ON THE EXISTENCE OF SINGULAR DIRECTIONS OF HOLOMORPHIC MAPS FROM THE UNIT DISK INTO Pn(C)

doi:10.1016/S0252-9602(10)60156-7

数学物理学报(英文版) ›› 2010, Vol. 30 ›› Issue (5): 1631-1639.doi: 10.1016/S0252-9602(10)60156-7

ON THE EXISTENCE OF SINGULAR DIRECTIONS OF HOLOMORPHIC MAPS FROM THE UNIT DISK INTO Pⁿ(C)

涂振汉, 张莎莎

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

收稿日期:2008-08-30 修回日期:2009-02-24 出版日期:2010-09-20 发布日期:2010-09-20
基金资助:
The project supported in part by the National Natural Science Foundation of China (10971156).

ON THE EXISTENCE OF SINGULAR DIRECTIONS OF HOLOMORPHIC MAPS FROM THE UNIT DISK INTO Pⁿ(C)

TU Zhen-Han, ZHANG Sha-Sha

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

Received:2008-08-30 Revised:2009-02-24 Online:2010-09-20 Published:2010-09-20
Supported by:
The project supported in part by the National Natural Science Foundation of China (10971156).

摘要/Abstract

摘要：

This article proves the existence of Julia directions of value distribution of holomorphic mapping f from the unit disk into the n-dimensional complex projective space Pⁿ(C) under the assumption limits_{r\rightarrow 1^{-}} T(r, f)/ log 1/1-r= +∞ for hypersurfaces in general position. A heuristic principle concerning the existence of Julia directions of holomorphic mappings from the unit disk into Pⁿ(C) is given also.

关键词: complex projective spaces, holomorphic mapping, Julia direction, Picards theorem, value distribution theory

Abstract:

Key words: complex projective spaces, holomorphic mapping, Julia direction, Picards theorem, value distribution theory

中图分类号:

32H25

涂振汉, 张莎莎. ON THE EXISTENCE OF SINGULAR DIRECTIONS OF HOLOMORPHIC MAPS FROM THE UNIT DISK INTO Pⁿ(C)[J]. 数学物理学报(英文版), 2010, 30(5): 1631-1639.

TU Zhen-Han, ZHANG Sha-Sha. ON THE EXISTENCE OF SINGULAR DIRECTIONS OF HOLOMORPHIC MAPS FROM THE UNIT DISK INTO Pⁿ(C)[J]. Acta mathematica scientia,Series B, 2010, 30(5): 1631-1639.

参考文献

[1] Aladro G. Applications of the Kobayashi metric to normal functions of several complex variables. Utilitas Math, 1987, 31: 13--24

[2] Aladro G, Krantz S G. A criterion for normality in Cⁿ. J Math Anal Appl, 1991, 161(1): 1--8

[3] Eremenko A. A Picard type theorem for holomorphic curves. Period Math Hung, 1999, 38(1/2): 39--42

[4] Eremenko A. Value distribution and potential theory. ICM, 2002, 2: 681--690

[5] Fujimoto H. Extensions of the big Picard's theorem. T\^{o}hokuMath J, 1972, 24(3): 415--422

[6] Green M. Holomorphic maps into complex projective space omitting hyperplanes. Trans Amer Math Soc, 1972, 169: 89--103

[7] Jin J L. Singular points of Julia type of meromorphic functions in unit disk. Journal of Nanjing University (Natural Sci), 2000, 36(1): 28--33 (in Chinese)

[8] Julia G. Sur queques properiétés nouvelles des fonctions entères ou méromorphes. Ann École Norm Sup, 1916, 36: 93--125. Ibid, 1920, 37: 165--218

[9] Lang S. Introduction to Complex Hyperbolic Spaces. New York: Springer, 1987

[10] Lu Q, Gu Y X. The singular point of the meromorphic function on the unit disc. Acta Math Sinica, 2000, 43(4): 751--756(in Chinese)

[11] Milloux H. Le théorème de Picard, suites de fonctions holomorphes, fonctions holomorphes et fonctions entières. J Math Pures Appl, 1924, 3: 345--401

[12] Nochka E. On the theory of meromorphic functions. Soviet Math Dokl, 1983, 27(2): 377--381

[13] Noguchi J, Ochiai T. Geometry Function Theory in Several Complex Variables. Tanslations Math Monographs, Vol 80. Providence, Rhode Island: Amer Math Soc, 1990

[14] Rubel L A. Four counterexamples to Bloch's principle. Proc Amer Math Soc, 1986, 98(2): 257--260

[15] Shabat B V. Distibution of Values of Holomorphic Mappings. Transl Math Monogr, Vol 61. Providence, Rhode Island: Amer Math Soc, 1985

[16] Tsuji M. Potential Theory in Modern Function Theory. Tokyo: Maruzen Co, 1959

[17] Tu Z H. On the Julia directions of the value distribution of holomorphic curves in Pⁿ(C). Kodai Math J, 1996, 19(1): 1--6

[18] Tu Z H. Normality criteria for family of holomorphic mappings of several complex variables into Pⁿ(C). Proc Amer Math Soc, 1999, 127(4): 1039--1049

[19] Tu Z H, Li P L. On the singular directions of value distibution of holomorphic curves in Pⁿ(C). Acta Math Sci, 2006, 26B(4): 702--710

[20] Valiron G. Directions de Boral des Fonctions Méromorphes. Mémor Sci Math Fasc 89. Paris: Gauthier-Villars et Cie, 1938

[21] Zalcman L. A heuristic principle in complex function theory. Amer Math Monthly, 1975, 82(8): 813--817

[22] Zalcman L. Normal families: new perspectives. Bull Amer Math Soc, 1998, 35(3): 215--230

ON THE EXISTENCE OF SINGULAR DIRECTIONS OF HOLOMORPHIC MAPS FROM THE UNIT DISK INTO Pⁿ(C)

ON THE EXISTENCE OF SINGULAR DIRECTIONS OF HOLOMORPHIC MAPS FROM THE UNIT DISK INTO Pⁿ(C)

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