Value Distribution Theory

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 Hallstrom-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.