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 D. VAN NOSTRAND COMPANY, INC. 120 Alexander St., Princeton, New Jersey (Principal office) 24 West 40 Street, New York 18, New York D. VAN NOSTRAND COMPANY, LTD. 358, Kensington High Street, London, W.14, England D. VAN NOSTRAND COMPANY (Canada), LTD. 25 Hollinger Road, Toronto 16, Canada 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. Xl