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

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

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 mrm.iiti@gmail.com
SWADESH KUMAR SAHOO*
Affiliation:
Discipline of Mathematics, Indian Institute of Technology Indore, Simrol, Khandwa Road, Indore 453 552, India email swadesh@iiti.ac.in
*
✉swadesh@iiti.ac.in
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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. 

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