The Kobayashi Pseudometric for the Fock–Bargmann–Hartogs Domain and Its Application

  • Enchao Bi
  • Guicong Su
  • Zhenhan TuEmail author


The Fock–Bargmann–Hartogs domain \(D_{n,m}\) in \(\mathbb {C}^{n+m}\) is defined by the inequality \(\Vert w\Vert ^2<e^{-\Vert z\Vert ^2},\) where \((z,w)\in \mathbb {C}^n\times \mathbb {C}^m\), which is an unbounded non-hyperbolic domain in \(\mathbb {C}^{n+m}\). This paper mainly consists of three parts. Firstly, we give the explicit expression of geodesics of \(D_{n,1}\) in the sense of Kobayashi pseudometric; secondly, using the formula of geodesics, we calculate explicitly the Kobayashi pseudometric on \(D_{1,1}\); lastly, we establish the Schwarz lemma at the boundary for holomorphic mappings between the non-equidimensional Fock–Bargmann–Hartogs domains by using the formula for the Kobayashi pseudometric on \(D_{1,1}\).


Fock–Bargmann–Hartogs domains Kobayashi pseudometric Boundary Schwarz lemma 

Mathematics Subject Classification

32F45 32H02 30C80 



We sincerely thank the referees for his useful comments. The project is supported by the National Natural Science Foundation of China (No. 11671306) and the Natural Science Foundation of Shandong Province, China (No. ZR2018BA015).


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Authors and Affiliations

  1. 1.School of Mathematics and StatisticsQingdao UniversityQingdaoPeople’s Republic of China
  2. 2.School of Mathematics and StatisticsWuhan UniversityWuhanPeople’s Republic of China

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