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Quotients of Hyperbolic Metrics

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Abstract

In this paper, precise versions of several intuitive properties of quotients of hyperbolic metrics are established. Suppose that \(\Omega _j\) is a hyperbolic region in \(\mathbb {C}_\infty = \mathbb {C}\cup \{\infty \}\) with hyperbolic metric \(\lambda _j\), \(j=1,2\), and \(\Omega _1 \subsetneq \Omega _2\). First, it is shown that \(\lambda _1/ \lambda _2 \approx 1\) on compact subsets of \(\Omega _1\) that are not too close to \(\partial \Omega _1\). Second, \(\lambda _1/ \lambda _2 \approx 1\) when z is near \((\partial \Omega _1 \;\cap \; \partial \Omega _2 ) {\setminus } F_b\), where \(F = \partial \Omega _1 \;\cap \;\Omega _2\) and \(F_b = {{\mathrm{cl}}}(F)\;\cap \;\Omega _2\). The main tools used in establishing these results are sharp elementary bounds for \(\lambda _1(z)/ \lambda _2(z)\) in terms of the hyperbolic distance relative to \(\Omega _2\) from z to \(\partial \Omega _1 \;\cap \;\Omega _2\) that were first established and employed in complex dynamics.

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References

  1. Beardon, A.F., Minda, D.: Normal families: a geometric perspective, Comput. Methods Funct. Theory, Gehring Memorial Volume 14, pp. 331–355 (2014)

  2. Beardon, A.F., Pommerenke, Ch.: The Poincaré metric of plane domains. J. Lond. Math. Soc. 2(18), 475–483 (1978)

    Article  MATH  Google Scholar 

  3. Carathéodory, C.: Theory of Functions, vol. II, 2nd edn. Chelsea, New York (1960)

  4. Epstein, A.L.: Towers of finite type complex analytic maps, Ph.D. thesis. City University of New York (1993)

  5. Heins, M.: Selected Topics in the Classical Theory of Functions of a Complex Variable, vol. 34, Athena Series: Selected Topics in Mathematics. Holt, Rinehart and Winston, New York (1962)

    Google Scholar 

  6. Hejhal, D.: Universal covering maps for variable regions. Math. Z. 137, 7–20 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  7. Landau, H.J., Osserman, R.: On analytic mappings of Riemann surfaces. J. Anal. Math. 7, 249–279 (1959/1960)

  8. Masumoto, M., Shiba, M.: Intrinsic disks on a Riemann surface. Bull. Lond. Math. Soc. 27, 371–379 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  9. Mihaljević-Brandt, H., Rempe-Gillen, L.: Absence of wandering domains for some real entire functions with bounded singular sets. Math. Ann. 357, 1577–1604 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Minda, D.: Bloch constants. J. Anal. Math. 41, 54–84 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  11. Minda, D.: Hyperbolic distortion for holomorphic maps. J. Anal. 18, 317–336 (2010)

    MathSciNet  MATH  Google Scholar 

  12. Schleicher, D., Zimmer, J.: Periodic points and dynamic rays of exponential maps. Ann. Acad. Sci. Fenn. Math. 28, 327–354 (2003)

    MathSciNet  MATH  Google Scholar 

  13. Yamashita, S.: The Pick version of the Schwarz lemma and comparison of the Poincaré densities. Ann. Acad. Sci. Fenn. Ser. AI Math. 19, 291–322 (1994)

    MATH  Google Scholar 

Download references

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Correspondence to David Minda.

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Communicated by Mario Bonk.

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Minda, D. Quotients of Hyperbolic Metrics. Comput. Methods Funct. Theory 17, 579–590 (2017). https://doi.org/10.1007/s40315-017-0195-1

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  • DOI: https://doi.org/10.1007/s40315-017-0195-1

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