Osaka Journal of Mathematics

Two theorems on the Fock-Bargmann-Hartogs domains

Akio Kodama and Satoru Shimizu

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In this paper, we prove two mutually independent theorems on the family of Fock-Bargmann-Hartogs domains. Let $D_1$ and $D_2$ be two Fock-Bargmann-Hartogs domains in $\mathbb{C}^{N_1}$ and $\mathbb{C}^{N_2}$, respectively. In Theorem 1, we obtain a complete description of an arbitrarily given proper holomorphic mapping between $D_1$ and $D_2$ in the case where $N_1 = N_2$. Also, we shall give a geometric interpretation of Theorem 1. And, in Theorem 2, we determine the structure of $\text{Aut}(D_1\times D_2)$ using the data of $\text{Aut}(D_1)$ and $\text{Aut}(D_2)$ for arbitrary $N_1$ and $N_2$.

Article information

Osaka J. Math., Volume 56, Number 4 (2019), 739-757.

First available in Project Euclid: 21 October 2019

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Mathematical Reviews number (MathSciNet)

Primary: 32A07: Special domains (Reinhardt, Hartogs, circular, tube)
Secondary: 32M05: Complex Lie groups, automorphism groups acting on complex spaces [See also 22E10]


Kodama, Akio; Shimizu, Satoru. Two theorems on the Fock-Bargmann-Hartogs domains. Osaka J. Math. 56 (2019), no. 4, 739--757.

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  • H. Alexander: Holomorphic mappings from the ball and polydisc, Math. Ann. 209 (1974), 249–256.
  • S. Bell: Analytic hypoellipticity of the $\bar\partial$-Neumann problem and extendability of holomorphic mappings, Acta Math. 147 (1981), 109–116.
  • H. Cartan: Sur les fonctions de plusieurs variables complexes: L'itération des transformations intérieures d'un domaine borné, Math. Z. 35 (1932), 760–773.
  • H. Cartan: Sur les fonctions de n variables complexes: les transformations du produit topologique de deux domaines bornés, Bull. Soc. Math. France 64 (1936), 37–48.
  • R.C. Gunning and H. Rossi: Analytic Functions of Several Complex Variables, Prentice-Hall, Englewood Cliffs, N.J., 1965.
  • H. Kim, V.T. Ninh and A. Yamamori: The automorphism group of a certain unbounded non-hyperbolic domain, J. Math. Anal. Appl. 409 (2014), 637–642.
  • S. Kobayashi: Hyperbolic Complex Spaces, Springer-Verlag, Berlin Heidelberg New York, 1998.
  • A. Kodama: On generalized Siegel domains, Osaka J. Math. 14 (1977), 227–252.
  • A. Kodama: A localization principle for biholomorphic mappings between the Fock-Bargmann-Hartogs domains, Hiroshima Math. J. 48 (2018), 171–187.
  • S. Murakami: On Automorphisms of Siegel Domains, Lecture Notes in Math., Vol. 286, Springer-Verlag, Berlin Heidelberg New York, 1972.
  • K. Peters: Starrheits\"stze für Produkte normierter Vektorräume endlicher Dimension und für Produkte hyperbolischer komplexer Räume, Math. Ann. 208 (1974), 343–354.
  • S.I. Pinchuk: On the analytic continuation of holomorphic mappings, Math. USSR Sb. 27 (1975), 375–392.
  • S. Shimizu: Automorphisms and equivalence of bounded Reinhardt domains not containing the origin, Tohoku Math. J. 40 (1988), 119–152.
  • N. Sibony: A class of hyperbolic manifolds, Ann. Math. Stud. 100 (1981), 357–372.
  • Z. Tu and L. Wang: Rigidity of proper holomorphic mappings between certain unbounded non-hyperbolic domains, J. Math. Anal. Appl. 419 (2014), 703–714.
  • A. Yamamori: The Bergman kernel of the Fock-Bargmann-Hartogs domain and the polylogarithm function, Complex Var. Elliptic Equ. 58 (2013), 783–793.