensen's formula (7) shows that f EM B implies f -1 EM B. For later reference we also observe that the first main theorem (10) gives (f_~)-l E MB iffE MB, for any complex number ~. The problem of characterizing OMB is: When does the field M B reduce to the field of constants?

6. Characterization of

OMB

6A. If the surface R is not a member of the class Oc of parabolic Riemann surfaces, then for a point a of R we consider the class of all positive superharmonic functions on R, with at least the singularIty -log Iz-al at a. It is known that this class is not empty and that it has a smallest member g(z, a) = gn(z, a), the Green's function on R (App. 1.5).

6B. We shall assume that fE MB on R, i.e., T(D.,j)=O(I). We infer from the definition of the characteristic that (59)

N(D.,j)

=

L f(a)

~

v(j,a)gn(zo,a)+n(-oo,oo)k n :::; M
- R' and similarly to a continuous map

1': R' -'>- 11. But since l' oland 101' are continuous maps of Rand R' into themselves fixing R elementwise, we see by (a) that these are identity maps of 11 and R'. Thus 1 =1'-1 is a homeomorphism of 11 onto R' fixing R elementwise.

Ie. Finally we shall establish the existence of the Cech compactijication R of R. For each f in G(R) we denote by If the closure feR) of the range set feR) in [ -00, <Xl]. Since [ -00,00] is compact, If is also a compact Hausdorff space. Consider the topological product of If' f E G(R), with the weak topology:

By Tychonoff's theorem ~ is again a compact Hausdorff space. We denote by 'TTf the projection map of ~ onto If. Let q; be the map of R into ~ defined by q;(Z) =

TI

fez),

fEC(R)

i.e., 'TTf(q;(z))=f(z). It is easy to see by the definition of the weak topology that q; is a homeomorphism of R onto q;(R). Hence it is sufficient to demonstrate the existence of the Cech compactification of q;(R). We shall show that the Cech compactification of q;(R) is the closure q;(R) of q;(R) in ~. Since q;(R) is a closed subset of a compact Hausdorff space ~, q;(R) itself is a compact Hausdorff space. Condition (a) is clearly

2C]

§1. THE EVANS-SELBERG POTENTIAL

101

satisfied. To prove that cp(R) satisfies (b) it is sufficient to show that for any F in C(cp(R)) there exists an F in C(cp(R)) such that F Icp(R) = F. Obviously F 0 cp =f E C(R) and therefore F has the representation F(cp(z)) = f(z) =7Tf(CP(Z)). For any x E L1 define the function F by F(x) = 7Tf(x).

Since 7Tf is the projection of L1 onto If, it is clearly continuous on L1 and hence on cp(R). Thus FE C(cp(R)). But on cp(R), F(cp(z)) =7Tf(CP(z)) = F(cp(z)), i.e., F Icp(R) = F.

2. Green's kernel on the tech compactification 2A. Let Q be a regular region on a parabolic Riemann surface R. Lemma. Let s be a superharmonic function on R - Q bounded from below and continuous on R-Q. Then s is continuous on R-Q and s;:::miniJns on R-Q. Proof. We can extend s continuously to all of R. The function s thus extended is continuous on R and consequently s is continuous on R - Q. Assume that s;:::c> -00 on R-Q. Let {RnH' be an exhaustion of R, i.e., a sequence of regular regions of R with Rn c Rn + 1 and U Rn = R. Suppose that Q c Rl and let Un be harmonic on Rn - Q with Un loRn = c and unl oQ=m=miniJns. Then by the maximum principle s;:::u n on Rn-Q. But since R is parabolic, Un -J>- m as n -J>- 00. Thus s;::: m on R - Q and hence on R-Q (cf. App.I.4). 2B. Let Ro be a regular region with connected complement R - Ro and let ga(z) be the Green's function on R - Ro with its pole at a E R - Ro. We set ga(z) =0 if one or both of z and a belong to oRo. It is bounded except in an arbitrary neighborhood of a. In view of Lemma 2A it is uniquely determined. For the existence of the Green's function and its properties we refer the reader to Ahlfors-Sario [1, pp. 188-189]. In particular, the function is symmetric: (1)

2C. We next show that the function (z, a) -J>- ga(z) is continuous on (R-Ro) x (R-Ro) -oRo x oRo. To this end we first prove: Lemma. Let D be a plane region and X a topological space. If h(z, x) is a real-valued function bounded from below such that z -J>- h(z, x) is harmonic in D and x -J>- h(z, x) is continuous on X, then (z, x) -J>- h(z, x) is continuous on

DxX.

102

IV. FUNCTIONS ON PARABOLIC RIEMANN SURFACES

[21)

Proof. We may assume that h > O. Fix a point (zo, x o) in D x X. Choo;.;e r so that the closed disk {z liz -zol ~ r} cD. Then by Poisson's formula r-p r+p - - h(zo, x) ~ h(z, x) ~ - - h(zo, x), r+ p r-p

p =

Iz-zol

and consequently lim(z.x)_(zo,xo)h(z, x) = h(zo, x o). From this lemma we conclude that (z, a) -+ ga(z) IS continuous at (zo, ao) with a o =F zoo Next as,mme that a o = Zo E R - Ro. Consider the cloR('d disks D: Iz-zol ~ 1/2 and D': Iz-zol ~ 1/4. Since ga(z) iii continuow.; on aD x D'. there exists a positive constant M such that ga(z)::o: M on aD x D'. For (z, a) E D x D' we have

1

ga(z) = log -I- I +h(z, a), z-a

where h(z, a) is harmonic with respect to z. Clearly -log Iz - a I ~ log 4 on and therefore

aD x D'

h(z, a) ::0: M-log4 aD x D'. By the harmonicity of z -+ h(z, a) in D the same is true on D x D', and we have

Oil

1

ga(z) ::0: log -I- I +M -log 4 z-a

for all (z, a)

E

D' x D'. We conclude that lim(z,a)~(Zo,Zo)ga(z)=CXJ=gzo(zo).

2D. By Lemma 2A, a -+ ga(z) is continuous on It - Ro. We denote by iJa(z) the extended function of a. Once we fix a in It - Ro, z -+ {fa(z) is again a real-valued function on R-Ro. We shall show: Lemma. The function z -+ {}a(z) is a strictly positive harmonic function on R - Ro, except perhaps at a, with continuously vanishing boundary values on aRo· Proof. Since {}a(z) = ga(z) if a E R - Ro, we have only to consider the case a=a E r. In view of Lemma 2A, iJ,,(z) > 0 for any z E R- Ro. Fix a point zoER-Ro and take the disks D: Iz-zol~2 amI D': Iz-zol~1/4 and_a subsurface F of R, with R - F a regular region such that Ro U Dc R - F. Then ga(z) is finitely continuous for (a, z) in aF x D'. By the harmonicity and boundedneiis of a -+ ga(z), zED', on F and by Lemma 2A we conclude that there exists a positive constant JI such that ga(z) < JI for (a, z) in F x D'. By using the Poisson representation as in the proof of Lemma 2C

we see that

2FJ

§I. THlC EI'ANS·SELBERG POTENTIAl,

10:,

where

, ( 1 - IZ-Z'1 1 + Iz-z'l p(lz-z I) = max 1-1 + Iz-z 'I' 1- Iz-",-'I Thus the family {ila(:::) I a E F} offnnctions z --+ ila(z) on D' is compact with respect to the parameter Ret F' and ila(z) converges uniformly to {}ii(Z) on D' as a --+ ii, a E F. \Ve conclude that ih(z) iR harmonic in D' and consequently in R - Ro. Next take a point Zo on c'R o and let D and D' be as above such that oRo n D={z Ilz-zol ::; 2, Im(z-zo) =O}. By the preceding argument there exists a constant N such that ga(z) < N for (a, z) E F x (D n (R - Ro)). Let ua(z) =ga(z) +N for (a, z) E F x (D n (R- Ro)) and ua(z) = -ga(;z)+N for (a, z) E F x (D n Ro). Then z --+ ua(z) is harmonic in D and 0 < ua(z) < 2N for (a, z) E F x D. By an argument similar to the one above we see that ua(z) converges uniformly on D' as a E F tcnds to ii. Thus in particular ga(z) --+{}a(z) uniformly on D' n (R-Ro) and we see that {}il vanishes on aRo.

2E. In view of Lemma 2D the function z --+ {}a(z) =limb~agb(z) with bE R - Ro is nonnegative harmonic on R - Ro for each a in R - Ro and hence it can be extended continuously to R - Ro. Definition. The Green's kernel G(x, y) on R - Ro is the function on (R - Ro) x (R - Ro) defined by the double limit G(x, y) = lim (lim gb(a)) a-x

b-y

with a, bE R-Ro. Clearly if we fix y in R - Ro, then G(z, y) = {}y(z) on R - Ro. Thus x --+ G(x, y) is the continuous extension of the function z --+ {}y(z) to R - Ro.

2F. We shall show: Lemma. The Green's kernel G(x, y) on R - Ro has the following properties: (a) G(z, t)=gt(z)for (z, t) in (R-Ro) x (R-Ro), (b) G(z, x) = G(x, z) for z in R- Ro and x in R - Ro, (c) G(x, y) is continuous in x on R-Rofor fixed y in R-Ro, (d) G(z, x) is continuous in (z, x) on (R - Ro) x (R - Ro) - aRo x oRo and finitely continuous in (z, x) on (R-Ro) x r, (e) G(z, x) is harmonic in z on R-Ro-xfor fixed x in R-Ro, (f) G(x, y) >0 (resp.=O) on R-Rofor any fixed y in R-Ro (resp. aRo), (g) fORo dG*(z, x) = 277 for any fixed x in R - Ro, where aRo is oriented positively with respect to Ro, (h) G(z, x) =0 on aRo for any fixed x in R - Ro.

104

IV. FUNCTIONS ON PARABOLIC RIEMANN SURFACES

[2F

Proof. Properties (a) and (c) are obvious by Definition 2E. Property (b) follows from Definition 2E and (1), i.e., G(z, x) = lim (lim !/b(a)) a-z

b-x

= lim gx(a) = fix(z) = lim !/b(Z) a-z

=

b-+x

lim (lim !/b(a)) b-x

a-z

= lim (lim !/a(b)) b-x

=

a-z

G(x, z),

where a, bER-Ro. Properties (e), (f), and (h) are obvious in view of Lemma 2D. Next we prove (d). The continuity of G(z, x) at (zo, xo) E (R - Ro) x (R-Ro)-oRo x oRo follows from 2C and (a). To prove the continuity at (zo, xo) in (R-Ro) x r let D'={zllz-zol ~ Ij4} and RI be a regular region such that Ro U D' C R I . In case Zo E oRo we moreover assume that oRo n D'={zllz-zol ~Ij4, Im(z-zo)=O} and we extend the harmonic function z ---* G(z, x), X E R - R I , to D' harmonically; this is possible by (h). Since G(z, x) = gx(z) we conclude in the same manner as in 2D that G(z, x) is bounded for (z, x) E D' X (R-R I ). By (e), (c), (b), and Lemma 2C, G(z, x) is finitely continuous at (zo, xo) E (R - Ro) x r. Finally we establish (g). First assume that a E R-Ro. Let {RnhXl be an exhaustion of R with Ro c R I , a E R I , and Da: Iz -al < 1 with 15ac RI Ro. Denote by Vn the harmonic function in Rn - Ro - Da such that Vn oRo U aDa = 1 and Vn loRn = O. Since R is parabolic, Vn ---* 1 on RRo - 15a and DRn -lio -15 a (v n ) ---* 0 as n ---* 00. By Green's formula

I

=

-

r

JORo

d!/~ -

r

JODa

d!/~,

where oRo and aDa are oriented positively with respect to Ro and Do.. Since !/a(z) = -log Iz-al +h(z) on Da with harmonic h,

r

JODa

while

d!/~ =

-217,

§1. THE EVANS·SELBERG POTENTIAL

si\]

105

as n -+ 00. Thus fORo dg: = 27T, i.e.,

r

(2)

= 27T

dG*(z, a)

JOR o

for a in R-Ro· Next let x E r. For a -+ x, G(z, a) converges uniformly to G(z, x) on Rl - Ro as in 2D. On letting a -+ x in (2) with a E R - Ro we obtain (g). 3. Transfinite diameter SA. Again let R be a parabolic Riemann surface and R (resp. r) the eech compactification (resp. boundary) of R. Given a nonempty compact set K in R - Ro we define Dn(K) by 1.··· .n

L

G(Xj, Xj)'

i<j

For arbitrary points Xl" . " xn+l in K 1.··· .n+l

L

j

k-l G(Xj, Xj)

=

0 the function Gc(z, t)=min (G(z, t), c) is finitely continuolls on x By the Stone-Weierstrass approxima_ tion theorem (see, e.g., Loomis [1, pp. 9-10]) there exists a function

an an.

m

CPn(z, t) =

L aJij(z)hj(t) 1

with aj real andfj, hj finitely continuous on

an such that

We infer that (10)

Observe that J J CPndfLnkdfLnk =

--+

~ a j ffjdfLnk f

hjdfLnk

~ a j JfjdfL J hjdfL =

;::: f J Gc(z, t) dfL(Z)

as k --+

00.

