Unicity of Meromorphic Mappings

by Hu, Pei-Chu; Li, Ping; Yang, Chung-Chun

• ISBN13: 9781402012198
• ISBN10: 1402012195
• Format: Hardcover
• Copyright: 2003-06-01
• Publisher: Kluwer Academic Pub

Summary

This book introduces value distribution theory starting with a survey of two Nevanlinna-type main theorems and defect relations for meromorphic mappings from parabolic manifolds into projective spaces. Then the unicity theory of meromorphic functions or mappings is discussed systematically and the discussion also covers value distribution theory of algebroid functions of several variables and its applications in unicity theory.Audience: Graduate students and researchers involved in the fields of analysis, complex function theory of one or several variables, value distribution theory and analysis on complex manifolds.

Author Biography

Pei-Chu Hu: Department of Mathematics, Shandong University, Jinan, China Ping Li: Department of Mathematics, University of Science and Technology of China, Hefei, Anhui Chung-Chun Yang: Department of Mathematics, The Hong Kong University of Science and Technology, China

Table of Contents

Preface	vii
1 Nevanlinna theory	1	(118)
1.1 Parabolic manifolds and Hermitian geometry	1	(8)
1.2 The first main theorem	9	(10)
1.3 Growths of meromorphic functions	19	(12)
1.4 The lemma of logarithmic derivative	31	(11)
1.5 Growth estimates of Wronskians	42	(5)
1.6 The second main theorem	47	(6)
1.7 Degenerate holomorphic curves	53	(10)
1.8 Value distribution of differential polynomials	63	(13)
1.9 The second main theorem for small functions	76	(10)
1.10 Tumura-Clunie theory	86	(15)
1.11 Generalizations of Nevanlinna theorem	101	(8)
1.12 Generalizations of Borel theorem	109	(10)
2 Uniqueness of meromorphic functions on C	119	(92)
2.1 Functions that share four values	119	(14)
2.2 Functions that share three values CM	133	(11)
2.3 Functions that share pairs of values	144	(6)
2.4 Functions that share four small functions	150	(7)
2.5 Functions that share five small functions	157	(6)
2.6 Uniqueness related to differential polynomials	163	(28)
2.7 Polynomials that share a set	191	(3)
2.8 Meromorphic functions that share the same sets	194	(5)
2.9 Unique range sets	199	(6)
2.10 Uniqueness polynomials	205	(6)
3 Uniqueness of meromorphic functions on Cm	211	(98)
3.1 Technical lemmas	211	(9)
3.2 Multiple values of meromorphic functions	220	(3)
3.3 Uniqueness of differential polynomials	223	(6)
3.4 The four-value theorem	229	(5)
3.5 The three-value theorem	234	(9)
3.6 Generalizations of Rubel-Yang's theorem	243	(11)
3.7 Meromorphic functions sharing one value	254	(8)
3.8 Unique range sets of meromorphic functions	262	(12)
3.9 Unique range sets ignoring multiplicities	274	(10)
3.10 Meromorphic functions of order less than 1	284	(9)
3.11 A note on the abc-conjecture	293	(8)
3.12 A note on Hall's conjecture	301	(8)
4 Uniqueness of meromorphic mappings	309	(70)
4.1 Notes on the first main theorem	309	(7)
4.2 The first main theorem for line bundles	316	(9)
4.3 The second main theorem for line bundles	325	(8)
4.4 Uniqueness of meromorphic mappings into Pn	333	(7)
4.5 Finiteness theorems	340	(7)
4.6 General meromorphic mappings	347	(6)
4.7 Dependence theorems	353	(12)
4.8 Propagation theorems	365	(9)
4.9 Uniqueness dealing with multiple values	374	(5)
5 Algebroid functions of several variables	379	(62)
5.1 Preliminaries	379	(6)
5.2 Techniques of value distribution	385	(4)
5.3 The second main theorem	389	(6)
5.4 Algebroid reduction of meromorphic mappings	395	(11)
5.5 The growth of branching divisors	406	(6)
5.6 Reduction of Nevanlinna theory	412	(10)
5.7 Generalizations of Malmquist theorem	422	(8)
5.8 Uniqueness problems	430	(7)
5.9 Multiple values of algebroid functions	437	(4)
References	441	(20)
Symbols	461	(2)
Index	463