A criterion for normality in \(\mathbb{C}^ n\)
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.
scientific article; zbMATH DE number 57287
scientific article