Abstract
The Fock–Bargmann–Hartogs domain \(D_{n,m}(\mu )\) (\(\mu >0\)) in \(\mathbb {C}^{n+m}\) is defined by the inequality \(\Vert w\Vert ^2<e^{-\mu \Vert z\Vert ^2},\) where \((z,w)\in \mathbb {C}^n\times \mathbb {C}^m\), which is an unbounded non-hyperbolic domain in \(\mathbb {C}^{n+m}\). Recently, Tu–Wang obtained the rigidity result that proper holomorphic self-mappings of \(D_{n,m}(\mu )\) are automorphisms for \(m\ge 2\), and found a counter-example to show that the rigidity result is not true for \(D_{n,1}(\mu )\). In this article, we obtain a classification of proper holomorphic mappings between \(D_{n,1}(\mu )\) and \(D_{N,1}(\mu )\) with \(N<2n\).
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Acknowledgements
The authors would like to thank Professor Xianyu Zhou for his helpful discussions, and thank the referees for useful comments. The first author was supported by the National Natural Science Foundation of China (No. 11671306), and the second author was partially supported by China Postdoctoral Science Foundation (No. 2016M601150).
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Tu, Z., Wang, L. Classification of Proper Holomorphic Mappings Between Certain Unbounded Non-hyperbolic Domains. J Geom Anal 29, 378–391 (2019). https://doi.org/10.1007/s12220-018-9995-4
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DOI: https://doi.org/10.1007/s12220-018-9995-4