Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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A GENERALIZED HURWITZ METRIC

  • Arstu, Arstu (Discipline of Mathematics Indian Institute of Technology Indore) ;
  • Sahoo, Swadesh Kumar (Discipline of Mathematics Indian Institute of Technology Indore)
  • Received : 2019.08.22
  • Accepted : 2020.03.26
  • Published : 2020.09.30
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Abstract

In 2016, the Hurwitz metric was introduced by D. Minda in arbitrary proper subdomains of the complex plane and he proved that this metric coincides with the Poincaré's hyperbolic metric when the domains are simply connected. In this paper, we provide an alternate definition of the Hurwitz metric through which we could define a generalized Hurwitz metric in arbitrary subdomains of the complex plane. This paper mainly highlights various important properties of the Hurwitz metric and the generalized metric including the situations where they coincide with each other.

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References

  1. D. Burago, Y. Burago, and S. Ivanov, A course in metric geometry, Graduate Studies in Mathematics, 33, American Mathematical Society, Providence, RI, 2001. https://doi.org/10.1090/gsm/033
  2. K. T. Hahn, Some remarks on a new pseudodifferential metric, Ann. Polon. Math. 39 (1981), 71-81. https://doi.org/10.4064/ap-39-1-71-81
  3. A. Hurwitz, Uher die Anwendung der elliptischen Modulfunktionen auf einen Satz der allgemeinen, Funktionentheorie. Vierteljahrschr. Naturforsch. Gesell. Zurich 49 (1904), 242-253.
  4. L. Keen and N. Lakic, A generalized hyperbolic metric for plane domains, in In the tradition of Ahlfors-Bers. IV, 107-118, Contemp. Math., 432, Amer. Math. Soc., Providence, RI, 2007. https://doi.org/10.1090/conm/432/08303
  5. L. Keen and N. Lakic, Hyperbolic geometry from a local viewpoint, London Mathematical Society Student Texts, 68, Cambridge University Press, Cambridge, 2007. https://doi.org/10.1017/CBO9780511618789
  6. C. D. Minda, The Hahn metric on Riemann surfaces, Kodai Math. J. 6 (1983), no. 1, 57-69. https://doi.org/10.2996/kmj/1138036663
  7. D. Minda, The Hurwitz metric, Complex Anal. Oper. Theory 10 (2016), no. 1, 13-27. https://doi.org/10.1007/s11785-015-0446-y
  8. Z. Nehari, Conformal Mapping, McGraw-Hill Book Co., Inc., New York, Toronto, London, 1952.
  9. K. Tavakoli, Coincidence of hyperbolic and generalized Kobayashi densities on plane domains, Ann. Acad. Sci. Fenn. Math. 34 (2009), no. 2, 347-351.