Holomorphic Automorphism Groups of (m,1)-Circular Domains

  • Shuxia Feng
  • Hongjun LiEmail author
  • Chunhui Qiu


A quasi-circular domain of weight (mp) in \({\mathbb {C}}^2 \) is in fact a (mp)-circular domain Cartan (J Math Pures Appl 96:1–114, 1931). In this article, we give isotropic subgroups and holomorphic automorphism groups of bounded (m, 1)-circular domains under linear equivalences by Lie group technique. Adding to Thullen (Math Ann 104:244–259, 1931), Xu (Acta Math Sin 13(3):419–432, 1963), holomorphic automorphisms of bounded (mp)-circular domains are determined.


Quasi-circular domain \((m{, }p)\)-circular domain Holomorphic automorphism group Isotropic subgroup 

Mathematics Subject Classification




The authors would like to thank Prof. Daniel Zhuang-Dan Guan and Prof. Yichao Xu for giving us useful discussions. The authors are also pleased to thank the referees for very helpful comments and suggestions. This research is supported by the National Natural Science Foundation of China (Grant Nos. 11571288, 11771357).


  1. 1.
    Cartan, H.: Les fonctions de deux variables complexes et le problème de la représentation analytique. J. Math. Pures Appl. 96, 1–114 (1931)zbMATHGoogle Scholar
  2. 2.
    Cartan, H.: Sur les transformations analytiques des domaines cerclés et semi-cerclés bornés. Math. Ann. 106, 540–573 (1932)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cartan, H.: Sur les fonctions de plusieurs variables complexes. Litération des transformations intérieures dun domaine borné. Math. Z. 35, 760–773 (1932)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cartan, H.: Sur les groupes de transformations analytiques. Act. Sci. Ind. Hermann, Paris (1935)Google Scholar
  5. 5.
    Deng, F., Rong, F.: On biholomorphisms between bounded quasi-Reinhardt domains. Ann. Mat. Pura Appl. (4) 195(3), 835–843 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Kaup, W.: Über das Randverhalten von holomorphen automorphismen beschränkter Gebiete. Manuscr. Math. 3, 257–270 (1970)CrossRefzbMATHGoogle Scholar
  7. 7.
    Li, H., Qiu, C., Xu, Y.: On bounded positive \((m, p)\)-circle domains. Chin. Ann. Math. 39(4), 665–682 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Rong, F.: On automorphisms of quasi-circular domains fixing the origin. Bull. Sci. Math. 140, 92–98 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Rong, F.: The degree of automorphisms of quasi-circular domains fixing the origin. Int. J. Math. 28(9), 1740008 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Shimizu, S.: Automorphisms of bounded Reinhardt domains. Proc. Jpn. Acad. Ser. A Math. Sci. 63(9), 354–355 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Shimizu, S.: Automorphisms and equivalence of bounded Reinhardt domains not containing the origin. Tohoku Math. J. 40(1), 119–152 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Shimizu, S.: Automorphisms of bounded Reinhardt domains. Jpn. J. Math. (N.S.) 15(3), 385–414 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Sunada, T.: Holomorphic equivalence problem for bounded Reinhardt domains. Math. Ann. 235, 111–128 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Thullen, P.: Zur den Abbildungen durch analytische Funktionen mehrerer komplexer Veränder-lichen. Math. Ann. 104, 244–259 (1931)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Xu, Y.: On the groups of analytic automorphisms of bounded positive \((m, p)\)-circular domains. Acta Math. Sin. 13(3), 419–432 (1963). (in Chinese)zbMATHGoogle Scholar
  16. 16.
    Xu, Y.: On the classification of symetric schlicht domains in several complex variable. Adv. Math. (China) 8, 109–144 (1965). (in Chinese)Google Scholar
  17. 17.
    Yamamori, A.: Automorphisms of normal quasi-circular domains. Bull. Sci. Math. 138, 406–415 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Yamamori, A.: On the linearity of origin-preserving automorphisms of quasi-circular domains in \({{{\mathbb{C}}}}^n\). J. Math. Anal. Appl. 426, 612–623 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Yamamori, A., Zhang, L.: On origin-preserving automorphisms of quasi-circular domains. J. Geom. Anal. 28, 1840–1852 (2018)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsHenan UniversityKaifengPeople’s Republic of China
  2. 2.School of Mathematical SciencesXiamen UniversityXiamenPeople’s Republic of China

Personalised recommendations