In (10) by first letting k --+

00

f J CPndfL dfL

dfL(t)-~

and then n --+

00

D(an) ;::: f f Gc(z, t) dfL{z) dfL(t).

Again for c --+

00

we deduce that D(an) ;:::

IIfLI12 ;:::

W(aQ).

we obtain

5.A.]

§1. THE EV ANS·SELBERG POTENTIAL

4E. We are now able to show (Nakai [1]): Theorem. If R is parabolic, then the Cech boundary E(r)

= D(r) =

r

113

of R is so small that

00.

proof. By Lemmas 3C, 3D, 4D, and 4C E(r)

D(r)

;?:

D(oO)

;?:

W(oO)

;?:

;?:

217

,

Dn-Ro(un )

where 0 is a regular region with Eo C O. Since R is parabolic, Q ---'.>- R and consequently Dn-Ro{u n ) ---'.>- O.

Un ---'.>-

0 as

5. Construction

SA. We denote by a co =aco(R) the Alexandroff ideal boundary point of R. Then the complements of regular regions form a base for the neighborhoods of a co in R u a co . First we show (Nakai [1], Kuramochi [3]): Theorem. Let R be a parabolic open Riemann surface and let Ro be a regular region of R with connected complement. Then there exists a positive harmonic function p(z) on R - Eo such that (a)

limz~a",

p(z) =

00,

(b) p(z) has continuously vanishing boundary values on oRo, (c) fORo dp*=217, where oRo is positively oriented with respect to Ro.

Proof. By Theorem 4E, E(r)=oo. Since En(r) ---'.>- E(r), we can find a sequence {nk}:=l such that E nk (r»2 k - 1 • Choose a system of points X k1 , ••• , X knk in r such that nk

(U)

inf

2: G(x,

XEr i

X ki )

> n k2 k - 1 •

== 1

We denote by ILk the measure on i= 1,· .. , n k.

r

with total measure 2 - k such that

f'k(x kt )=nk 12-k,

Let

() Jrr G( z, y ) dILk ()Y

Pk z

=

~ G(z, xktl

= i~

nk2k .

Then it is easy to see by Lemma 2F that Pk(Z) is positive and harmonic on R-Eo, vanishes on oRo, and is continuous on R-Ro. Hence by (II) Pk(X) >

t

on

r,

and we can find a regular region Ok in R with Eo C Ok such that (12)

114

IV. FUNCTIONS ON PARABOLIC RIEMANN SURFACES

Next let fL= L:r' fLk· Then fL is a unit Borel measure on

p(Z)

=

r.

Put

L

G(z, y) dfL(Y)·

By Lemma 2F, G(z, y) is finitely continuous on (l!-R o) x r; consequently p(z) is continuous on R - Ro anJ a fortiori on R - Ro. Again by Lemma 2F, p(z) vanishes on oRo and is harmonic on R - Ro. Thus p(z) satisfies (b). Clearly <Xl

p(z)

=

L Pk(Z). 1

For arbitrary m let Km=D. 1_ u···u D. m , where Ok is as in (12). Then K m is compact in R with Km ~ Ro and by (12)

on R-Km. This shows that (a) holds for thisp(z). Finally by (g) of Lemma 2F

5B. Now let R be an arbitrary open Riemann surface. The function p(z, a) on R is called the Evans-Selberg potential on R with its pole at a E R if it satisfies the following three conditions: (a) a -* p(z, a) is a harmonic function on R -a,

(b) p(z, a)-log Iz-al is harmonic at a, (c) limz~a",p(z, a)=oo.

5C. We are ready to state our main result (Nakai [1], Kuramochi [3]): Theorem. An open Riemann surface R is parabolic if and only if the Evans-Selberg potential exists on R. Proof. First assume that the Evans-Selberg potential p(z, a) exists on R. If we take t sufficiently large, then Da={zIP(z, a)< -t} is a relatively compact disk by (b), (c), and (a). The function u(z) =p(z, a) +t is positive harmonic on R-Da' vanishes on aDa, and limz~a", u(z) =00. Let Rn

= {ZIZER-Da'u(z) < n}uDa·

Then {Rn}r' is an exhaustion of R. Let un(z) be harmonic on Rn-Da with Un I oDa=O and Un I oRn=l. We have un(z)=u(z)/n -* 0 on R-Da' and R is parabolic (App. 1.4). Conversely suppose R is parabolic and consider the disks Ro: Iz-al - 00 is a planar covering surface. By construction the set of all points covered by S infinitely often is precisely S while no point of K is covered at all.

Ie. We proceed to prove that the covering surface S is parabolic. Let Sn with n> 0 be the part of Sn cS over the subregion Sn of the base surface S. For n=O we denote by So the region So on So cS. It is easy to see that Sn is a relatively compact region of S and that SncSn+1 for every n:2':O. Moreover, {Sn}O' exhausts S. Let wn be the harmonic measure of fJS n with respect to Sn -So and form the subharmonic function wn on Sn such that wn = Wn on 8 n -So and wn =0 onS o. Denote bYffJ the projection of S on the extended ~-plane and by Dn(w n 0 ffJ) the Dirichlet integral of wn 0 ffJ over the region Sn -So. By the Dirichlet principle

n-1

11

(N(v) + I)Dn(w n),

v=l

because Sn covers each point of Sn exactly n~,:-; (N(v) + 1) times. From this it follows by (1) that limn~oo Dn(w n) =0, and we conclude that ,9 is parabolic (App. 1.4).

ID. Since S is planar there exists a bijective conformal map / of S onto a region R of the extended z-plane with E =C R of capacity zero The function f=ffJ 0/- 1 satisfies the conditions of the theorem. In fad. let zo be a point of E and let r be a positive number such that the circle c: Iz-zol =r does not meet E. Since the image/- 1 (c) of con S is a compact set, there exists an n such that Rn contains j -l(C), and the disk Iz -zol < r contains the image of at IC'ast one component of S -lin undC'r /. Every component of R-Sn contains as a subrcgion at least one replica of S,!: for k= 1,· . " N(m-l) and m>n. It follows that in an arbitrary neighborhood U(zo) of Zo E E, f takes on each valuc contained in the complement of K with respect to the extended ~-plane infinitdy often. Thus our theorem is established. Remark. Modifying the above construction we can also prove the theorC'm under the assumption that the given set K is a countable union of compact sets of capacity zero. For the details of the construction we refer the reader to Matsumoto [1].

§1. INFINITE PICARD SETS

20]

123

As a consequence of this extension we see that Picard sets cannot only be uncountably infinite but may also have considerable extent. Take an uncountable compact set K of capacity zero in the ~-plane. Let Kc be the translation of K with respect to the vector ~. The union of the sets Kc for all rational complex numbers ~ is an example.

2. Sets of positive capacity 2A. We shall now show that even relatively small sets of singularities can permit Picard sets of positive capacity (Matsumoto): Theorem. There exists a meromorphic function with a set of essential singularities of vanishing linear measure and with a Picard set of positive capacity at each singularity. The proof will be given in 2B to 2D. 2B. The Cantor set on the interval 10: [ - 2 -1, 2 -1] of length 1 is constructed by first centrally removing from loa segment of length 1 - gl with 0 < gl < 1. The remaining set 11 consists of equal segments 1 1.1 and 11 ,2 of total length gl' Inductively one removes centrally from each segment Ink of In with n= 1,2, ... and k= 1" . " 2n a segment of length 2-n(1- gn+l) n~ gi with 0 < gn+l < 1 so as to obtain a set In+l of equal segments of total length n~ + 1 gi' It is known (Nevanlinna [22, p. 155]) that the capacity of the Cantor set In vanishes if and only if

nr

~ log g;;l =

of

00

2n

.

The choice gn = 2l n with a fixed 0 < l < 1/2 provides us with segments of equal lengths In(n + 1)/2 and with a Cantor set K of positive capacity. Clearly K is of linear measure zero. 2C. Consider the complement S of K with respect to the extended ~-plane. Choose an no such that

(1 -lno + 1 )lno (no + 1 J/2
ll- (no + p) 5(I-lno +p+ l)lno +p

p-l ~ 2 no+ p . TI (2 no+v+I) < TI (2 no+v+I)·

=

v =0

v= 0

p

TI

li(no+v)

v = Po + 1

for a suitable Po and any P:::::Po, where we have set A = f1~~0(2no +v+ I). It follows that for N > Po N

L log fLno +

L N

p

-log v(N) >

p=1

P=Po

log ll(no+p)

+1

+ 2- 1 (N -Po)(N +2n o +po+ I) log l2-1og A =

l 2- 1 (N -Po)(N +2n o +po+ I) log r-Iog A, 1

31\] SO

125

§2. FINITE PICARD SETS

that

;~~ {~1 log fLno +p-Iog v(N)}

=

00.

By the criterion of Pfluger [2]-Mori [2] or of Kuroda (App. 1.16-19) we conclude that S belongs to the class 0 AB' We have taken the slits Lnk on the real axis of the ~-plane. Therefore there exists a bijective conformal mapping j of S onto the complement R of a compact set E on the real axis with respect to the extended z-plane. Since S belongs to the class 0 AB we conclude by a theorem of Kametani [1] and Ahlfors-Beurling [1] that E is of linear measure zero (ef. AhlforsSario [1, pp. 252-254]). Denote by rp the projection of S into the extended ~-plane. We see by the same reasoning as in 1D that the function f=rp oj-l satisfies all conditions of our theorem. The proof is herewIth complete.

§2. FINITE PICARD SETS We turn to the following problem: Is there a perfect set E in the z-plane such that every function f meromorphie in the complementary region of E with an essential singularity at each point of E has at most a finite number of Picard values at each singularity? Recently Carleson [2] and Matsumoto [2], [4] gave an affirmative solution to this problem by establishing sufficient conditions for sets E to have the above property. In particular, Carles on exhibited sets of positive capacity permitting at most three Picard values. In the sequel we shall derive a sufficient condition, due to Matsumoto [2], [4], that substantially sharpens earlier results.

3. Generalized Picard theorem 3A. Let E be a perfect set of the extended z-plane, i.e., a closed set containing no isolated points. Clearly there exists an exhaustion {Rn} of the complement R of E with the additional property that each component Rnm of Rn - Rn -1 is doubly connected. The regularity of the exhausting regions is modified only in that multiple points of the components of oRn are permitted where Rnm branches off into two or more components of Rn+l- Rn- We shall call this exhaustion normal. Set an-l.m=oRnm (\ oR n _ 1 and finm=oR nm (\ oRn. Let Unm be the harmonic function in Rnm with boundary values 0 on an -l.m and a constant P,nm on finm such that the flux is JtJr nm dU~m = 21T. Here finm is oriented

126

V. PICARD SETS

l3B

positively with respect to Rnm. Similarly let Un be the harmonic function in Rn - R n- 1 with boundary values zero on oR n- 1 and a constant an on oR n such that fORn du~ = 27T. For a suitable choice of the additive constant of u~ the function un + iu*n maps Rnm cut along a level line of u~ onto a rectangle 0 < Un < an' bm< u~ < am + bm with N(n)

2: am = 27T

m=l

and

with 1 <m:5.N(n). Consequently un+iu~ maps Rn-Rn-1 with suitable slits bijectively onto a slit rectangle 0 < Un < an' 0 < u~ < 27T. We define the function u+iu* by un+iu~+Lj;Jaj on each Rn-Rn_1,n~l, with ao =0. It maps R - Ro which has at most a countable number of suitable slits onto a strip 0 < u < L, 0 < u* < 27T with 00

L

=

2: aj :5.

