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Normality criteria for families of holomorphic
mappings of several complex variables into $P^N(C)$

Author: Zhen-han Tu
Journal: Proc. Amer. Math. Soc. 127 (1999), 1039-1049
MSC (1991): Primary 32A17, 32H25, 32H30; Secondary 30D35, 30D45
MathSciNet review: 1469438
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Abstract: By applying the heuristic principle in several complex variables obtained by Aladro and Krantz, we shall prove some normality criteria for families of holomorphic mappings of several complex variables into $P^N(C)$, the complex N-dimensional projective space, related to Green's and Nochka's Picard type theorems. The equivalence of normality to being uniformly Montel at a point will be obtained. Some examples will be given to complement our theory in this paper.

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  • 1. G. Aladro and S. G. Krantz, A criterion for normality in $C^n$, J. Math. Anal. Appl. 161(1991), 1-8. MR 92j:32004
  • 2. A. Bloch, Sur les systèmes de fonctions holomorphes à variétés linéaires lacunaires, Ann. Ecole. Norm. Sup. 43(1926), 309-362.
  • 3. R. Brody, Compact manifolds and hyperbolicity, Trans. Amer. Math. Soc. 235(1978), 213-219. MR 57:10010
  • 4. H. Cartan, Sur les zéros des combinaisons linéaires de p fonctions holomorphes données, Mathematica 7(1933), 5-31.
  • 5. J. B. Conway, Functions of One Complex Variable, 2nd Ed., Springer-Verlag, New York, 1978. MR 80c:30003
  • 6. D. Drasin, Normal families and the Nevanlinna theory, Acta Math. 122(1969), 231-263. MR 40:2835
  • 7. H. Fujimoto, On families of meromorphic maps into the complex projective space, Nagoya Math. J. 54(1974), 21-51. MR 51:3543
  • 8. M. Green, Holomorphic maps into complex projective space omitting hyperplanes, Trans. Amer. Math. Soc. 169(1972), 89-103. MR 46:7547
  • 9. M. Green, The hyperbolicity of the complement of $2n+1$ hyperplanes in general position in $P_n$ and related results, Proc. Amer. Math. Soc. 66(1977), 109-113. MR 56:15994
  • 10. K. T. Hahn, Asymptotic behavior of normal mappings of several complex variables, Canad. J. Math. 36(1984), 718-746. MR 86b:32028
  • 11. W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964. MR 29:1337
  • 12. S. Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings, Dekker, New York, 1970. MR 43:3503
  • 13. S. Kobayashi, Intrinsic distances, measures and geometric function theory, Bull. Amer. Math. Soc. 82(1976), 357-416. MR 54:3032
  • 14. S. Lang, Introduction to Complex Hyperbolic Space, Springer-Verlag, New York-Berlin-Heidelberg, 1987. MR 88f:32065
  • 15. P. Montel, Lecons sur les familles normales de fonctions analytiques et leurs applications, Gauther-Villars, Paris, 1927.
  • 16. E. Nochka, On the theory of meromorphic functions, Soviet Math. Dokl. 27(1983), 377-381. MR 85i:32038
  • 17. J. Noguchi and T. Ochiai, Geometric Function Theory in Several Complex Variables, Translations Math. Monographs Vol. 80, Amer. Math. Soc., Providence, R.I., 1990. MR 92e:32001
  • 18. M. Ru and W. Stoll, The Cartan conjecture for moving targets, Proceedings of Symposia in Pure Math. 52(1991), part 2, 477-508. MR 93f:32028
  • 19. L. A. Rubel, Four counterexamples to Bloch's principle, Proc. Amer. Math. Soc. 98(1986), 257-260. MR 87i:30064
  • 20. Z.-h. Tu, Normality criterions for families of holomorphic mappings into $P^N(C)$, Geometric Complex Analysis (edited by Junjiro Noguchi et al.), World Scientific Publishing Co., 1996, 623-627. CMP 97:14
  • 21. Z.-h. Tu, On the Julia directions of the value distribution of holomorphic curves in $P^n(C)$, Kodai Math. J. 19(1996), No.1, 1-6. MR 97d:32033
  • 22. H. Wu, Normal families of holomorphic mappings, Acta Math. 119(1967), 193-233. MR 37:468
  • 23. L. Zalcman, A heuristic principle in complex function theory, Amer. Math. Monthly 82(1975), 813-817. MR 52:757

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Additional Information

Zhen-han Tu
Affiliation: Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong; Department of Mathematics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, People’s Republic of China

Keywords: Complex projective spaces, holomorphic mappings, hyperplanes in general position, normality criteria and Picard type theorems
Received by editor(s): February 3, 1997
Received by editor(s) in revised form: July 14, 1997
Communicated by: Steven R. Bell
Article copyright: © Copyright 1999 American Mathematical Society