Value Distribution Theory

246 Pages·1966·4.873 MB·English

by  Leo Sario, Kiyoshi Noshiro (auth.)

Value Distribution Theory
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, MICHEIr-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
N ACHBIN, LEOPOLDG--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.
Palue Dtstrtbutton
Theory
LEO SARlO
Professor of Mathematics
University of California
Los Angeles, California
AND
KIYOSHI NOSHIRO
Professor of Mathematics
Nagoya University
Nagoya, Japan
in collaboration with
T ADASHI KURODA KIKUJI MATSUMOTO MITSURU NAKAI
D. VAN NOSTRAND COMPANY, INC.
PRINCETON, NEW JERSEY
TORONTO NEW YORK LONDON
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COPYRIGHT © 1966, BY
D. VAN NOSTRAND COMPANY, INC.
SOFTCOVER REPRINT OF THE HARDCOVER 1ST EDITION 1966
Published simultaneously in Canada by
D. VAN NOSTRAND COMPANY (Canada), LTD.
No reproductiun in any form of this book, in whole or in
part (except for brief quotation in critical articles or rcviews),
may be made without written authorization from the publishcrs.
ISBN 978-1-4615-8128-4 ISBN 978-1-4615-8126-0 (eBook)
DOI 10.1007/978-1-4615-8126-0
DEDICATED TO
The University of California
which gave us the opportunity to discover
CONTENTS
ACKXOWLEDG~IEXTS Xl
PREFACE 1
IXTRODL"CTIO)l" 5
1. Historical 5
2. New metric 5
3. The fundamental A-, B-, and C-functions 7
4. Method of areal proximity 8
5. Summary 10
CHAPTER I .:\IAPPINGS INTO CLOSED RIEMANN SURF ACES
§1. .i\IAPPINGS OF ARBITRARY RIEMA)l"N SURFACES 11
1. The proximity function s(~,a) 12
2. The fundamental functions A, B, and C 13
3. Euler characteristic 16
4. Areal proximity 18
5. Main theorem 19
6. N ondegeneracy 21
7. Exceptional points 23
8. Ramification 24
§2. MEROMORPHIC FUNCTIONS ON ARBITRARY RIE~iA)l"N SURFACES 25
9. Main theorems 25
10. Sharpness of even bounds 27
11. Sharpness of arbitrary bounds 29
12. The class of Rp-surfaces 30
§3. SURFACES Rs AND CONFORMAL METRICS 32
13. Metric 32
14. The fundamental functions 33
15. Preliminary form of the second main theorem 35
16. Evaluations 36
17. Exceptional intervals 38
18. Second main theorem 39
19. Picard points 40
Vll
nil CONTENTS
CHAPTER II MAPPINGS INTO OPEN RIEMANN SURFA CES
§1. PRINCIPAL FUNCTIONS 42
1. Preliminaries 43
2. Auxiliary functions 44
3. Linear operators 47
4. An integral equation 49
§2. PROXIMITY FUNCTIONS ON ARBITRARY RIEMANN SURFACES 51
5. Boundedness of auxiliary functions 52
6. Uniform boundedness from below of S ("a) 53
7. Symmetry of s(',a) 54
8. Conformal metric 57
§3. ANALYTIC :\rAPPINGS 59
9. Main theorems 60
10. Affinity relation 62
11. Existence of mappings 63
12. Area of exceptional sets 65
13. Decomposition of S ("a) in subregions 67
14. Joint continuity of s(',a) 68
15. Consequences 70
16. Capacity of exceptional sets 71
CHAPTER III FUNCTIONS OF BOUNDED CHARACTERISTIC
§1. DECOMPOSITION 74
1. Generalization of Jensen's formula 75
2. Decomposition theorem 78
3. Extremal decompositions 82
4. Consequenc6s 85
§2. THE CLASS OMB 86
5. Preliminaries 87
6. Characterization of OMB 89
7. Decomposition by uniformization 92
8. Theorems of Heins, Parreau, and Rao 94
CHAPTER IV FUNCTIONS ON PARABOLIC RIEMANN
SURFACES
§1. THE EVANS-SELBERG POTENTIAL 98
1. The Cech compactification 99
2. Green's kernel on the Cech compactification 101
3. Transfinite diameter 105
CONTENTS IX
4. Energy integral 109
5. Construction 113
§2. ]\fERO:\IORPHIC Fl:XCTIOXS IX A BOuXD.\RY XEIGHBORHOOD 115
6. The af Hiillstrom-Tsuji approach 115
7. Exceptional sets 118
CHAPTER V PICARD SETS
§1 . IKFIXITE PICARD SETS 120
1. Sets of capacity zero 120
2. Sets of positive capacity 123
§2. FEITE PIC.\RD SETS 125
3. Generalized Picard theorem 125
4. Auxiliary rcsults 127
5. Proof of the generalized Picard theorem 129
6. Classes of sets 'iyith the Picard property 132
n
CHAPTER RIE}IAXXIAN DIAGES
§1 . }IEAX SHEET XL:\IBERS 136
1. Base surface 137
2. Coyering of subregions 139
3. Covering of curves 141
§2. ELLER CHARACTERISTIC 144
4. Preliminaries 144
5. Cross-cuts and regions 145
6. Main theorem on Euler characteristic 147
7. Extension to positive genus 151
§3. ISLAXDS AKD PEKIXSULAS 152
8. Fundamental inequality 152
9. Auxiliary estimates 153
10. Proof of the fundamental inequality 155
11. Defects and ramifications 157
§4. ?dEROMORPHIC FuXCTIONS 158
12. Regular exhaustibility 159
13. Application of the fundamental inequality 160
14. Role of the inverse function 163
15. Localized second main theorem 164
16. Localized Picard theorem 165
§5. MAPPIKGS OF ARBITRARY RIEMANN SURFACES 167
17. Conformal metrics 167
x CONTENTS
18. Main theorem for arbitrary Riemann surfaces 169
19. Integrated form 171
20. Algebroids 173
21. Sharpness of nonintegrated defect relation 175
22. Direct estimate of M (p) 176
23. Extension to arbitrary integers 178
ApPENDIX 1. BASIC PROPERTIES OF RIEMANN SURFACES 179
ApPENDIX II. GAUSSIAN MAPPING OF ARBITRARY MINIMAL SURFACES 194
1. Triple connectivity 194
2. Arbitrary connectivity 195
3. Arbitrary genus 196
4. Arbitrary genus and connectivity 196
5. Gaussian mapping 197
6. Picard directions 198
7. Islands and peninsulas 199
8. Regular exhaustions 199
9. Open questions 200
BIBLIOGRAPHY 201
SUBJECT INDEX 231
AUTHOR INDEX 235
ACKNOWLEDGMENTS
We are deeply gratdul to the U. S. Army Research Office-Durham,
in general, and to Drs. John W. Dawson and A. S. Galbraith in partic
ular, for Deveral U.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 arc due to Professor S. S. Chern for the inclusion
of our book in this distinguished series and for his continued stimula
tion.
Weare indebted to many colleagues who read the manuscript, in
particular our collaborator M. Nakai, who made substantial contribu
tions to several parts of the theory and scrutinized the entire manu
script; our collaborator K. Matsumoto, who contributed his conclusive
results on Picard sets; our collaborator T. Kuroda, who with Matsu
moto 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; 11. 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.
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