00.

j=l

This slit strip is the graph of R associated with the exhaustion {Rn} in the sense of Noshiro [5]. The number L is called the length of this graph. By theorems of Sario [2] and Noshiro [5], R is the complement of a compact set of capacity zero if and only if there exists a graph of R whose length L is infinite (App. 1.14-15).

I

3B. Let f3r be the level line {z u(z) = r, 0 < r < L} on R. Except for a countable set of values r which we shall exclude, f3r consists of a finite number of Jordan curves f3rm, m= 1,2, ... , n(r). We shall call each component of the open set Rn-Rk' n>k~O, an R-chain. For every f3rm consider the longest doubly connected R-chain R(f3rm) such that f3rm is contained in R(f3rm) or is one of the two components of oR(f3rm). Denote by log fL(f3rm) the modulus of this R-chain and set

Generally Rnm may branch off at f3nm into a finite number of regions Rn + 1.k' If every Rnm branches off into at most p regions Rn + 1.k' we say that the exhaustion {Rn} branches off at most p times everywhere.

4i\]

§2. FINITE PICARD SETS

127

3e. We are rcady to state our main criterion (Matsumoto [2], [4]): Theorem. Let E be a totally disconnected compact set in the z-plane and let

R be its complementary region. If there exists a normal exhaustion {Rn} of R which branches off at most p times everywhere and if lim fL(r)

(2)

= 00,

T~L

then every function merom orphic in R with an essential singularity at each point of E has at most p + 1 Picard values at each singularity. 3D. We shall actually prove more. Let f be merom orphic in R with at least one essential singularity in E, not necessarily at each point of E. We shall say that f has a Picard value ~ in R at an essential singularity Zo E E if there is a neighborhood U(zo) of Zo such thatf(z),., ~ in R n U(zo). Such a value ~ could be taken by f at points of E near Zo where f is meromorphic. We claim (Matsumoto [4]): Theorem. Under the same conditions on R as in Theorem 3C, every function meromorphic in R with at least one essential singularity in E has at most p+ 1 Picard values in R at each singularity. If a meromorphic function in R omits p + 2 values in R it must consequently be rational. In the case p = 1 the set E reduces to a point and our assertion is true by Picard's theorem. We shall give the proof for p=2. The general case is analogous if we take p+2 points ~i instead of 4 in Lemma 4A below.

4. Auxiliary results 4A. Before proving the theorem we shall establish two lemmas. Consider the Riemann sphere ~ with radius 1/2 tangent to the ~-plane at the origin. For ~ and ~' in the extended ~-plane wc denote by [~, their chordal distance. Further let C( ~; 0) 'with 0> be the spherical open disk with center ~ and chordal radius o. Let f be a meromorphic function in an annulus 1 < Izl < e" omitting four values ~" i = 1, ... ,4, and let 0> be so small that the spherical disks C(~i; 0) are disjoint by pairs. We recall the Bohr-Landau theorem (e.g., Tsuji [19, p. 268]): If g is analytic in Izl < 1 and g(z),., 0, 1 there, then

