Value Distribution Theory

ISBN-13: 9781461581284 / Angielski / Miękka / 2012 / 236 str.

T. Kuroda; K. Matsumoto; M. Nakai

Opis

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 over lapping with any existing monograph on merom orphic 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, pre dominantly 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."

Spis treści

1. Historical.- 2. New metric.- 3. The fundamental A-, B-, and C-functions.- 4. Method of areal proximity.- 5. Summary.- I Mappings Into Closed Riemann Surfaces.- §1. Mappings of Arbitrary Riemann Surfaces.- 1. The proximity function s(?,a).- 2. The fundamental functions A, B, and C.- 3. Euler characteristic.- 4. Areal proximity.- 5. Main theorem.- 6. Nondegeneracy.- 7. Exceptional points.- 8. Ramification.- §2. Meromorphic Functions on Arbitrary Riemann Surfaces.- 9. Main theorems.- 10. Sharpness of even bounds.- 11. Sharpness of arbitrary bounds.- 12. The class of Rp-surfaces.- §3. Surfaces RS and Conformal Metrics.- 13. Metric.- 14. The fundamental functions.- 15. Preliminary form of the second main theorem.- 16. Evaluations.- 17. Exceptional intervals.- 18. Second main theorem.- 19. Picard points.- II Mappings Into Open Riemann Surfaces.- §1. Principal Functions.- 1. Preliminaries.- 2. Auxiliary functions.- 3. Linear operators.- 4. An integral equation.- §2. Proximity 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.- §3. Analytic 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.- III Functions of Bounded Characteristic.- §1. Decomposition.- 1. Generalization of Jensen’s formula.- 2. Decomposition theorem.- 3. Extremal decompositions.- 4. Consequences.- §2. The Class OMB.- 5. Preliminaries.- 6. Characterization of OMB.- 7. Decomposition by uniformization.- 8. Theorems of Heins, Parreau, and Rao.- IV Functions on Parabolic Riemann Surfaces.- §1. The Evans-Selberg Potential.- 1. The ?ech compactification.- 2. Green’s kernel on the ?ech compactification.- 3. Transfinite diameter.- 4. Energy integral.- 5. Construction.- §2. Meromorphic Functions in a Boundary Neighborhood.- 6. The af Hällström-Tsuji approach.- 7. Exceptional sets.- V Picard Sets.- §1. Infinite Picard Sets.- 1. Sets of capacity zero.- 2. Sets of positive capacity.- §2. Finite Picard Sets.- 3. Generalized Picard theorem.- 4. Auxiliary results.- 5. Proof of the generahzed Picard theorem.- 6. Classes of sets with the Picard property.- VI 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. Fundamental inequality.- 9. Auxiliary estimates.- 10. Proof of the fundamental inequality.- 11. 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.- 18. Main theorem for arbitrary Riemann surfaces.- 19. Integrated form.- 20. Algebroids.- 21. Sharpness of nonintegrated defect relation.- 22. Direct estimate of M(p).- 23. Extension to arbitrary integers.- Appendix I. Basic Properties of Riemann Surfaces.- Appendix II. Gaussian Mapping of Arbitrary Minimal Surfaces.- 1. Triple connectivity.- 2. Arbitrary connectivity.- 3. Arbitrary genus.- 4. Arbitrary genus and connectivity.- 5. Gaussian mapping.- 6. Picard directions.- 7. Islands and peninsulas.- 8. Regular exhaustions.- 9. Open questions.- Author Index.