A criterion for normality in \(\mathbb{C}^ n\)
type: Item
instance of: scholarly article
title: A criterion for normality in \(\mathbb{C}^ n\)
zbMATH Open document ID: 0749.32001
DOI: 10.1016/0022-247X(91)90356-5
author: Gerardo Aladro
author: Steven G. Krantz
published in: Journal of Mathematical Analysis and Applications
publication date: 1992-09-27T00:00:00Z
review text: The main result of the paper is the following Theorem 3.1. A family \({\mathcal F}=\{f_ \alpha\}_{\alpha\in A}\) of holomorphic mappings of a hyperbolic domain \(\Omega\subseteq\mathbb{C}^ n\) to a complete complex Hermitian manifold \(M\) is not normal if and only if there exist a compact set \(K\Subset\Omega\) and sequences \(\{p_ j\}\subset K\), \(\{f_ j\}\subset{\mathcal F}\), \(\{\rho_ j\}\) with \(\rho_ j>0\) and \(\rho_ j\to 0^ +\) and \(\{\xi_ j\}\subset\mathbb{C}^ n\) Euclidean unit vectors, such that \(g_ j(\zeta){\buildrel{\text{def}} \over =}f_ j(p_ j+\rho_ j\xi_ j\zeta)\); \(\zeta\in\mathbb{C}\), converges uniformly on compact subsets of \(\mathbb{C}\) to a nonconstant entire function. It is a generalization of Zalcman's criterion for normality of families of holomorphic functions on the disk in \(\mathbb{C}\) to several complex variables.
Mathematics Subject Classification ID: 32A17
Mathematics Subject Classification ID: 32A30
Mathematics Subject Classification ID: 30D45
Mathematics Subject Classification ID: 32Q45
zbMATH DE Number: 57287
zbMATH Keywords: normal families of holomorphic mappings
zbMATH Keywords: hyperbolic domain
MaRDI profile type: Publication
cites work: Q3710804
cites work: Compact Manifolds and Hyperbolicity
cites work: Asymptotic properties of normal and nonnormal holomorphic mappings
cites work: Asymptotic Behavior of Normal Mappings of Several Complex Variables
cites work: Q3285751
cites work: Q5183860
cites work: A Heuristic Principle for a Nonessential Isolated Singularity
cites work: Metamathematical problems
cites work: Q5622543
cites work: Four Counterexamples to Bloch's Principle
cites work: Normality criteria for families of meromorphic functions
cites work: Normal families of holomorphic mappings
cites work: Some theorems on projective hyperbolicity
cites work: A Heuristic Principle in Complex Function Theory
full work available at URL: 0022-247x(91)90356-5
OpenAlex ID: W2045027528
description: scientific article; zbMATH DE number 57287