°

n

°

~~~ Ig(z)1

:0;

exp

(D log i'~(~)' +2))

for all r < 1, with D a positive constant independent of g. Using this inequality (a precise form of Schottky's theorem) we shall establish the fOllowing auxiliary result:

V. PICARD SETS

128

[4B

Lemma. There is a positive constant S' > 0 such that, if f takes a value outside of C(~i; S) for some i at a point on Izl =e,,/2, then the image of Izl '== eU/2 under f lies completely outside the concentric spherical disk C(~i; S'). Here S' depends only on a, S, and the points ~i' not on f. Proof. By the Bohr-Landau theorem we see that if g is an analytic func_ tion in 1 < Izl < e" such that

g(z) =1= 0, 1

and

min

Ig(z)1

< M

for some

M > 0,

Izl=e"12

then there is a constant M' > 0 depending only on M and a and satisfying the inequality max Ig(z)l::;:; M'. Izl = e"12

Denote by t = T;km with distinct i, j, k the linear transformation which maps ~i' ~j' and ~k to the point at infinity, the origin, and the point t = 1, respectively. Since the number of such Th is finite, there exists a positive M so large that for each T;k the image of the exterior of C(~i; S) is contained in It I <M. For this M, t=T;k(f(Z)) has the same properties as g stated above, and hence

with M' > 0 depending only on M and a. The image of the exterior V of It I ::;:; M' under (T;k) -1 is an open disk containing ~i' Denote by d;k the chordal distance between ~i and the boundary of (T;k) -1( V). The minimum under all distinct i, j, k S' = mindh is positive and obviously satisfies all conditions of the lemma. 4B. The following estimate is a revised form of Carleson's [2]. Lemma. Let f be a meromorphic function in an annulus 1::;:; Iz I : ;:; el'. If f takes no value in a spherical disk C(~o; S), then there exists a positive constant A depending only on S such that the diameter of the image of Izl =el'/2 under f in terms of the chordal distance is dominated by Ae -1'/2 for sufficiently large /L.

In particular, if S is sufficiently close to 1, i.e., the spherical disk C( -1/~0; d) complementary to C(~o; S) has a sufficiently small radius d, then A < Bd, where B is a positive constant. Proof. We may assume without loss of generality that the center ~o of C(~o; S) is the point at infinity, for otherwise we can map to this point by the linear transformation (1 + '0')/(' - '0)' under which the chordal

'0

§2. FINITE PICARD SETS

5A]

129

distance remains invariant. Let I~I >M be the region in the ~-plane corresponding to C(~o; 8). Then

If(z)1

:os; M

on

1:os;

Izl

:os; eU.

By Cauchy's integral formula we have f'(z)

=

_1

27Ti

{r

)111=0"

Izl =eU/ 2 and hence, if /L"?2,

for every z on ,

If (z)1

M {27Te U 27T} 2e 2 :os; 27T (eU_eU/2)2+ (eU/2_1)2 :os; (e-1)2 Me- U.

It follows that the length of

I

JJ!L dt- r JJ!L dt} (t_Z)2 )111=1 (t_Z)2

Izl=0"'2

If'(z)lldzl

:os;

Izl = eU/ 2 under f

has the bound

2e2 Me- u ·27Teu /2 (e-l)

--2

=

47Te 2 Me- u /2. (e-1)

--2

We conclude that the diameter of the image of Izl =eU / 2 with respect to the metric Id~l, and consequently with respect to the chordal distance, is dominated by 27Te 2(e-l)-2Me- U/2. We can choose A=27Te2(e-l)-2M as M depends only on 8. If d < 1/2, then M < 2d and

satisfies our condition. Our lemma is herewith established.

5. Proof of the generalized Picard theorem SA. Theorem 3D will be proved by contradiction. Suppose there exists a mel'Omorphic function fin R with at least one essential singularity in E and with more than three Picard values at an essential singularity 20 E E. Then there is a neighborhood U(zo) of Zo such that f omits four values ~i' i= 1, .. ·,4, in U(zo) n R. We take a positive 8 so small that the spherical disks C(~i; 8) are disjoint by pairs. For this 8 and a a> 0, Lemma 4A determines 8' > 0. We take this 8' as 8 of Lemma 4B and choose /Lo so large that Ae- uo/ 2
such that

°

/L(r) > /Lo+2a

for all

r

with

ro < r < L.

130

V. PICARD SETS

L5B

5B. The level line ,Br ={z I u(z) =r} consists of a finite number of Jordan curves ,Brie with k= 1, .. " n(r), and one of them, say ,Br.1> encloses zoo For r sufficiently near L the longest doubly connected R-chain R(,Br.1) = D 1.1 defined in 3B is contained in U(zo). The modulus of D 1 • 1 is greater than fLo + 2 but is not infinite, for otherwise Zo would have to be isolated and f could not have four Picard values at zoo Therefore D 1 •1 must branch off. Now suppose D 1 • 1 is a component of the open set Rn -Rn' with n> n', and branches off into two regions R n + 1 •m and R n + 1 •m ,. Consider the longest doubly connected R-chains D 2.1 and D 2.2 containing R n + 1,m and R n + 1,m" respectively. They both have moduli greater than fLo + 2 and one of them, say D 2 ,l' separates Zo from D 1 ,l' Its modulus is finite for the same reason as above. Hence D 2 ,l is a component of the open set R il - Rn for some n and branches off into two regions R il + 1.m and R il + l,m" We denote by D 3 ,l and D 3 ,2 the longest doubly connected R-chains containing them. If the modulus of D 2,2 is infinite, one of the boundary components of D 2,2 is a point Zl E E andjis meromorphic at Zl' If the modulus is finite we obtain two R-chains D 3 •3 and D 3 ,4 in the same manner as above. Thus we have at most 22 R-chains D 3 ,q such that their harmonic moduli are greater than fLo+2, and one of them encloses zoo Moreover, each of them branches off into two regions if the modulus is finite, or has a point Zl E E as one of its boundary components at whichjis meromorphic if the modulus is infinite. Continuing inductively we obtain a set of R-chains Dpq with p = 1,2, ... and q = 1, . , . , Q(p) ~ 2 P -1, which has the following properties: (a) U;'=l U~~{ Jjpq~;)., where;). is the intersection of R with the set bounded by the Jordan curve ,Br,l' (b) the modulus of each Dpq is greatcr than fLo+2, (c) each Dpq branches off into two Dp+ l,q if its modulus is finite, or (c') each Dpq has a point Zl E E as one of its boundary components and j is meromorphic at Zl if the modulus of Dpq is infinite. In this casc we shall denote the point Zl by Zpq and the value j(zpq) by ~pq.

5C. Each Dpq is conform ally equivalent to the annulus 1 < It I <e". If fL < 00 we denote by D~q, D;q, and Dgq the subregions of Dpq corresponding to the annuli 1 < It I < e", e" < It I < e" -" and e" -" < It I < e", respectively, and by ,B~q, ,B~q, and ,Bgq the closed curves corresponding to It I=e,,/2, It I=e"/ 2 , and It I=e"-"/2, respectively. We shall see that for each ,B~q the diameter of its image under j with respect to the chordal distance is dominated by K =min (1/24, 8'/3). In fact, for z' E ,B~q and z" E ,Bgq the images j(z') and j(z") lie outside of at least one C(~i; 8), say C(~l; 8), and hence on applying Lemma 4A to D~q and Dgq we see that the images of ,B~q and ,Bgq, and, as a consequence of the maximum principle, the image of the annular l'egion bounded by

§2. FINITE PICARD SETS

5EJ

131

them, lies outside of C(~1; S'). Our assertion follows from Lemma 4B, because the modulus of D~q is greater than fLo. SD. Every D p+1.q' with p?1 has in common with another D p+1.qH a

D p.q branching off into them, and we shall denote by l::!pq the triply connected region bounded by (3~q, (3~+1,q" and (3~+1,qH, where (3~+l,q.=Zp+1,q· or (3~+ 1,q" = zp + 1,q" if Dp + 1,q' or Dp +1,q" has infinite modulus. For ~ E f((3~q), Ef((3~+1,q·), and ~"Ef((3~+1,q") we consider spherical disks Ca; K). C(~'; K), and C(~"; K), which of course contain f((3~q), f((3~+1,q.), and f((3~+ 1,q")' respectively. Since K < S' /3, there is at least one ~l' say ~1' not contained in the disks, and we conclude that each disk meets the union of the other two disks. In fact, if this were not the case, there would exist a z* E l::!pq such that f(z*) can be joined to ~1 by a curve A in the exterior of the union of these three disks. We would be led to the contradiction that the element of the inverse function f -1 corresponding to z* can be continued analytically along A up to a point arbitrarily near to ~1 so that f takes the value ~1 in l::!pq. We conclude: (ex) For every l::!pq there is a spherical disk with the chordal radius 3K containing its image f(l::!pq). Next consider (3~q for p?2. The region l::!pq and some l::!P_1,q' have (3~q as the common boundary, and D pq c l::!pq U (3~q U l::!p _ 1,q" In view of (ex) the images of l::!pq U (3~q U l::!p -1,q' and consequently of D~q C Dpq are contained in a spherical disk with chordal radius 6K < 1/2. On applying Lemma 4B to D~q for d=6K we see that the diameter off((3~q) is < 6KBe- Il o /2 < K/2, For p?2 each boundary component of l::!pq thus has an image with diameter less than K/2. By the same reasoning as above we infer: ((3) For p? 2 the image of every l::!pq is contained in a spherical disk with chordal radius 3K/2. By induction we deduce for every n: (y) For p? n the image of every l::!pq is contained in a spherical disk with chordal radius 3K/2 n - 1.

r

SE. Let l::!' be the intersection of R and the region bounded by the Jordan curve (3t1 and let z* be a point of (3'i,1' Then it follows from property (a) of {Dpq} that 00

l::!' c

Q(p)_

U U l::!pq, p=1 q=1

and consequently for any z' E l::!' there is a l::!p.q. whose closure contains z'. From (y) we have for a chain of l::!pq joining l::!1,1 and l::!p.q., p'

rJ(z'),J(z*)] ::;;

3K

2: diamf(l::!pq) < 2 p=l 2: 2P-1 p=1 00

1

=

12K < 2'

132

v.

PICARD SETS

[5F

where the diameter is in terms of the chordal distance. By means of a linear transformation we conclude that f is bounded in tl'. On the other hand, on applying the criterion of Pfluger [2]-Mori [2] or of Kuroda (App. 1.16-19) to the annular regions {Dpq} we see easily that the part E' of E contained in the region bounded by fJi.l has a complement of cla;;;s oAB. Hence each point of E' must be a removable singularity of a bounded function.f. This contradicts our assumption that Zo E E' is an essential singularity of f, and we conclude that f cannot omit four values in R at Zoo

This completes the proof of Theorem 3D .

.'iF. For Cantor sets wc have the following immediate consequence: Corollary. Let E be a Cantor set on the closed interval [0, 1] with successive ratios f" satisfying the condition (3)

lim fn = n~

o.

00

Then every function merom orphic in R = CE with at least one essential singularity in E has at most three Picard values in R at each singularity. This result, due to Matsumoto [4], is somewhat sharper than thc original theorem of Carleson [2], where the condition . log f;; 1 11m ---

n~oo

log n

= 00

was used. It is significant that in both cases the capacity of E may be positive, i.e., the extra condition 2:1 2 -n log f;;l < 00 can be imposed on E (cf. 2B).

6. Classes of sets with the Picard property

6A. We shall denote by tff n' n = 1, 2, ... , the class of totally disconnected compact sets E in the z-plane such that every meromorphic function in CE with E as the set of essential singularities has at most n + 1 Picard values at each singularity. It is clear that

We claim that these inclusions are strict (Matsumoto [2]): Theorem. There exists a set E in the class tffn such that CE carries a merornorphic function with exactly n + 1 exceptional values at each singularity. By Picard's theorem the set E consisting of a single point meets the requirement of the above thcorem for n = 1. We shall give an example for n = 2. The construction is similar in the case n > 2.

§2. FINITE PICARD SETS

6CI

133

6B. We delete the origin and the point ~ = I from the ~-plane and denote the resulting region by S. By induction wc shall construct covering surfaces Sn of the ~-plane and define an exhaustion {Sk} of their limiting surface S. Take a set {A, B; C, D, E}oo of analytic Jordan curves satisfying the following conditions with respect to the points 00, 0, and I: (a) A and B each separate 00 from 0 u I, (b) A is tangent to B at one point, (c) C separates A from 00, (d) D encircles 0 and E encircles I, (e) D and E are tangent to each other, forming with B the boundary of a doubly connected region (B, DUE). Here we are using the general notation (C l , C 2 ) for a doubly connected region with contours C l , C2 . We introduce sets {A, B; C, D, E}o and {A, B; C, D, Eh defined as above after cyclically permuting 00, 0, I in (a), (c), and (d). We also consider a set {F, G; H, 1}oo of analytic Jordan curves with the following properties: (a) F separates 00 from 0 U I, (f3) G is homotopic to zero with respect to S and is tangent to F at one point, (')I) H separates F from 00, while 1 separates H from 00. Sets {F, G; H, 1}0 and {F, G; H, 1h are again defined by cyclically permuting 00, 0, I in (a) and (')I). 6C. First take a replica Sl of S. There exists a set

such that the moduli of the doubly connected regions

