Classification of Proper Holomorphic Mappings Between Certain Unbounded Non-hyperbolic Domains

  • Zhenhan Tu
  • Lei Wang


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 \(N<2n\).


Fock–Bargmann–Hartogs domains Local biholomorphisms Proper holomorphic mappings 

Mathematics Subject Classification

Primary 32A07 32H35 32M05 



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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Authors and Affiliations

  1. 1.School of Mathematics and StatisticsWuhan UniversityWuhanPeople’s Republic of China
  2. 2.School of Mathematics and StatisticsHuazhong University of Science and TechnologyWuhanPeople’s Republic of China

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