The Apollonian Metric of a Domain in ℝ n

Beardon, A. F.

doi:10.1007/978-1-4612-0605-7_8

A. F. Beardon⁵

683 Accesses
30 Citations

Abstract

This paper introduces a new Möbius invariant metric α _D, which we call the Apollonian metric, of a domain D with the following desirable properties:

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Chapter: USD 29.95; Price excludes VAT (USA)

eBook: USD 84.99; Price excludes VAT (USA)

Softcover Book: USD 109.99; Price excludes VAT (USA)

Hardcover Book: USD 109.99; Price excludes VAT (USA)

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

References

Ahlfors, L.V., Conformal invariants, McGraw-Hill, 1973.
MATH Google Scholar
Beardon, A.F., The geometry of discrete groups, GTM Vol.91, Springer Verlag, 1983.
Book Google Scholar
Beardon, A.F. and Pommerenke, Ch., On the Poincaré metric of plane domains, J. London Math. Soc. (2), 18 (1978), 475–483.
MathSciNet MATH Google Scholar
de la Harpe, P., Hilbert’s metric for simplices, Geometric Group Theory, (G.A. Niblo and M.A. Roller, eds.), London Math. Soc. Lecture Notes, Vol. 182, Cambridge Univ. Press, 1993, 97–119.
Google Scholar
Gehring, F.W. & Palka, B.P., Quasiconformally homogeneous domains, Journal d’Analyse Math., 30 (1976), 172–199.
Article MathSciNet MATH Google Scholar
Hayman, W.K., Multivalent functions, 2nd edition, Cambridge Univ. Press, 1994.
Book MATH Google Scholar
Hilbert, D., Ueber die gerade Linie als kürzeste Verbindung zweier Punkte, Math. Ann. 46 (1895), 91–98.
Article Google Scholar
Keogh, F.R., A characterisation of convex domains in the plan. Bull. London Math. Soc., 8 (1976), 183–185; see also Math. Review,57 (1979), # 6392.
Google Scholar
Kirwan, W.E. and Schober, G., Extremal problems for meromorphic univalent functions, J. Analyse Math., 30 (1976), 330–348.
Article MathSciNet MATH Google Scholar
Köhn, J., Die Harnacksche Metrik in der Theorie der harmonischen Funktionen, Math. Zeit., 91 (1966), 50–64.
Article MATH Google Scholar
Leutwiler, H., On a distance invariant under Möbius transformations in Ann. Acad. Sci. Fenn., 12 (1987), 3–17.
MathSciNet MATH Google Scholar
Vuorinen, M., Conformal geometry and quasiregular mapping,Spring-er Lecture Notes 1319, 1988.
Google Scholar
Weitsman, A., Symmetrization and the Poincaré metric, Annals of Math., 124 (1986), 159–169.
Article MathSciNet MATH Google Scholar

Download references

Author information

Authors and Affiliations

Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge, CB2 1SB, England
A. F. Beardon

Authors

A. F. Beardon
View author publications
You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

Department of Mathematics, University of Michigan, 4074 East Hall, Ann Arbor, MI, 48109-1109, USA
Peter Duren
Department of Mathematics, University of Michigan, 525 East University Street-East Hall, Ann Arbor, MI, 48109-1109, USA
Juha Heinonen
Department of Mathematics, Stanford University, Stanford, CA, 94305-2125, USA
Brad Osgood
Department of Mathematics, University of Texas, Austin, TX, 78712-1082, USA
Bruce Palka

Rights and permissions

Reprints and permissions

Copyright information

About this chapter

Cite this chapter

Beardon, A.F. (1998). The Apollonian Metric of a Domain in ℝⁿ . In: Duren, P., Heinonen, J., Osgood, B., Palka, B. (eds) Quasiconformal Mappings and Analysis. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0605-7_8

Download citation

DOI: https://doi.org/10.1007/978-1-4612-0605-7_8
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-6836-9
Online ISBN: 978-1-4612-0605-7
eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

The Apollonian Metric of a Domain in ℝn