are not less than 2. In fact, first choose the curves a2.2 and a2.3' then al,2 with mod (al,2' a2,2 U a2,3) ;:::: 2. Next select

al,l

and a2,l such that mod

(al,l,

a2,l) ;:::: 2.

The region bounded by al,l U al.2 is taken as Sl and the region bounded by a2.l U a2,2 U a2,3 as S2' Choose So with So c:.S l and such that Sl -So is a doubly connected region with modulus ;:::: 1.

V. PIOARD SETS

134

l6D

Next take three replicas 8 i , i = 1, 2, 3, of 8. Draw in Sl and

in 8 1 as follows. First take

(X4.3, (X5.3,

mod

and

((X4,3' (X5,3

U

(X5.4

{(X3.1, (X3.2; (X4.1, (X5.1}.,

in 8 1 with

(X5,4) :;:::

5.

Then determine {(X3,l, (X3,2; (X4,l, (X5,l}'" in Sl such that (X3,l U (X3,2 is contained in the end part of Sl bounded by (X2,l and does not intersect the same curve as (X4,3 drawn in Sl. Moreover, we require that

and

Last trace (X4,2 and (X5,2 such that the region bounded by tains the same curve as (X3,2 drawn in 8 1 and that mod

((X4,2, (X5,2) :;:::

(X4,2

U

(X4,3

con-

5.

We connect 8 1 with Sl crosswise across a slit in the region bounded by (X3,2' If we choose this slit sufficiently small, we have

In a similar manner we draw

{(X3,3, (X3,4; (X4,4, (X5,5}O

and

in Sl, {(X4,5, (X4,6; (X5,6, (X5,7, (X5,S}O in 8 2 , and {(X4,8' (X4,9; (X5,10, (X5,lV (X5,12h in 8 3 , We connect 8 2 and 8 3 with Sl across suitable slits in regions bounded by (X3,4 and (X3,6' The resulting surface is denoted by S2. We take as S3 the region of S2 bounded by Ur= 1 (X3,i, as S4 that bounded by Ur=l (X4,i' and as S5 that bounded by Ut~l (X5,i'

6D. Suppose that Sn and Sic for 0 ~ k ~ 3n -1 are constructed with Sn consisting of 4 n -1 sheets, and ()S3n -1 of 3· 4 n -1 analytic Jordan curves (X3n -l,i each separating one of the three points from the other two. Furthermore, suppose each component of SIc+ 1 -Sic' which is a doubly connected region, has a modulus not less than k + 1 for 0 ~ k ~ 3n - 2. Then we take 3· 4 n -1 replicas Si with 1 ~ i ~ 3 . 4 n -1 of 8 and connect each 8 i with Sn crosswise across a suitable slit in the end part of Sn bounded by (X3n -l,i as follows. It suffices to consider only the case where (X3n -1,1 encircles the point at infinity. In the other cases we simply replace 00 by 0 or 1. In the same

135

§2. FINITE PICARD SETS

6F]

manner as above we choose {IX3n.2i-1, IX3n.2i; end part of Sn bounded by IX3n _ 1.i and

IX3n+1.3i-2, IX3n+2.4i-3}""

in the

{a3n + l,3i - U CX3n + l,3i; CX3n +2.4i - 2; CX3n +2.4i -1' CX3n +2,4i}oo

in Si such that the moduli of the doubly connected regions ( IX 3n+1.3i-2' IX3n+2.4i-3),

( IX 3n+1.3i-1, IX3n+2.4i-2),

and are not less than 3n+2, and that mod (IX3n.2i-1, IX3n+1.3i-2) ;::: 3n+1, mod (IX3n-1.i' IX3n.2i-1 U IX3n.2i) ;::: 3n. Then we connect Si with Sn crosswise across a slit in the region bounded by IX3n.2i' choosing it so small that mod

(IX3n.2i, IX3n+1.3i-1

U

IX3n+1.3i) ;:::

3n+ 1.

On the surface Sn+1 thus obtained we take S3n' S3n+1' and S3n+2 as the regions bounded by UiIX3n.i' UiIX3n+1.i, and UiIX3n+2.i, respectively. It is easily seen that Sn + 1 and Sic with k :::; 3n + 2 possess all properties listed above for Sn and Sic' 0:::; k:::; 3n -1, with n replaced by n + 1. The limiting surface S is a planar covering surface of the extended ~-plane.

°: :;

6E. We map S bijectively and conform ally onto the complement R of a totally disconnected compact set E in the extended z-plane. Let / be the mapping function and cp the projection of S into the extended ~-plane. By the same argument as in 1D we see thatf=cp 0/- 1 is merom orphic in R, has an essential singularity at each point of E, and possesses three Picard values, 0, 1, and 00, at each singularity. Moreover, E satisfies the conditions of Theorem 3C. In fact, the exhaustion of R by the regions R Ic =/-l(SIc) with k=O, 1"" is clearly normal in the sense of 3A and branches off at most twice everywhere. Furthermore, the moduli of the components of R Ic -RIc - 1 , k;::: 1, dominate k, and we have lim fL(r) = 00, r-L

where L is the length of the graph associated with {Ric}'

6F. The set E thus constructed for n> 1 is a perfect set. In this context the question arises: Does tfi\ contain any perfect sets? This can be answered affirmatively by a very recent result of Matsumoto [5] (cf. Corollary 5F): If Sn+1 =o(s~), then every function meromorphic in CE with E as the set of essential singularities has at most two Picard values at each singularity.

CHAPTER VI RIEMANNIAN IMAGES A milestone in the evolution of value distribution theory was Ahlfors' [11] discovery that the main theorems are not based on the analyticity of the mapping functions but can, in essence, be derived from purely metrictopological properties of covering surfaces. We begin the present chapter by following rather closely Ahlfors' original proof arrangement, unsurpassed in elegance by other presentations. The theory culminates in his fundamental inequality, often referred to as the nonintegrated form of the second main theorem on meromorphic functions. We shall also give this inequality and Picard's theorem as localized by Noshiro [2] to a transcendental singularity of the inverse function. We then present a generalization by Noshiro [4], [6], and Sario [7] of the nonintegrated form of the second main theorem to mappings of arbitrary Riemann surfaces into closed Riemann surfaces. For surfaces Rp carrying capacity functions with compact level lines (1.12 and IV) we show, following Noshiro-Sario [1], that an integrated form can be derived from this nonintegrated form. The chapter closes with Noshiro's recent results on algebroids and Sario's [7] and Rodin's [2] examples to prove the sharpness of the" nonintegrated" bound for the number of Picard points. The presentation is divided into two parts: §§1 to 3 deal with the general theory of orientable covering surfaces, while §§4 and 5 are devoted to Riemannian images under analytic mappings of the plane and of arbitrary Riemann surfaces. For earlier literature on the subject we refer the reader to Dufresnoy [11], Hayman [4], Kunugui [3], Nevanlinna [22], Tamura [1], T6ki [2], Tsuji [19], Tumura [4], Valiron [46], and others listed in the Bibliography.

§l. MEAN SHEET NUMBERS After some preliminaries on lengths and areas we introduce the mean sheet numbers of covering surfaces and derive Ahlfors' covering theorems both for regions and curves. 136

IA]

§1. MEAN SHEET NUMBERS

137

1. Base surface lAo By a finite surface we shall understand a triangulated closed or compact bordered surface. We also fix a sequence of successive subdivisions of the given triangulation. In IA to Ie we shall consider the former, in ID, the latter case. Given a closed triangulated surface So we endow it with a metric satisfying the following conditions: (a) There exists a set {y} of simple curves y on So such that to each curve " there corresponds a finite positive number Iyl as its length. Furthermore, {y} contains every curve which appears in the triangulation of So and in the successive subdivisions of it. (b) For any two points Sl and S2 on So there exists at least one y which joins Sl to S2. We define the distance between Sl and S2 as the infimum of lengths Iyl for all such curves y. We assume that the distance between Sl and {2 vanishes if and only if the points coincide. (c) The topology induced by this distance is equivalent to the original topology of So· (d) Any region ~ bounded by a Jordan curve y has a finite positive area I~I·

(e) The length Iyl and the area I~I are extended to be .finitely additive measures on the finitely additive classes generated by {y} and {~}, respectively. (f) To each S E So there corresponds a simply connected open neighborhood N({) with an exterior point and a positive number k(s) such that for any Jordan curve ycN(O that is the boundary of a region ~ cN(O, the inequality (1)

is valid. Property (f) shall be referred to as the regularity of the metric. Lemma. There exist two positive numbers d and k such that any Jordan curve y with Iyl < d on So bounds a region ~ whose area is

For the proof denote by 8> 0 the shortest distance from a point S to the boundary of N(O and let V(O be a neighborhood of S contained in the 1l/2-neighborhood of Since So is compact, there exists a finite covering V(Sl), ... , V(Sm) of So. We denote by d the minimum of the corresponding numbers 81 /2, ... , Il m /2, and by k the maximum of the corresponding constants k({l)'···' k(sm). Every Jordan curve y on So with Iyl and by Lemma IB we conclude that (15)

VI. RIEMANNIAN IMAGES

142

On replacing Sv

n

~

by the set

~-Sv

n

~

[3B

we obtain

(16) with the same constant k. Division of (15) and (16) by

I~I

gives

(17)

and (18)

We observe that these inequalities continue to hold if 2Ifv(~) is replaced by 2Ifh)= ISv n yi/lyi. In fact by Lemma lB, l/lyl :::;k/l~1 and consequently

Similarly l - 2Ifv(y)

k

1

= TY1 (lyl-ISv n yi) :::; ]Lfi (lyl-ISv n yi) k

:::; ]Lfi (iyl-ISv n yl + Lv)· It follows that (19)

and (20) From these inequalities we deduce that

By Lemma Ie (see also ID) we have min (ISv n yl, lyl-ISv n yi) :::; k"Lv and therefore (22) where k' is a constant depending only on y. Summing (22) for v= 1,· .. , n gives

§1. MEAN SHEET NUMBERS

3C]

143

:By virtue of Theorem 2C we obtain

1M -M(y)1

(24)

~

fL.

3e. In the general case we deduce by using arguments analogous to those in 2C that (25)

IM(y)-M(y')1

~ 1:1

L,

where y' is an arc of any regular curve y and k is a constant depending only on y. For the sake of completeness we shall supply a proof. Clearly IS(y')1 = L: IS. n y'l and IS. n y'l ~ IS. n YI. Hence M ( ')

= IS. n y'l

Iy'l

•y

~

IS. n yl Iy'l

and

Accordingly (26) Since y' n (So -8.) is contained in y n (So -8.),

ly'I-IS. n y'l

~

Iyl-IS. n yl,

and a fortiori I-M(') < •y

-

lyl-IS.nyl . Iy'l

On the other hand, I-M ( ) •y

=

Iyl-IS. n yl < Iyl-IS. n yl Iyl Iy'l

and we have (27) From (26) and (27) it follows that IM.(y)-M.(y')1

~ 1:'1

min (IS. n yl, Iyl-IS. n yj).

By virtue of Lemma 1C (28) where k is a constant depending only on y. On summing (28) for all v we obtain (25).

VI. RIEMANNIAN IMAGES

144

[3D

3D. After these preliminaries let y be a regular curve on So. Take an arbitrary point ~o on the curve y, a small simply connccted open neighbor_ hood N(~o), and an arc y' of y in N(~o). Form a regular Jordan curve y* in N(~o) by adding a suitable arc to y'. This is certainly possible by (g) of 3A. It is clear that y* decomposes So. For y and y' we have by (25)

IM(y)-M(y')1
0 and there existed a Po with 0 < Po < Po such that L(r) > qM(r) for Po < r < Po. It would follow that

f"p !:!.... J" l(r)

< 47r fOP dM(r) < 41T fOp dM(r) < 47r _1_, -

0

L(r)2

q2

J"

M(r)2 -

q2 M(p)

VI. RIEMANNIAN IMAGES

170

and therefore

f

pp

M(p)

p

[l8C

dr 477 l(r):O::; q2'

in violation of (98). 18C. Let e(p) be the Euler characteristic of Rp. We apply the generaliza. tion (51) of Ahlfors' fundamental inequality to the Riemannian image of R under f and collect our results in the following main statement (Sario [7], Noshiro): Theorem. For an analytic mapping of an arbitrary Riemann surface into a closed Riemann surface,

(99)

(eo+q)M(p)
O. We change our notations slightly by setting for O:o::;p and let

°

~nm

=

I

{p p > Po, (103) false}.

VI. RIEMANNIAN IMAGES

172

Then

i

l1 nm

[19C

d 1 (n) 4 ('" dC(p) og p ~ 7T jpo C(p)Pm_1(C(p))(10g(m)C(p))2

47T

= log(m)c(po)
2

2:

1=1

M mi·

22A]

§5. MAPPINGS OF ARBITRARY RIEMANN SURFACES

177

Under the mapping 8 = e the rectangle RmJ becomes a k-sheeted annulus with outer radius Z

(120) and inner radius R-1. The function t=(8+i}/(8-i} maps the annulus onto the k-sheeted complement of two Steiner circles encircling t = 1 and t= -1, respectively, symmetrically placed about the real and imaginary t-axes and intersecting the real axis at the images of 8= ± iR, ± iR-l, i.e., at distances .8+1 t1 = - (121) R-I and tIl from t=O. The function ~={lt maps the k-sheeted complement of the two Steiner disks onto the I-sheeted complement of the 2k images of the disks, which appear as distorted disks encircling points ~·=etlJ!., rpv=v1T/k, v= 1, .. " 2k, and are located in the annulus (I22) The function 7J=Wh gives as the final image 7J(R m;} of RmJ the h-sheeted complement of 2k/h distorted disks encircling points 7J:= eta., a v = Vh1T/k, v= 1, .. " 2k/h, and located in the annulus (I23) By definition the mean sheet number M mj of the image of Rmj in the stereographic 7J-metric is the 1T- 1 -fold area of 7J(R mf }. By omitting the annulus (I23) we obtain

M mj

> ~ ((2" (', r drp dr (2" (00 r drp dr) 1T Jo Jo (I+r2}2+ Jo Jr,' (l+r2}2

(1

1) = l+r21' 2hri

-h -+r 12 - 1 --1-+r l 2 1 On setting (124) We

E

.

m,

=

2 exp (21TVm2-p}-1

have tl = 1 + Emf and

M Here

2h(I+Emj}-2hlk h(I }-2hlk mf > 1 (' + Emj . + ~ +EmJ) 2hl k >

