Skip to main content
SpringerLink
Log in

Geometric Characterizations of Symmetric Maps in the Complex Plane

  • Published:
Bulletin of the Malaysian Mathematical Sciences Society Aims and scope Submit manuscript

Abstract

In this paper, we establish several characterizations of planar symmetric maps, such as preserving Euclidean circles, preserving k-Apollonius circles for some \(k\in (0,\infty )\), and preserving geometric moduli of all pairs of disjoint continua in the extended complex plane.

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

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Beardon, A., Minda, D.: Sphere-preserving maps in inversive geometry. Proc. Am. Math. Soc. 130(4), 987–998 (2002)

    Article  MathSciNet  Google Scholar 

  2. Beurling, A., Ahlfors, L.V.: The boundary correspondence under quasiconformal mappings. Acta Math. 96, 125–142 (1956)

    Article  MathSciNet  Google Scholar 

  3. Guan, T., Ponnusamy, S., Zhou, Q.: A note on \(\partial \)-biLipschitz mappings in quasiconvex metric spaces. Bull. Sci. Math. 176, 103128, 19 (2022)

    Article  MathSciNet  Google Scholar 

  4. Hariri, P., Klén, R., Vuorinen, M.: Conformally Invariant Metrics and Quasiconformal Mappings. Springer Monographs in Mathematics, p. xix+502. Springer, Cham (2020)

    Book  Google Scholar 

  5. Haruki, H.: A new characteristic of Möbius transformations by use of Apollonius points of triangles. J. Math. Anal. Appl. 197(1), 14–22 (1996)

    Article  MathSciNet  Google Scholar 

  6. Hästö, P., Ibragimov, Z.: Apollonian isometries of planar domains are Möbius mappings. J. Geom. Anal. 15(2), 229–237 (2005)

    Article  MathSciNet  Google Scholar 

  7. Hästö, P., Ibragimov, Z., Minda, D., Ponnusamy, D., Sahoo, S.K.: Isometries of some hyperbolic-type path metrics, and the hyperbolic medial axis. In: The tradition of Ahlfors–Bers. IV, 63–74, Contemp. Math, vol. 432. American Mathematical Society, Providence (2007)

  8. Hästö, P., Lindén, H.: Isometries of the half-Apollonian metric. Complex Var. Theory Appl. 49(6), 405–415 (2004)

    MathSciNet  Google Scholar 

  9. Hästö, P., Ponnusamy, S., Sahoo, S.K.: Inequalities and geometry of the Apollonian and related metrics. Rev. Roumaine Math. Pures Appl. 51(4), 433–452 (2006)

    MathSciNet  Google Scholar 

  10. Piyapong, N.: A note on the characteristics of Möbius transformations. J. Math. Anal. Appl. 248(1), 203–215 (2000)

    Article  MathSciNet  Google Scholar 

  11. Tukia, P., Väisälä, J.: Quasisymmetric embeddings of metric spaces. Ann. Acad. Sci. Fenn. Ser. A I Math. 5, 97–114 (1980)

    Article  MathSciNet  Google Scholar 

  12. Tukia, P., Väisälä, J.: Rings and bilipschitz maps in normed spaces. Ann. Fenn. Math. 46(1), 587–591 (2021)

    Article  MathSciNet  Google Scholar 

  13. Yang, S.: On the properties of sphere-preserving maps. J. Math. Anal. Appl. 419(2), 748–755 (2014)

    Article  MathSciNet  Google Scholar 

  14. Zhou, Q., Ponnusamy, S.: Gromov hyperbolicity in the free quasiworld. I. Stud. Math. 268(1), 23–49 (2023)

    Article  MathSciNet  Google Scholar 

  15. Zhou, Q., Ponnusamy, S., Guan, T.: Gromov hyperbolicity of the \({\widetilde{j}}_G\) metric and boundary correspondence. Proc. Am. Math. Soc. 150(7), 2839–2847 (2022)

    Article  MathSciNet  Google Scholar 

  16. Zhou, Q., Rasila, A.: Quasimöbius invariance of uniform domains. Stud. Math. 261, 1–24 (2021)

    Article  Google Scholar 

  17. Zhou, Q., Rasila, A.: Teichmüller’s problem on Gromov hyperbolic domains. Israel J. Math. 252(1), 399–427 (2022)

    Article  MathSciNet  Google Scholar 

  18. Zhou, Q., Rasila, A.: Geometric characterization of Gromov hyperbolic Hölder domains. Forum Math. 34(6), 1621–1640 (2022)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

We are grateful for the constructive remarks from the referees.

Funding

This work is supported by Research Fund of Guangdong-Hong Kong-Macao Joint Laboratory for Intelligent Micro-Nano Optoelectronic Technology (No. 2020B1212 030010) and Natural Science Foundation of China (No.11901061).

Author information

Authors and Affiliations

  1. School of Mathematics and Big Data, Foshan University, Foshan, 528000, People’s Republic of China

    Qianghua Luo

  2. School of Information Science and Technology, Xiamen University Tan Kah Kee College, Zhangzhou, 363105, Fujian, People’s Republic of China

    Bo Jiao

  3. School of Sciences, Changzhou Institute of Technology, Changzhou, 213032, People’s Republic of China

    Xuejing Ren

Authors
  1. Qianghua Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bo Jiao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xuejing Ren
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xuejing Ren.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Communicated by Saminathan Ponnusamy.

Publisher's Note

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

Qianghua Luo was supported by Research Fund of Guangdong-Hong Kong-Macao Joint Laboratory for Intelligent Micro-Nano Optoelectronic Technology (No. 2020B1212030010). Xuejing Ren was supported by NSFC(No.11901061).

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Luo, Q., Jiao, B. & Ren, X. Geometric Characterizations of Symmetric Maps in the Complex Plane. Bull. Malays. Math. Sci. Soc. 47, 73 (2024). https://doi.org/10.1007/s40840-024-01665-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40840-024-01665-9

Keywords

Mathematics Subject Classification

Search

Navigation