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MAPPING PROPERTIES OF A SCALE INVARIANT CASSINIAN METRIC AND A GROMOV HYPERBOLIC METRIC

Part of: Real functions Geometric function theory Riemann surfaces Real and complex geometry

Published online by Cambridge University Press:  18 August 2017

MANAS RANJAN MOHAPATRA  and
SWADESH KUMAR SAHOO Open the ORCID record for SWADESH KUMAR SAHOO [Opens in a new window]
MANAS RANJAN MOHAPATRA
Affiliation:
Discipline of Mathematics, Indian Institute of Technology Indore, Simrol, Khandwa Road, Indore 453 552, India email [email protected]
SWADESH KUMAR SAHOO*
Affiliation:
Discipline of Mathematics, Indian Institute of Technology Indore, Simrol, Khandwa Road, Indore 453 552, India email [email protected]
*
✉[email protected]
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Abstract

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We consider a scale invariant Cassinian metric and a Gromov hyperbolic metric. We discuss a distortion property of the scale invariant Cassinian metric under Möbius maps of a punctured ball onto another punctured ball. We obtain a modulus of continuity of the identity map from a domain equipped with the scale invariant Cassinian metric (or the Gromov hyperbolic metric) onto the same domain equipped with the Euclidean metric. Finally, we establish the quasi-invariance properties of both metrics under quasiconformal maps.

Keywords

Möbius maphyperbolic-type metricsthe scale invariant Cassinian metricthe Gromov hyperbolic metricuniform continuityquasiconformal map

MSC classification

Primary: 51M10: Hyperbolic and elliptic geometries (general) and generalizations
Secondary: 26A15: Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) 30C20: Conformal mappings of special domains 30C65: Quasiconformal mappings in $^n$, other generalizations 30F45: Conformal metrics (hyperbolic, Poincaré, distance functions)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 97 , Issue 1 , February 2018 , pp. 141 - 152
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