VI. RIEMANNIAN IMAGES

178

[2M.

and we find by (119) that m-1

Mm> 2

2:

h(l+em)- 2h lk

=

2(m-l)h(l+em)-2hlk.

j=l

For the Euler characteristic we have as before

em~4m(k-I).

r

em l' 2m(k-l) lmsup-Mm :::; lmsu p (m -1)h(1 +e m )-2h1k m--t-a)

m--t-a)

Hencc

2(k-l) h

In the special case h = 1 the value is 2(k -I), in perfect agreement with the result in 21C.

23. Extension to arbitrary integers 23A. In conclusion we give Rodin's [2] important modification of (116) to show that (114) is sharp for all integers g~O. Take k copies of the finite z-plane, each slit along the rays

(125)

Ij

=

{z I x < 0, y

=

27Tj},

j

=

0,

± I, ± 2, .. '.

For a fixed j the edges of the slits of I j are identified on the sheets so as to obtain a branch point of multiplicity k at z = 27Tji. This forms a covering surface R of the z-plane. As in 21B we now choose the metric dp = IdzI/27Tklzl, Po = (27Tk) -1 log Izl,,Bp the k-sheeted circle Izl = e2 J!kP, and l(p) = 1. On this surface consider the function (126)

J(z)

=

Yez-

-.

eZ_1

23B. This function is single-valued and meromorphic on the covering surface R. Clearly J omits S=0 and the k valucs e2niilk, j =0, I, .. " k-1. The poles are at z = 27Tji. Let Ll be a small disk containing the point at infinity. Using notations of 21 we have nm(Ll)~2m. Moreover em'" 2(k -I)m and g:::; k -1. It follows that the number k + I of exceptional values is at most 2+ g, i.e., k-I:::; Consequently g=k-I and we infer that the nonintegrated form (114) is sharp for every nonnegative integer g.

r

APPENDIX I BASIC PROPERTIES OF RIEMANN SURFACES In this appendix we shall derive ab ovo some properties of Riemann surfaces to which we have made reference. For the convenience of the reader we list here the headings. §l. Characterization of parabolicity.-l. Capacity function.-2. Harmonic measure.-3. Positive superharmonic functions.--4. Characterization of OG.-5. Proof.-6. Sets of vanishing capacity.-7. Functional capacity.-S. Equivalence.-9. Polar sets. §2. Modular OG- and 0 AB-tests.-1O. Modulus.-ll. Geometric meaning. -12. Modulus of a regular open set.-13. Modular inequality.-14. OGtest.-15. Graph.-16. OAB-test.-17. Modular form of the test.-lS. Integrated form of the test.-19. Wide exhaustions. §3. Directed nets.-20. Directed sets.-21. Fatou's lemma.-22. Examples. The grouping of these topics is as follows. In 1 to 5 we enumerate various ways of defining parabolicity and show their equivalence. Point sets of vanishing capacity are studied in 6 to 9. Basic properties of moduli are the topic of 10 to 13, modular OG-tests of Sario [2] and Noshiro [5] are derived in 14 and 15, and 0 AB-tests of Pfluger [21, Mori [2], and Kuroda in 16 to 19. Fatou's lemma for directed nets, together with examples, is discussed in 20 to 22.

§l. CHARACTERIZATION OF PARABOLICITY 1. Capacity function. Given an open Riemann surface R and a point R choose an arbitrary but then fixed parametric disk D containing zoo Let n with boundary f3n be a regular subregion of R containing D. For any set E we denote by H(E) the family of harmonic functions on E. The capacity function Pn of n with singularity at Zo is defined by the conditions Pn E H(D.-z o), Zo E

Pn I D = log

Iz-zol +h(z),

and Pn I f3n=k n (const.), where hE H(JJ) with h(zo) =0. The coefficient of log Iz-zol is immaterial; we have sometimes used 1/271" to obtain unit flux, while here 1 will be more convenient. 179

180

APP. 1. BASIC PROPERTIES OF RIEMANN SURFACES

[2

The Green's function fln of Q with singularity at Zo is the function in H(n-zo) with

I

fln D = -log

Iz-zol +h(z),

I

where hE H(D) and fln fJn=O. Trivially (1)

and h(zo) = kn is the Robin constant of Q with respect to Zo and D. It gives the capacity cp" = e - k" of fJn with respect to Zo and D. For QeQ' the maximum principle applied to fln'-fln shows that h(zo) increases with Q and a fortiori kn ::; k n ,. Therefore the directed limit

exists; it could be called the Robin constant of R with respect to Zo and D. It gives the capacity cp

=

e-k~ =

lim cp" n~R

of the ideal boundary fJ of R with respect to Zo and D. An open Riemann surface R is parabolic or hyperbolic according as cp=O or cp>O. We include closed surfaces in the class OG of parabolic surfaces so defined. It is easily seen that the directed limit flR(Z)

=

lim fln(z) n~R

exists if and only if R singularity zoo

r; 0G'

We call flR the Green's function of R with

2. Harmonic measnre. Given an open Riemann surface R choose a regular region Q o of R with boundary fJo and with connected complement. Let Q be a regular region of R containing The function Un E H(n - Q with Un fJn = 1, un fJo = 0, is the harmonic measure of fJn with respect to Q

-no·

I

no.

I

o)'

For Q e Q' the maximum principle gives 0 < un'::; un, and we can define the harmonic measure U R of fJ with respect to R as the directed limit

no

UR(z) = lim un(z). n~R

In the case uR=O we say that the harmonic measure of fJ vanishes. We include in this category the harmonic measure of the ideal boundary of a closed surface, since this boundary is empty.

§1. CHARACTERIZATION OF PARABOLICITY

5]

181

3. Positive superharmonic functions. Consider a positive superharmonic function v defined on R - no which is continuously extendable to R - 0 0 • If v ~ m (const.) on flo implies the same inequality in R - 0 0 for any v, then we say that the maximum principle is valid on R. A closed surface R is a trivial example. Let .F =.F(R; zo) be the family of positive superharmonic functions v on R such that (v+log Iz-zol) D is bounded and superharmonic. We include in the family the constant function 00. It can be easily seen that oF forms a Perron family and consequently infvE$Ov(z) is either identically 00 or a positive function in H(R-zo) with a logarithmic singularity at zoo As a consequence of the maximum principle the latter case can occur if and only if R ¢ 0G, and then gR(Z) = inf v(z). VE$O

I

4. Characterization of 0G. We are now ready to state (cf. Ahlfors [15], Ohtsuka [1]): Theorem. The following properties are equivalent: (a) R belongs to the class 0G of surfaces with vanishing capacity of fl, (b) the harmonic measure of f3 vanishes, (c) the maximum principle is valid on R, (d) R does not carry Green's functions, (e) there are no nonconstant positive superharmonic functions on R. In particular, (e) assures that the definition of parabolicity does not depend on the choice of Zo and D. Obviously all closed surfaces enjoy the above properties and we have only to prove the theorem for open surfaces. 5. Proof. Suppose R has property (a) and hence (d). To see that this implies (b) we first note that the minimum mn of gn on flo diverges to 00

as

n~R

E

0G. By Green's formula foo gn du~=27T and consequently Dn

=

r

JOg

undu~ =

r

Joo

du~::s;

27T,

mn

where Dn is the Dirichlet integral of Un over O-no. Therefore Dn ~ 0 as R, and the function Un converges to zero in view of Un flo = o. We conclude that U R = 0, and condition (b) is satisfied. Suppose now (b) holds. If v Iflo~m, then by the maximum principle V-m(l-un)~O onn-O o. On letting 0 ~ R we havev-m~O on R-no, and condition (c) is met. The proof that (c) implies (d) is by contradiction. Assume indeed that there exists a Green's function gR of R and let D,,={z z E R, gR(Z) > p}. For sufficiently large p, D" is a relatively compact "disk" about zo, and D" ~ Zo as p ~ 00. By (c), gR~P on R-D". On letting p ~ 00 we obtain gg=oo on R-zo, a contradiction.

n_

I

I

182

APP. 1. BASIC PROPERTIES OF RIE},IANN SURFACES

[6

We proceed to show that (e) iR a consequence of (d). Let v be a non con_ stant positive superharmonic function on R. We may assume that RUPRV> 1 > infRv. Since min(l, v) is again a non constant pORitive Huper_ harmonic function on R, we may also suppose that v = 1 in a relatively compact disk 0 0 about some point Zo in R and that infRv < l. Clearly v~l-un on 0-0 0 and therefore v~l-uR' whence uR>O in R-O o. Take a local parameter z such that Oo={z Ilz-zol::o:; l}c{z Ilz-zol S2} and set a=(minlz_zol=2UR(Z))-1. Consider the function

q(z) = {a log 2-log Iz-zol a log 2 - auR(z) log 2 It is easy to see that q E :F(R; zo) and we infer that gR exists.

The remaining statement, (a) implied by (e). is immediate. In fact, if R rf= OG' then as remarked in 1, gR exists and is a non constant superharmonic function on R. The proof of Theorem 4 is herewith complete. 6. Sets of vanishing capacity. By definition a compact set K of the extended z-plane has logarithmic capacity zero or simply capacity zero if the complement CK belongs to the class 0G' A general set E in the extended z-plane is said to have capacity zero, or more precisely inner capacity zero, if every compact subset of E has capacity zero. Let E be a subset of a Riemann surface R. Suppose that EnD has capacity zero for every parametric disk D of R. Then we say that E has capacity zero. 7. Fnnctional capacity. Given a Riemann surface R and a positive continuous function k in R x R in the extended sense (cf.IV.IA) we put for a compact set K c R

f,ct K ) = inf

Jk(Zl' Z2) dfJ-(Zl) dfJ-(Z2)'

where fJ- runs over all Borel measures whose supports are contained in K, and fJ-(K) = l. If f,AK) = 00, then we say that K has k-capacity zero. The extension to an arbitrary set E is similar to 6. Consider two functions kl and k2 such that kl - k2 is bounded on K x K. It is readily seen that K has k1-capacity zero if and only if it has k 2 capacity zero. In particular, the concept of gR-capacity zero is equivalent to that of log-capacity zero. 8. Eqnivalence. Let K be a compact set in the finite z-plane. Assume that flog (K) < 00. We can construct a nonconstant positive superharmonic

III

188

§2. MODULAR 0G· AND 0AB-TESTS

function on CK (cf. Tsuji [19, p. 60]). Thus CK ¢: Oc and K has positive capacity. Conversely suppose CK ¢: Oc. Then U CK > o. By Lemma IV.4A we conclude that 19 CD (K)0. o . We set un=1 on R-D, where Un refers to Do. Since Un(Zl) -+ 0 for a fixed point Zl E R-0. o -K, there exists an exhaustion {Dn} of R - K such that 00

L u nn

v =

n=l

is a positive superharmonic function in R - 0. 0 . Clearly v = 00 on K and thus K is a polar set. We have shown: A compact set is a polar set if and only if it has vanishing capacity.

§2. MODULAR 0G" AND OAB-TESTS 10. Modulus. Consider a regular region D of a Riemann surface R. We assume that the boundary aD of D consists of at least two contours and that the contours are divided into two sets 0: and (3. We also allow the degenerate case where the contours are piecewise analytic and not necessarily simple. Let U E Ii (0.) with U 0: = 0, U (3 = log fL, where fL> 1 is a constant deter-

I

I

mined by the condition J,du* = 217. The number log fL is called the modulus of the configuration (D,

0:,

(3) and is denoted by

mod (D,

0:,

(3)

=

mod D

=

log fL.

The function U is, by definition, the modulus function. Sometimes fL itself is called the modulus, and log fL is referred to as the logarithmic modulus or harmonic modulus. However, we have used the term modulus for log fL throughout the book.

,=

11. Geometric meaning. Let U* be the (multiple-valued) conjugate harmonic function of u. The analytic mapping given by eU + iu* indicates the geometric meaning of the modulus log fL. If D is doubly connected,

184

APP. I. BASIC PROPERTIES OF RIEMANN SURFAOES

[12

then ~ maps n conform ally onto the annulus with radii 1 and jL, and the modulus is the logarithm of the ratio of the radii of this image. If Q is planar but a consists of one contour and f1 of two contours f1! and f12' then we cut Q along some level line of u* joining f1! and f12· The function ~ maps the resulting doubly connected region onto a radial slit annulus with radii 1 and jL. Next consider the case where aQ consists of two contours a and f1 but the genus of Q is 1. Cut Q along a level line of u* which is a Jordan curve not dividing Q. The resulting surface is planar and is again mapped by ~ onto a radial slit annulus with radii 1 and jL. By using both methods of cutting, one for contours and the other for genus, we obtain from an arbitrary Q a planar region which is mapped by ~ onto a radial slit annulus with radii 1 and jL. We conclude: The modulus is the logarithm of the ratio of the radii of the image radial slit annulus. 12. Modulus of a regular opeu set. If n consists of a finite number of disjoint regular regions Qi, then we call Q a regular open set. Assume that each Qi has more than one contour and that the contours are divided into two sets ai and (1i. We set a=Ua i and f1=Uf1i. We also allow the case where the contours are piecewise analytic and not necessarily simple. In the same manner as in 10 we define the modulus log jL of the configuration (Q, a, (1). By virtue of inequality (3) to be now established we see that (2)

