View in EDS

Normality criteria for families of holomorphic mappings of several complex variables into \(P^N(\mathbb{C})\)

Saved in:
Bibliographic Details
Title: Normality criteria for families of holomorphic mappings of several complex variables into \(P^N(\mathbb{C})\)
Authors: Tu, Zhen-han
Publisher Information: American Mathematical Society (AMS), Providence, RI
Subject Terms: Nevanlinna theory, Picard-type theorems and generalizations for several complex variables, heuristic principle, complex projective spaces, normal families, Picard type theorems, Montel normality criteria, Value distribution of meromorphic functions of one complex variable, Nevanlinna theory, Special families of functions of several complex variables, holomorphic mappings, hyperplanes in general position, Normal functions of one complex variable, normal families, hyperplanes, Value distribution theory in higher dimensions
Description: By modifying the heuristic principle in several complex variables obtained by \textit{G. Aladro} and \textit{S. G. Krantz} [J. Math. Anal. Appl. 161, 1-8 (1991; Zbl 0749.32001)], the author proves some normality criteria for families of holomorphic mappings of a domain \(D\subset \mathbb{C}^n\) into the complex \(N\)-dimensional projective space \(\mathbb{P}^N(\mathbb{C})\) related to the Green's and Nochka's Picard type theorems. The equivalence of normality to being uniformly Montel at a point is obtained: Let \(F\) be a family of holomorphic mappings of a domain \(D\subset \mathbb{C}^N\) into \(\mathbb P^N(\mathbb C).\) Then \(F\) is normal at a point \(z_0\in D\) iff \(F\) is uniformly Montel at the point \(z_0.\) Two examples are included to illustrate the author's theory.
Document Type: Article
File Description: application/xml
DOI: 10.1090/s0002-9939-99-04610-9
Access URL: https://zbmath.org/1245428
Accession Number: edsair.c2b0b933574d..fd8e83f4e89ca3700775f2395b1bc6dd
Database: OpenAIRE
Description
Abstract:By modifying the heuristic principle in several complex variables obtained by \textit{G. Aladro} and \textit{S. G. Krantz} [J. Math. Anal. Appl. 161, 1-8 (1991; Zbl 0749.32001)], the author proves some normality criteria for families of holomorphic mappings of a domain \(D\subset \mathbb{C}^n\) into the complex \(N\)-dimensional projective space \(\mathbb{P}^N(\mathbb{C})\) related to the Green's and Nochka's Picard type theorems. The equivalence of normality to being uniformly Montel at a point is obtained: Let \(F\) be a family of holomorphic mappings of a domain \(D\subset \mathbb{C}^N\) into \(\mathbb P^N(\mathbb C).\) Then \(F\) is normal at a point \(z_0\in D\) iff \(F\) is uniformly Montel at the point \(z_0.\) Two examples are included to illustrate the author's theory.
DOI:10.1090/s0002-9939-99-04610-9