Montel-type theorems in several complex variables with continuously moving targets
type: Item
instance of: scholarly article
title: Montel-type theorems in several complex variables with continuously moving targets
zbMATH Open document ID: 1207.32004
author: Shasha Zhang
author: Zhenhan Tu
published in: Chinese Annals of Mathematics. Series B
publication date: 2010-11-17T00:00:00Z
review text: Let \(P^N(\mathbb C)\) be the complex projective space of dimension \(N,\) and \( \rho: \mathbb C^{N+1}\to P^N(\mathbb C)\) be the standard projective mapping. Let \(h\) be a holomorphic mapping from a domain \(D\) in \(\mathbb C^n\) into \(P^N(\mathbb C)\). For any \(a \in D\), the mapping \(h\) has a reduced representation \(\widetilde{h}^{(p)}(z)=(h^{(p)}_0(z), \dots, h^{(p)}_N(z))\) on some neighborhood \(U\) of \(a\) in \(D,\) i.e., \(h^{(p)}_0(z), \dots, h^{(p)}(z)\) are holomorphic on \(U\) without common zeros such that \(h(z)=\rho(\widetilde{h}(z))\) on \(U\). A family \(F\) of holomorphic mappings of a domain \(D\) in \(\mathbb C^n\) into \(P^N(\mathbb C)\) is said to be normal on \(D\) if any sequence in \(F\) contains a subsequence which converges uniformly on compact subsets of \(D\) to a holomorphic mapping of \(D\) into \(P^N(\mathbb C),\) where a subsequence \(\{f^p\}\subset F\) is said to converge uniformly on compact subsets of \(D\) to a holomorphic mapping \(f\) of \(D\) into \(P^N(\mathbb C)\) iff, for any \(a \in D,\) each \(f^{(p)}(z)\) has a reduced representation \(\widetilde{f}^{(p)}(z)=(f^{(p)}_0(z), \dots, f^{(p)}_N(z))\) on some fixed neighborhood \(U\) of \(a\in D\) such that \(\{f^{(p)}_i(z)\}_{i=1}^\infty\) converge uniformly on compact subsets of \(U\) to a holomorphic function \(f_i(z)\) \((i=0, \dots, N)\) on \(D\) with the property that \(\widetilde{f}(z):=(f_0(z), \ldots, f_N(z))\) is a reduced representation of \(f(z)\) on \(U\). Let \(E\) be a subset of \(\mathbb C^n\). For \(z\in E\), define the linear form \[ L(z)(Z_0,\ldots,Z_N):=\sum_{j=0}^Na_j(z)Z_j, \] where \(a_j(z)\) \((j=0,\ldots,N)\) are continuous functions on \(E\) without common zeros. Then \[ H(z):= \rho(\{(Z_0,\ldots,Z_N)\in\mathbb C^{N+1}\setminus \{0\} : L(z)(Z_0,\ldots,Z_N)=0\}) \] is called a continuous moving hyperplane in \(P^N(\mathbb C)\) corresponding to the linear form \(L(z)\) \((z \in E)\). Continuous moving hyperplanes in \(H_j(z)\) (where \(j=1,\ldo…
Mathematics Subject Classification ID: 32A19
Mathematics Subject Classification ID: 32H25
Mathematics Subject Classification ID: 32H30
zbMATH DE Number: 5816604
zbMATH Keywords: holomorphic mappings
zbMATH Keywords: normal families
zbMATH Keywords: Picard-type theorems
zbMATH Keywords: value distribution theory
reviewed by: Peter Dovbush
MaRDI profile type: Publication
full work available at URL: s11401-009-0009-5
OpenAlex ID: W2067048883
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DOI: 10.1007/S11401-009-0009-5
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description: scientific article; zbMATH DE number 5816604