1 mod (Q, a, (1)

13. Modular inequality. Let harmonic measure of f1, i.e., un

1

=

~ mod (Qi, ai, f1i)·

n E

be a regular open set, and Un the H(D.) with Un a=O, and Un f1= 1. By

I

I

Green's formula fadu~=Dn(un) and in view of (log jL)un=u (3)

Let y be a finite set of disjoint analytic Jordan curves in Q separating a from f1 and dividing Q into two open sets Q! and Q 2 such that a e aQ! and f1e aQ2. Let V l be the harmonic measure of y in Ql' and V 2 that of f1 in Q2. We define the functionf" on

D. by

15]

185

§2. MODULAR Oa· AND 0AB·TESTS

where ,\ is a real number between 0 and 1. By Green's formula Dn(f!.) = Dn(uo. ) + Do.(u o. - fl.) and therefore

Do.(u o. ) ::; Dn(f!.)

=

,\2 Do.l (v I )+ (1- ,\)2 Do.2 (V 2).

We choose ,\=Do.2(V2)/(Do.l(VI)+Do.2(V2)) and conclude that 1

1

1

----- > + . Do.(u o. ) - Do.l (VI) Do.2 (v 2) By (3) this is the desired inequality

(4)

mod (Q, a,

13)

:?: mod (Q I , a, y)+mod (Q 2 , y,

13).

14. 0G"test. Let {Rn}O' be an exhaustion of an open Riemann surface R, with log fLn=mod (Rn-Rn-I' oR n _ I , oRn) for n= 1,2,·· .. Here and in the sequel we again allow the case where the contours of Rn are piecewise analytic and not necessarily simple. We shall prove the following modular criterion of parabolicity (Sario [2] and Noshiro [5]): Theorem. An open Riemann surface R belongs to OG if and only if there exists an exhaustion of R such that 00

(5)

2: log fLn =

00.

n=l

Proof. Let Vm be the harmonic measure of aRm with respect to Rm - R o. Then by (3) and (4) 217" m D:?: log fLn' m

2:

n=l

where Dm is the Dirichlet integral of Vm over Rm - Ro. Therefore (5) implies that Dm -+ 0 as m -+ 00. This shows that the harmonic measure of the ideal boundary of R vanishes, i.e., R E 0G. Conversely assume that R E OG and take an arbitrary exhaustion {Qn}O' of R. Fix an integer m:?: 1. Let Vn and Vnm be the harmonic measures of oRn with respect to Q n and Q n - m , n > m, respectively. Since Vn > Vnm and Vn -+ 0 as n -+ 00, we have Vnm -~ 0 and therefore by (3)

no

lim log fLnm =

n_oo

n

00,

where log fLnm = mod (Q n - nm , oQ m , oQn). Thus we can choose a suitable subsequence {Rn}O' of {Q n} for which (5) holds. 15. Graph. Take an exhaustion {Rn}O' of an open Riemann surface R. Let Un and log fLn be the modulus function and the modulus of (Rn-Rn-v eRn-I' eRn), respectively. The function un+iu~ maps

186

APP. I. BASIC PROPERTIES OF RIEMANN SURFACES

[16

Rn - En -1 with a finite number of suitable slits conform ally onto a slit rectangle 0 < Un < log /Ln' 0 < u~ < 27T. Let

~ = g+i1) = un+iu~+

(6)

n-1

2:o log /L;,

/Lo

= 1,

on Rn-Rn-1' n=l, 2,···. The function ~ maps R-Eo with at most a countable number of suitable slits conformally onto a countably slit strip 0< g < L, 0 < 1) < 27T with L = 'if' log /Lt. This strip is the graph of R associated with {Rn} and L is the length of the graph (Noshiro [5]). In terms of the graph Theorem 14 is restated as follows: Theorem. An open Riemann surface R belongs to Oa if and only if there exists an exhaustion such that the length of the graph is infinite. 16. 0 AB·test. We turn to modular tests for a given R to belong to the class 0 AB of Riemann surfaces without nonconstant bounded analytic functions. Let YA be the level line g=..\, where g is the real part of the function in (6) and ..\ E [0, L). For each ..\, YA consists of a finite number of piecewise analytic Jordan curves y~, j = 1, ... , m(..\). We consider the variation

A.(..\) J

of

=

f

g* and set A("\)

1 1.

dg*

= max

Aj(..\).

l:5j:5m(A)

Using this maximum variation and the quantity M("\)

= max

m(p)

O:5P:5A

we first state the following criterion of Pfluger [2]: Theorem. If the maximum variation satisfies the condition (7)

li~~sLup (47T

f ~~)

-log M(..\))

= 00,

then R belongs to 0 AB. Proof. Suppose there exists a nonconstant AB-functionf on R; we may assume If I < 1. We denote by S(..\) the area of the multisheeted Riemannian image f(Ro U {z I g(z) < ..\}) and by l{ the length of the Riemannian image f(y{), both in the Poincare metric of the unit disk. Since

and

§2. MODULAR 0G· AND OAB-TESTS

17]

187

we obtain by Schwarz's inequality

for j

= 1, ... , m('\). On summing these inequalities we obtain

m~)

(l{)2

~

A('\)

d~~'\).

1

Since the Poincare metric has constant curvature -4, the corresponding isoperimetric inequality takes the form (Schmidt [1], [2, p. 753, footnote 12]) where 8 j ('\) is the area of the multisheeted domain bounded by the Riemannian image f(y{). Clearly 8('\) ~ IT w 8 j ('\) and therefore

Consequently 417

3:.L

m(A)

m(A)

1

1

L 48;('\){17+8;('\)) ~ L (l{)2.

~

< d8(p) _

A(p) -

8(p)

d8(p) 8(P)+17M('\)

for 0 ~ p ~'\, and we obtain by integration

fA

417

Jo

d'\

A('\)

~ log

(8('\) 8(0)+17M('\)) 8('\) +17M(,\)· 8(0)

I 8(0)+17M('\) < I (M('\) 8(0)+17) ~ og 8(0) - og 8(0)

= log M('\)+log

8(0) +17 8(0) .

This contradicts (7). 17. Modular fonn of the test. Let {Rn}O' be an exhaustion of R and let Rn - Rn -1 consist of regions R nk , k = 1, ... , k n. We denote by log jtn and log itnk the moduli of (Rn - R n- 1 , eRn_I, eRn) and (Rnk' (eRn_I) n (eR nk ), (oRn) n (eR nk )), and by T; the jth partial length of the graph, i.e., TO

=

o.

188

APP. I. BASIC PROPERTIES OF RIEMANN SURFACES

[17

For the minimum modulus we use the symbol log Vn

min log {tnk

=

l$kSk n

and set

N(n)

=

max k j . l::5isn

The following modular form of Pfluger's criterion is due to Mori [2]: Theorem. If there exists an exhaustion of R with doubly connected surface

fragments Rnk whose sum of minimum moduli grows so rapidly that

li~~~up (~ log Vj-t log N(n))

(8)

=

CfJ,

then R belongs to 0 AB. Proof. Let u j and u jk be the modulus functions corresponding to log ftf and log {tjk. Then Uj=CjkUjk in Rjk with cjk=(log {tj)j(log {tjk) and consequently Aj(A) ::s;

J

for

y~

C

J

dg* =

~=Mn~k

dUf = cjk

~=Mn~k

J

dUfk = 21TCjk

~=Mn~k

R jk . Since k is arbitrary, A(A)::s; 21TC jk and a fortiori

for Tj _ 1 ::s; A::s; Tj. From this and from

A Jor ~A(A) -

±J'I *' 1

'I -1

~+JA ~ A(A) A(A) 'I

it follows for O::s; Tj::S; A::s; Tj.,.l that (9)

41T

i

A

a

dA logvj+1 A(A)?;2L.logvj+2l . (A-Tj). 1 og {tJ + 1

In particular

Since the Rnk are annuli, we have M(Tn)=N(n). Therefore

41T

r' Jon

dA n ) A(A)-logM(T,,) 2': 2 ( ~logvj-tlogN(n) ,

and we infer that (8) implies (7).

18]

189

§2. MODULAR OG- AND OAB-TESTS

18. Integrated form of the test. The following variant of Pfluger's test was established by Kuroda: Th~orem. If the maximum variation has the property

(10)

li~_~up {f_l exp (47T

f ~~J

dA

= 00,

then R belongs to 0 AB' Proof. Suppose there exists a nonconstant analytic function f = h + ih* with If I < 1 on R. Denote by D(A) the Dirichlet integral of f over Ro U {z I g(z) < A},

Let h j be a constant such that

Jf (h-h f ) dg*=O. YA

Then the Fourier

expansion of h( g*) - hf takes the form

for O:=:; g*:=:; Aj(A). On differentiating both sides with respect to g* =7] we obtain the Fourier coefficients {Af(A) -127Tkak , Aj( A) -127Tkb k }f for oh( g*)/o7]' Therefore

and

Jyi ("8oh)2 7]

dt* !,

47T 2 ~ k 2( 2 b2) = A.(A)2 L... ak+ k' J 1

A comparison of the right-hand sides of these equalities gives

By Schwarz's inequality we have

190

APP.1. BASIC PROPERTIES OF RIEMANN SURFACES

[19

It follows that D(A) ~ (41T) -1 A(A)D'(A), and by integration we obtain D(O) exp

(4TT

f ~~)) ~

D(A).

On the other hand,

!:... f

JYA

dA

h 2 dg* = 2

f

h oh dg* = 2D(A)

JYA

N

and consequently

{1_1

exp (41T

f ~~))

dA

~ 2~(O) (l,1 h2dg*- l,j-1 h2d g*) ~

D70)'

in violation of (10). 19. Wide exhaustions. Condition (8) can be weakened if we require that the surface fragments Rjk are wide enough in the sense that Illog Vj= 0(1). On the other hand, we no longer require that the Rjk be doubly connected or even planar. The test takes the following form (Kuroda): Theorem. If there exists an exhaustion of R such that

log Vi > S > 0

for j = 1,2, ... and that

li~...s~p (~1 log

(II)

Vj-!

log kn)

=

00,

then R belongs to 0 AB. Proof. Integration of (9) gives J"

J'n'n

=

;?:

-1

(fA dA) Jo A(A) dA

exp 41T

I ( n-1 ) 2°lg ILn (exp (2 log vn) -I) exp 2 ~ log Vi . og Vn 1

From (2) it follows that I

kn

log ILn and for this reason (log ILn)/(log I n

~

=

vn)~

I/k n. We conclude that

t(I-exp(-2S))exp

This shows that (II) implies (10).

I

~ log ILn/ (2~IOgvi-IOgkn).

§.3. DIRECTED SETS

21]

191

§3. DIRECTED SETS 20. Directed sets. A semiordered set A = {A} with ordering > is, by definition, a directed set, if for every pair of elements A1, A2 E A there exists an element A3 E A such that A1 ::; A3 and A2 ::; A3. A directed set A is called integral if it contains a subset Ao with the following properties: (a) Ao is a sequence {An};" of elements An E A, (b) Am < An for m < n, (c) for every AE A there exists a An E Ao with An> A. An example of directed sets appearing frequently in our book is the set {a} of regular subregions 0 of an open Riemann surface R adjacent to a fixed parametric disk Ro of R. The inclusion relation gives the ordering (cf.1.2A). This directed set is integral. In fact, the set {On};" of regions On = Rn Ro, where {Rn}"

(12)

Here lim inf" stands, as usual, for sup"inf.!,>.!.

192

APP.1. BASIC PROPERTIES OF RIEMANN SURFACES

[22

Proof. Let Ao={AnH" be a subset of A with properties (a), (b), and (c) in 20. Clearly f f/!)." dp.? f inf)., >). f/!).' dp. for every A" > A; hence

lim}nf

Jf/!). dp. ? Jinf).,>).9').' dp.

for every A, and (13)

lim}nf

Jf/!). dp. ? s~p finf).,>).9').' dp..

Set -Pn=inf).'>)..f/!).' and note that inf).'>).f/!).' increases with A. For this reason sup).f inf).'>).f/!).' dp.?f -Pndf" and by (13) we conclude that (14) The last equality is assured by the usual Fatou lemma. By 20(c) there exists an n such that -Pn?inf).'>).f/!).' for a given A; hence limn-Pn?inf).'>).f/!).' for every A, and finally limn-Pn?lim inf).f/!).' On the other hand, inf)., >).f/!).'? -Pn for A> An and therefore lim inf).f/!).?-Pn for every n. We infer that lim inf).f/!).=limn-Pn and that lim inf).f/!). is measurable. This together with (14) yields the validity of (12).

22. Examples. First we remark that the function Bk(a)

=

r

JPk

s(f(z), a) du*

considered in II.12B and a fortiori B(k, a) and A(k, a)=C(k)-B(k, a) are finitely continuous with respect to a on S. This follows from Theorem II.14B. In fact, if a o if(f3k)' then the continuity of Bk at a o is a direct consequence of the finite joint continuity of s. If ao E f(f3k)' then by II.(79) we have only to establish the continuity of

r

JPk

gG(f(z), a) du*

at ao, where gG is the Green's function of a regular region G containing f(f3k)' Let f -l(a o) n f3k = {Zl' .. " zn}; let f3kV be a small open subarc of f3k containing ZV' v= 1, .. " n, and set f3~ =f3k -Uvf3kV' Since the function (z, a) -gG(f(z), a) behaves logarithmically near (Z., ao),

r

J

PkY

gG(f(z), a) du* -

r

J

gG(f(z), ao) du*

=

O(e)

Pky

for a given e > 0 and for (z, a) in the vicinity of (Z., ao), if we choose f3kv sufficiently small. By the continuity of fp~ gG(f(z), a) du* at a=a o we

22]

§3. DIRECTED SETS

193

conclude on that of IOk gG(f(z), a) du* at a Q • The continuity of A(k, a) can also be easily deduced from its very definition. Let /L be a Borel measure on S. The measure w in II.SA is an example of /L. As a consequence of Theorem 21 we state: The function Bk(a) is continuous, infn'=>nBIc.(a} with k'=k(Q') is upper aemicontinuous, and lim infnBIc(a} with k=k(Q) is Borel measurable. Moreover

with k = k(Q).

APPENDIX II GAUSSIAN MAPPING OF ARBITRARY MINIMAL SURFACES A smoothly immersed oriented surface R in Euclidean 3-space E3 is minimal, by definition, if its mean curvature vanishes. The natural metric of E3 induces a conformal structure making R into a Riemann surface. Similarly an oriented unit sphere 2: in E3 is a Riemann surface, and the radius of 2: parallel to the normal n(z) of R at z E R gives a conformal mapping n=n(z) of R into 2:. This is the Gaussian mapping. In 1 to 4 of this appendix we shall give an explicit construction of complete minimal surfaces of arbitrary finite or infinite connectivity and genus, smoothly immersed in the Euclidean 3-space. In 5 to 8 we study the distribution of normals to an arbitrary oriented minimal surface R. Given a point a of the unit sphere in E3, we ask how frequently the radii parallel to the unit normals on a regular subregion Q exhausting R touch a, and how close in the mean the parallels to norma13 on oQ come to a. A simple illustration of Theorem 5 is the distribution of normals to a catenoid: the omission of (two) directions is compensated for by a close proximity to those directions by the normals on oQ. For the number of Picard directions in the general case we now have the explicit expression (9) for YJ in the bound (10). At the end of the appendix we give the nonintegrated forms of the main theorems. These forms are especially suitable for the study of complete minimal surfaces, on which the natural induced metric is always available to explicitly test regular exhaustibility. A list of open questions is included. The result in 1 is due to Osserman [2], and those in 2 to 8 to Klotz-Sario [1], [2]. 1. Triple connectivity. Osserman's proof that there exist triply connected minimal surfaces in E3 is based on the following theorem [2]: Suppose there exists a Riemann surface F, a meromorphic function f on F, and a harmonic function h on F such that (a) the zeros of the differential w =