References

Anderson, G. D., Vamanamurthy, M. K. and Vuorinen, M. K., Conformal Invariants, Inequalities, and Quasiconformal Maps (Wiley, New York, 1997).Google Scholar
Beardon, A. F., Geometry of Discrete Groups (Springer, New York, 1995).Google Scholar
Beardon, A. F., ‘The Apollonian metric of a domain in ℝ n ’, in: Quasiconformal Mappings and Analysis (Ann Arbor, MI, 1995) (eds. Duren, P., Heinonen, J., Osgood, B. and Palka, B.) (Springer, New York, 1998), 91–108.Google Scholar
Berger, M., Geometry I (Springer, Berlin, 1987).Google Scholar
Duren, P. L., Univalent Functions (Springer, Heidelberg, 1983).Google Scholar
Gehring, F. W. and Hag, K., ‘The Apollonian metric and quasiconformal mappings’, Contemp. Math. 256 (2000), 143–163.Google Scholar
Gehring, F. W. and Osgood, B. G., ‘Uniform domains and the quasihyperbolic metric’, J. Anal. Math. 36 (1979), 50–74.Google Scholar
Gehring, F. W. and Palka, B. P., ‘Quasiconformally homogeneous domains’, J. Anal. Math. 30 (1976), 172–199.Google Scholar
Graham, I. and Kohr, G., Topics in Geometric Function Theory in One and Higher Dimensions (Marcel Dekker Inc., New York, 2003).Google Scholar
Gromov, M., ‘Hyperbolic groups’, in: Essays in Group Theory, Mathematical Sciences Research Institute Publications, 8 (Springer, New York, 1987), 75–263.Google Scholar
Hästö, P., ‘The Apollonian metric: uniformity and quasiconvexity’, Ann. Acad. Sci. Fenn. Math. 28(2) (2003), 385–414.Google Scholar
Hästö, P., ‘Gromov hyperbolicity of the $j_{G}$ and $\tilde{j}_{G}$ metrics’, Proc. Amer. Math. Soc. 134(4) (2005), 1137–1142.Google Scholar
Hästö, P., Klén, R., Sahoo, S. K. and Vuorinen, M., ‘Geometric properties of 𝜑-uniform domains’, J. Anal. 24(1) (2016), 57–66.Google Scholar
Hayman, W. K., Subharmonic Functions, Vol. 2, London Mathematical Society Monographs, vol. 20 (Academic Press (Harcourt Brace Jovanovich, Publishers), London, 1989).Google Scholar
Ibragimov, Z., ‘On the Apollonian metric of domains in ℝ n ’, Complex Var. Theory Appl. 48(10) (2003), 837–855.Google Scholar
Ibragimov, Z., ‘The Cassinian metric of a domain in ℝ n ’, Uzbek. Mat. Zh. 1 (2009), 53–67.Google Scholar
Ibragimov, Z., ‘Hyperbolizing metric spaces’, Proc. Amer. Math. Soc. 139(12) (2011), 4401–4407.Google Scholar
Ibragimov, Z., ‘A scale-invariant Cassinian metric’, J. Anal. 24(1) (2016), 111–129.Google Scholar
Ibragimov, Z., Mohapatra, M. R., Sahoo, S. K. and Zhang, X.-H., ‘Geometry of the Cassinian metric and its inner metric’, Bull. Malays. Math. Sci. Soc. 40(1) (2017), 361–372.CrossRefGoogle Scholar
Kim, K. and Langmeyer, N., ‘Harmonic measure and hyperbolic distance in John disks’, Math. Scand. 83 (1998), 283–299.CrossRefGoogle Scholar
Klén, R., Li, L. and Vuorinen, M., ‘Subdomain geometry of hyperbolic type metrics’, Trans. Inst. Math. Natl. Acad. Sci. Ukr. 10(4–5) (2013), 190–206.Google Scholar
Klén, R., Mohapatra, M. R. and Sahoo, S. K., ‘Geometric properties of the Cassinian metric’, Math. Nachr. 290 (2017), 1531–1543.Google Scholar
Klén, R., Vuorinen, M. and Zhang, X.-H., ‘Quasihyperbolic metric and Möbius transformations’, Proc. Amer. Math. Soc. 142(1) (2014), 311–322.Google Scholar
Kumaresan, S., Topology of Metric Spaces, 2nd edn (Alpha Science International, Oxford, UK, 2011).Google Scholar
Lehto, O. and Virtanen, K. I., Quasiconformal Mappings in the Plane (Springer, New York, 1973).CrossRefGoogle Scholar
Mohapatra, M. R. and Sahoo, S. K., ‘A Gromov hyperbolic metric vs the hyperbolic and other related metrics’, Preprint, 2017, arXiv:1705.08574.Google Scholar
Rudin, W., Principles of Mathematical Analysis, 3rd edn (McGraw Hill, USA, 1976).Google Scholar
Seittenranta, P., ‘Möbius-invariant metrics’, Math. Proc. Cambridge Philos. Soc. 125 (1999), 511–533.Google Scholar
Väisälä, J., Lectures on n-dimensional Quasiconformal Mappings (Springer, Berlin–Heidelberg–New York, 1971).Google Scholar
Väisälä, J., ‘Gromov hyperbolic spaces’, Expo. Math. 23 (2005), 187–231.Google Scholar
Vuorinen, M., ‘Conformal invariants and quasiregular mappings’, J. Anal. Math. 45 (1985), 69–115.Google Scholar
Vuorinen, M., Conformal Geometry and Quasiregular Mappings, Lecture Notes in Mathematics, 1319 (Springer, Berlin, 1988).Google Scholar
Vuorinen, M., ‘Metrics and quasiregular mappings’, in: Quasiconformal Mappings and their Applications (New Delhi, India, 2007) (eds. Ponnusamy, S., Sugawa, T. and Vuorinen, M.) (Narosa Publishing House, New Delhi, 2007), 291–325.Google Scholar