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Holomorphic Maps with Large Images

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Abstract

We show that each pseudoconvex domain \(\Omega \subset {\mathbb {C}}^n\) admits a holomorphic map \(F\) to \({\mathbb {C}}^m\) with \(|F|\le C_1 e^{C_2 \hat{\delta }^{-6}}\), where \(\hat{\delta }\) is the minimum of the boundary distance and \((1+|z|^2)^{-1/2}\), such that every boundary point is a Casorati–Weierstrass point of \(F\). Based on this fact, we introduce a new anti-hyperbolic concept—universal dominability. We also show that for each \(\alpha >6\) and each pseudoconvex domain \(\Omega \subset {\mathbb C}^n\), there is a holomorphic function \(f\) on \({\Omega }\) with \(|f|\le C_\alpha e^{C_\alpha ' \hat{\delta }^{-\alpha }}\), such that every boundary point is a Picard point of \(F\). Applications to the construction of holomorphic maps of a given domain onto some \({\mathbb C}^m\) are given.

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Acknowledgments

Research supported by the Key Program of NSFC No. 11031008. We would like to thank Professor Franc Forstnerič for numerous comments on this paper. We also thank Dr. Qi’an Guan for catching an inaccuracy in the proof of Proposition 1.3. Finally, we wish to thank the referee for a very careful reading and many valuable comments.

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  1. Department of Mathematics, Tongji University, Shanghai , 200092, China

    Bo-Yong Chen & Xu Wang

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  1. Bo-Yong Chen
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  2. Xu Wang
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Correspondence to Bo-Yong Chen.

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Communicated by Der-Chen Edward Chang.

Dedicated to the memory of Shoshichi Kobayashi.

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Chen, BY., Wang, X. Holomorphic Maps with Large Images. J Geom Anal 25, 1520–1546 (2015). https://doi.org/10.1007/s12220-014-9482-5

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  • DOI: https://doi.org/10.1007/s12220-014-9482-5

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