East Asian mathematical journal

The Youngnam Mathematical Society (영남수학회)
권호 표지
DOI QR코드

DOI QR Code

AN INTRINSIC CRITERION FOR THE TYPE OF AUTOMORPHISMS OF THE UNIT DISC

  • Lee, Kang-Hyurk (Lee Department of Mathematics and Research Institute of Natural Science, Gyeongsang National University)
  • Received : 2021.04.08
  • Accepted : 2021.05.31
  • Published : 2021.05.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we deal with a problem to determine the type of automorphisms of the unit disc in ℂ in terms of intrinsic geometry. We will characterize the hyperbolicity and parabolicity of automorphism by the distance function of the Poincaré metric.

Keywords

Acknowledgement

The research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (No. NRF-2019R1F1A1060891).

References

  1. L. V. Ahlfors, An extension of Schwarz's lemma, Trans. Amer. Math. Soc., 43 (1938), pp. 359-364. https://doi.org/10.1090/S0002-9947-1938-1501949-6
  2. K.-T. Kim and H. Lee, Schwarz's lemma from a differential geometric viewpoint, vol. 2 of IISc Lecture Notes Series, IISc Press, Bangalore; World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011.
  3. V. V. Kisil, Geometry of Mobius transformations, Imperial College Press, London, 2012. Elliptic, parabolic and hyperbolic actions of SL2(R), With 1 DVD-ROM.
  4. S. Kobayashi, Hyperbolic manifolds and holomorphic mappings, vol. 2 of Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 1970.
  5. S. Kobayashi, Hyperbolic complex spaces, vol. 318 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1998.
  6. P. Koebe, Uber die Uniformisierung reeller algebraischer Kurven., Nachr. Ges. Wiss. Gottingen, Math.-Phys. Kl., 1907 (1907), pp. 177-190.
  7. S. Lang, SL2(R), Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1975.
  8. K.-H. Lee, On parabolic and hyperbolic automorphisms of the unit disc, Internat. J. Math. Anal., 9 (2015), pp. 1405-1413. https://doi.org/10.12988/ijma.2015.53114
  9. H. Poincare, Sur l'uniformisation des fonctions analytiques, Acta Math., 31 (1908), pp. 1-63. https://doi.org/10.1007/BF02415442