Skip to main content
SpringerLink
Log in

Classification of a Class of Bounded Anti-circular Domains

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

Cartan (J Math Pures Appl 96:1–114, 1931) firstly introduced bounded anti-circular domains centered at the origin in \({\mathbb {C}}^2\). In this paper, we give the classification and realization of a class of bounded anti-circular domains centered at the origin with the Thullen condition. Their isotropic subgroups and holomorphic automorphism groups are also determined.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Cartan, H.: Les fonctions de deux variables complexes et le problème de la représentation analytique. J. Math. Pures Appl. 96, 1–114 (1931)

    MATH  Google Scholar 

  2. Cartan, H.: Sur les transformations analytiques des domaines cerclés et semi-cerclés bornés. Math. Ann. 106, 540–573 (1932)

    Article  MathSciNet  Google Scholar 

  3. Cartan, H.: Sur les fonctions de plusieurs variables complexes. L’itération des transformations intérieures d’un domaine borné. Math. Z. 35, 760–773 (1932)

    Article  MathSciNet  Google Scholar 

  4. Cartan, H.: Sur Les Groupes de Transformations Analytiques (Exposés mathámatiques IX). Actual. Scient. et Industr., Hermann, Paris (1935)

    Google Scholar 

  5. Feng, S., Li, H., Qiu, C.: Holomorphic automorphism groups of \((m,1)\)-circular domains. J. Geom. Anal. 30, 3035–3063 (2020)

    Article  MathSciNet  Google Scholar 

  6. Kaup, W.: Über das randverhalten von holomorphen automorphismen beschränkter gebiete. Manuscr. Math. 3, 257–270 (1970)

    Article  Google Scholar 

  7. Kritikos, N.: Über analytische Abbildungen eiens Klasse von vierdimensionalen Gebieten. Math. Ann. 99, 321–342 (1928)

    Article  MathSciNet  Google Scholar 

  8. Li, H., Qiu, C.: The holomorphic automorphism group of higher dimension Thullen domain. Acta Math. Sci. 36B(5), 1358–1368 (2016)

    Article  MathSciNet  Google Scholar 

  9. Li, H., Qiu, C., Xu, Y.: On bounded positive \((m, p)\)-circle domains. Chin. Ann. Math. Ser. B 39(4), 665–682 (2018)

    Article  MathSciNet  Google Scholar 

  10. Naruki, I.: The holomorphic equivalence problem for a class of Reinhardt domains. Publ RIMS Kyoto Univ. 4A(2), 527–543 (1968)

    Article  MathSciNet  Google Scholar 

  11. Rong, F.: On automorphisms of quasi-circular domains fixing the origin. Bull. Sci. Math. 140, 92–98 (2016)

    Article  MathSciNet  Google Scholar 

  12. Rong, F.: The degree of automorphisms of quasi-circular domains fixing the origin. Int. J. Math. 28(9), 1740008 (2017)

    Article  MathSciNet  Google Scholar 

  13. Sunada, T.: On bounded Reinhardt domains. Proc. Jpn. Acad. 50(2), 119–123 (1974)

    MathSciNet  MATH  Google Scholar 

  14. Sunada, T.: Holomorphic equivalence problem for bounded Reinhardt domains. Math. Ann. 235(2), 111–128 (1978)

    Article  MathSciNet  Google Scholar 

  15. Thullen, P.: Zu den abbildungen durch analytische funktionen mehrerer komplexer veränderlichen die invarianz des mittelpunktes von kreiskörpern die invarianz des mittelpnnktes von kreiskörpern. Math. Ann. 104, 244–259 (1931)

    Article  MathSciNet  Google Scholar 

  16. 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)

    MATH  Google Scholar 

  17. Xu, Y., Wang, D.: On the classification of bounded semi-circular domains in \(\mathbb{C}^2\). Acta Math. Sin. 23(3), 372–384 (1980). (in Chinese)

    MATH  Google Scholar 

  18. Yamamori, A.: Automorphisms of normal quasi-circular domains. Bull. Sci. Math. 138, 406–415 (2014)

    Article  MathSciNet  Google Scholar 

  19. 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)

    Article  MathSciNet  Google Scholar 

  20. Yamamori, A., Zhang, L.: On origin-preserving automorphisms of quasi-circular domains. J. Geom. Anal. 28, 1840–1852 (2018)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Prof. Yichao Xu for giving us useful discussions. The authors are very grateful to the referees for providing many valuable suggestions. This research is supported by the National Natural Science Foundation of China (Grant Nos. 11971401, 11971415, 11701494), China Postdoctoral Science Foundation(Grant No. 2019M652513), Postdoctoral Research Foundation of Henan Province (Grant No. CJ3050A 0670218), and the Nanhu Scholars Program for Young Scholars of XYNU.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Henan University, Kaifeng, 475004, People’s Republic of China

    Hongjun Li

  2. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, People’s Republic of China

    Chunhui Qiu

  3. School of Mathematics and Statistics, Xinyang Normal University, Xinyang, 464000, People’s Republic of China

    Hongchuan Xia

Authors
  1. Hongjun Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chunhui Qiu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hongchuan Xia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hongjun Li.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, H., Qiu, C. & Xia, H. Classification of a Class of Bounded Anti-circular Domains. J Geom Anal 31, 5010–5032 (2021). https://doi.org/10.1007/s12220-020-00465-4

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12220-020-00465-4

Keywords

Mathematics Subject Classification

Search

Navigation