East Asian mathematical journal

  • 제39권1호
  • /
  • Pages.11-21
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  • 2023
  • /
  • 1226-6973(pISSN)
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  • 2287-2833(eISSN)
영남수학회 (The Youngnam Mathematical Society)
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A CHARACTERIZATION OF AUTOMORPHISMS OF THE UNIT DISC BY THE POINCARÉ METRIC

  • Kang-Hyurk, Lee (Department of Mathematics and Research Institute of Natural Science, Gyeongsang National University) ;
  • Kyu-Bo, Moon (Department of Mathematics and Research Institute of Natural Science, Gyeongsang National University)
  • 투고 : 2022.09.23
  • 심사 : 2022.10.31
  • 발행 : 2023.01.31
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초록

Non-trivial automorphisms of the unit disc in the complex plane can be classified by three classes; elliptic, parabolic and hyperbolic automorphisms. This classification is due to a representation in the projective special linear group of the real field, or in terms of fixed points on the closure of the unit disc. In this paper, we will characterize this classification by the distance function of the Poincaré metric on the interior of the unit disc.

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과제정보

This work was supported by Samsung Science and Technology Foundation under Project Number SSTF-BA2201-01.

참고문헌

  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). 
  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. K.-H. Lee, An intrinsic criterion for the type of automorphisms of the unit disc, East Asian Math. J., 37 (2021), pp. 307-317.  https://doi.org/10.7858/EAMJ.2021.021
  10. H. Poincare, Sur l'uniformisation des fonctions analytiques, Acta Math., 31 (1908), pp. 1-63. https://doi.org/10.1007/BF02415442