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A Landau’s theorem in several complex variables. (English) Zbl 1366.32008

One can show, by example, that the Landau theorem and the related result of Brody-Zalcman is not true in several variables.
In this paper, the author introduces a class of holomorphic maps for which one can get a Landau theorem and a Brody-Zalcman theorem in several complex variables.
Denote by \(B^k=\big\{z\in \mathbb C^k : \sum_{j=1}^k|z_j|^2<1\big\}\) the unit ball in \(\mathbb C^k\), \(k>1\). Let \(\Phi: B^k \rightarrow \mathbb C^k \) be a holomorphic map. Let \(\Phi^\prime(a)\) denote the Jacobian matrix of \(\Phi\) computed in the point \(a\in B^k.\)
The author proves the following theorems.
Theorem 2.1. (Brody-Zalcman-type theorem) Let \(C\) be a positive constant. Consider a family of holomorphic maps \(\Phi\) satisfying \[ ||\Phi^\prime(a)|| \cdot ||\Phi^\prime(a)^{-1}|| \leq C,\quad \forall a \in B^k. \] If the family is not normal, then, after reparametrization, we can extract a subsequence converging to a non-degenerate holomorphic map \(\Psi : \mathbb C^k \rightarrow \mathbb C^k\).
Assume that \(\Phi: B^k \rightarrow\mathbb C^k \), \(\Phi(0) = 0\), and \(\Phi^\prime(0) = \mathrm{Id}\).
Theorem 3.1. (Landau-type theorem) Let \(C\) be a positive constant such that \[ ||\Phi^\prime(z)|| \cdot ||\Phi^\prime(z)^{-1}|| \leq C,\quad \forall z \in B^k. \] Then there exists \(\rho>0\), depending only on \(C\), such that \(\Phi(B^k)\) contains a ball of radius \(\rho>0\). There exists also a domain \(U\) such that \(\Phi(U) = B(a,\rho)\).

MSC:

32H99 Holomorphic mappings and correspondences
32A18 Bloch functions, normal functions of several complex variables
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