(hx-ihy) dz 194

APPENDIX II. GAUSSIAN MAPPING

2]

195

coincide in location and multiplicity with the zeros and poles of f on F, and (b) for every closed curve C on F,

Then there exists a conformal immersion X of F into E3 as a minimal surface X (F) on which

To construct a triply connected complete minimal surface (of zero genus) let F be the plane Izl < ro punctured at -1 and 1. The functions 1 V2 1 V2 f= z-I+(z-I)2+ z +1+(z+I)2' h

give

W=

f

=

log Iz2-11-2V2Re~1

z -

dz, and condition (a) is satisfied. For a closed curve C in F

L7 L =

dz

=

Lf Lrdz w =

0, =

0,

r

as the residues of at -1, 1 vanish. A conformal immersion X (F) in E3 thus exists. We have ds::::ldzl/2 on F, while IfI2~2/Iz+114 at -1 and If12~ 2/1z-114 at 1. One concludes that the images on X(F) of the paths in F to -1, 1, and ro have infinite length, and the surface X(F) is complete. 2. Arbitrary connectivity. We can now construct a complete minimal surface FC of arbitrary finite or infinite connectivity c and zero genus. In view of the plane and the catenoid we may take c> 3. Make a slit a in the surface F of 1 along the real axis from -1 to 1 and, if c < ro, take c - 2 copies F l' . . . , Fe _ 2 of such slit surfaces. Join the copies into one surface FC by identifying the lower edge of a on Fi for 1 sis c - 3 with the upper edge of a on Fi + l' and the lower edge of a on Fe _ 2 with the upper edge of a on Fl' If c = ro, form Fe by taking infinitely many copies ···F -2' F -1' Fa, F 1 , F 2 , ' " of the slit surface F and by identifying the lower edge of a on Fi with the upper edge of Fi + 1 for all i. In each case the surface FC has zero genus and the desired connectivity c. The natural extension Xc of X to Fe gives a conformal immersion Xe(Fc) of Fe as a complete minimal surface in E3.

196

APPENDIX II. GAUSSIAN MAPPING

[3

3. Arbitrary genus. If 1:0 requirements are made on the number of boundary components, the simplest way to construct a complete minimal surface Gg of given finite or infinite genus g is to use two copies F;, j = 1, 2, of the surface Fe constructed above, with c = 2g + 4. For g < 00 each F; consists of 2g + 2 copies F ij , i = 1, ... , 2g + 2, of F slit along a. We cut each Fi along g + 1 slits f3kj' k = 1, ... , g + 1, each consisting of the lower half of the imaginary axis on F 2k - l . j and the upper half of the imaginary axis on F 2k •j . The desired surface Gg is obtained by identifying the left edge of f3kl with the right edge of f3k2' and vice versa. If g = 00, both copies of Fj consist of infinitely many duplicates F ij , i=··· -2, -1,0,1,2"", and the same construction gives Goo. In both cases the natural extension of X to G9 gives a conformal immersion of Gg into E3 as a complete minimal surface. 4. Arbitrary genus and connectivity. For short we shall refer to the number of components of the ideal boundary (see, e.g., Ahlfors-Sario, [1, p. 67]) as the connectivity of the surface even if the genus is positive. The connectivity of G9 constructed above is uniquely determined by the genus g and increases with it. To construct a surface of arbitrary connectivity (:?: 4) and genus g we shall first form a surface H4.9 of connectivity 4 and genus g. If g < 00, let F~, m = 1, ... , g + 1, be duplicates of F\ each consisting of two copies F im , i = 1, 2, of F joined along a. On each F~ make two slits on the real axis: Ylm on F 1m from -00 to -1 and Y2m on F 2m from 1 to 00. For each i identify the lower edge of Yim' 1:-:::; m:-:::; g, with the upper edge of Yi.m + 1, and the lower edge of Yi.9 + I with the upper edge of Yil' If g = 00, we use infinitely many copies F~, m = .. " - 2, -1, 0, 1, 2, ... , and identify for each m and i the lower edge of Yim with the upper edge of Yt.m+l' The resulting surface H4.9 continues to have 4 boundary components. The genus g is, by definition, the largest number of disjoint singular I-cycles i5 n on H4.9 whose union has a connected complement. On each F~, m = 1, ... , g, choose for 15 m the simple closed curve consisting of the circle Iz + 11 = 1 on F im and the circle Iz -11 = 1 on F 2m' The f'urve has the shape of a horizontal figure 8, with the two loops, including the points at z = 0, lying on different sheets of F~. A surface HC.9 of any connectivity c:?: 4 and arbitrary genus g is obtained from Jl4.9 by replacing the copy Ft +1 in it by Fe +4 slit along a. Here the slit YI.g+I is made in the first copy FI of the c+2 duplicates of F that constitute Fc +4. The slit Y2.9+ I is similarly taken in the second copy!' 2' We have arrived at the following result (Klotz-Sario [1]): Theorem. There exist complete minimal surfaces, smoothly immersed in E3, of any connectivity c and any genus g. More accurately, for g = 0, C can be arbitrary, while for g> we can prescribe any c:?: 4.

°

APPENDIX II. GAUSSIAN MAPPING

5]

197

5. Gaussian mapping. Let L be the unit sphere in E3, intersecting the complex ~-plane along the circle I~I = I, and oriented by choosing its inner normal. To study the distribution of normals to an arbitrary oriented minimal surface R we apply the theory of complex analytic mappings to the Gaussian mapping n=n(z) of R into L, both viewed as Riemann surfaces. As in 1.1 and 1.9 we use ~ as the local parameter of L, and retain the meanings of to, So, and (1)

The mass element (2)

is now the area element of L. In terms of Rh ofL2A we let v(h, a) be the number of times, with multiplicity, the normal n to R takes a given direction a E L in R. The Afunction (3)

A(h, a)

= 417

J:

v(h, a) dh

again reflects the frequency of a-points in R; for a Picard direction it vanishes identically. We use u, Qh, and Ph of L2A and denote by (n, a) the central angle between nand a. Since s(~, a) = -2 log (2 sin «~, a)j2)), the B-function takes the form (4)

B(h, a) = -2

r

Jp,,-po

2

log sin (n a) du*.

It is the mean proximity of n on Ph -Po to a in terms of s(~, a). The C-function is simply (5)

C(h)

= -

where K(h) is the total curvature

J:

-fH"

K(h) dh, dw(n(z)) of R h •

Theorem. For the Gaussian mapping of an arbitrary minimal surface (6)

A(k, a)+ B(k, a)

=

C(k).

The meaning of the theorem as discussed in L2E is particularly fascinating in view of the present concrete significance of the A-, B-, and Cfunctions. We again refer to the illuminating simple example of the catenoid and the two directions omitted by n.

APPENDIX II. GAUSSIAN MAPPING

198

[6

6. Picard directions. We ask in general how many directions can the normals omit? Given a function fJ> of h E [0, k] we again let fJ>o stand for fJ> in the purely notational formula fJ>i(h) =

f

fJ>t-l(h) dh,

i= 1,2,3. It is the use of these multiply integrated quantities that permits the extension of the value distribution theory to the Gaussian mapping of arbitrary minimal surfaces. We denote by e(h) the Euler characteristic of Q h and state Theorem 1.9B in the present setup: For any regular subregion Q of an arbitrary minimal surface R, complete or not, q

(7)

(2 -q)K3(k)
••• , ~q

of ~ we have

(13)

For greater accuracy the sum 47T Lvb(~v) for the orders b of the branch points above ~v can be subtracted on the right. The theorem again shows that only few regions ~v can be sparsely covered. For a more precise meaning we now ask: When is L negligible compared with K? 8. Regular exhaustions. We consider a complete minimal surface R. Let Zo E RcE3 be given and let p(z, zo) be the distance along R from z E R to Zo in the Euclidean metric dp of E3. Set

{3p

= {z Iz E R, p(z, zo) = p},

Op

= {z Iz E

R, p(z, zo) < p},

0< p < 00. For p ~ 00, Op exhausts R. Let L(p), K(p) be the quantities L, K for {3p, Op, respectively. The exhaustion Op ~ R is by definition regular if (14)

· . f L(p) I1m In - K () p_oo p

= 0.

APPENDIX II. GAUSSIAN MAPPING

200

[9

Denote by l(p) the length of (3p in E3. The test VL18B reads in the present explicit case: The exhaustion is regular if the negative of the total curvature K(p) increases so rapidly that (15)

-00.

We let (16)

S=

. 47T e+(p) hm sup - K() p_oo p

and recall (VLlOD) that the regions Ll. can be replaced by points avo Theorem. For every regularly exhaustible complete minimal surface the number P of Picard directions has the bound (17)

P ~ 2+s.

The simplest example is again the catenoid, for which L ~ 0, K ~ -47T, LjK ~ 0, e+ =0, and s=O. It follows that P~2, and we know that the bound is taken. Remark. In accordance with the plan of this book we restricted our discussion of Gaussian mappings to applications of the main theorems in our general theory. For other aspects we refer the reader to Osserman [2] to [4], Voss [1], and other literature on Gaussian mappings listed in the Bibliography. 9. Open questions. We close by suggesting some further problems that appear significant in the study of minimal surfaces, known and new, complete or not. (a) What can be said about specific minimal surfaces in the light of the (A + B)-affinity (6) 1 (b) Are there complete minimal surfaces whose normals omit a given number of directions 1 (c) Are there minimal surfaces with a given 7]~0 of (9) 1 (d) Which surfaces are nondegenerate in the sense of (8) and thus have at most 2+7] omitted directions 1 (e) Is the bound 2+7] sharp in the sense that for any integer 7]~0 there are minimal surfaces with this 7] and with 2 + 7] omitted directions 1 (f) Which complete minimal surfaces are regularly exhaustible in the sense of (14) and thus satisfy (17)1 (g) Questions (c) and Ie) for complete minimal surfaces with 7] replaced by

s.

BIBLIOGRAPHY

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AHLFORs,

L. V.; BEURLING, A. [1] Conformal invariants and function. theoretic null-sets. Acta Math. 83 (1950), 101-129.

AHLFORs,

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AHLFORS,

AHMAD,M.

[1] A note on a theorem of Borel. Math. Student 25 (1957), 5-9. [2] On exceptional values of entire and meromorphic functions. Compositio Math. 13 (1958), 150-158. ALANDER, M.

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201

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a

point singulier essential isole. Mem.

CALUGAREANO, G. [1] Sur la determination des valeurs exeeptionnelles des fonetions entieres et meromorphes d'ordrefini. C.R. Acad. Sci. Paris 188 (1929), 37-39. [2] Sur la determination des valeurs exceptionnelles des fonetions entieres et meromorphes de genre fini. Bull. Sci. Math. (2) 54 (1930), 17-32. [3] Une yeneralisation du theoreme de AI. Borel sur les fonetions meromorphes. C.R. Acad. Sci. Paris 192 (1931), 329-330. [4] Sur un complement au theoreme de AI. Borel. Mathcmatica 6 (1932), 25-30. Bull. Cluj 6 (1932), 299-304. [5] Sur les valeurs exeeptionnelles, au sens de AI. Picard et de M. Nevanlinna, des fonetions meromorphes. C.R. Acad. Sci. Paris 195 (1932), 22-23. [6] Sur Ie theoreme de M. Borel dans Ie cas des fonetions meromorphes d'ordre infini. Mathematica 7 (1933), 61-69. Bull. Cluj 7 (1933), 174-182.

204

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S.·S. [1] Complex analytic mappings of Riemann surfaces. I. Amer. J. Math. 82 (1960), 323-337. [2] The integrated form of the first main theorem for complex analytic mappings in several complex variables. Ann. of Math. (2) 71 (1960), 536-551.

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J. [1] The derivative of a meromorphic function. Proc. Amer. Math. Soc. 7 (1956), 227-229. [2] On functions meromorphic in the unit circle. J. London Math. Soc. 32 (1957), 65-67. [3] On integral and meromorphicfunctions. Ibid. 37 (1962), 17-27.

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E. F. [1] Sur quelques theoremes de R. Nevanlinna. C.R. Acad. Sci. Paris 179 (1924), 955-957. [21 Sur une theoreme de M. Valiron. Ibid. 182 (1926), 40-42. r3] On meromorphic and integral functions. J. London Math. Soc. 5 (1930), 4-7. [4] Sur certaines ensembles definis pour les fonctions meromorphes. C.R. Acad. Sci. Paris 227 (1948),615-617.

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