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Conformality of the Apollonian Metric

Conformality of the Apollonian Metric The Apollonian metric aD of a domain \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$D\subset {\overline R}^{n}$\end{document} is rarely conformal. In fact, if it is conformal at one point then D is, up to a Möbius transformation, a complement of a convex body of constant width and if it is conformal at two points then D is a ball. We consider a quantity that measures the deviation of aD from being conformal. This quantity is essential in comparing the Apollonian metric to hyperbolic and quasihyperbolic metrics. We show that this quantity is invariant under Möbius transformations and compute it for some standard domains. We then use it to obtain sharp estimates between any two of the Apollonian, hyperbolic and quasihyperbolic metrics on such domains. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

Conformality of the Apollonian Metric

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Publisher
Springer Journals
Copyright
Copyright © Heldermann  Verlag 2003
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/bf03321045
Publisher site
See Article on Publisher Site

Abstract

The Apollonian metric aD of a domain \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$D\subset {\overline R}^{n}$\end{document} is rarely conformal. In fact, if it is conformal at one point then D is, up to a Möbius transformation, a complement of a convex body of constant width and if it is conformal at two points then D is a ball. We consider a quantity that measures the deviation of aD from being conformal. This quantity is essential in comparing the Apollonian metric to hyperbolic and quasihyperbolic metrics. We show that this quantity is invariant under Möbius transformations and compute it for some standard domains. We then use it to obtain sharp estimates between any two of the Apollonian, hyperbolic and quasihyperbolic metrics on such domains.

Journal

Computational Methods and Function TheorySpringer Journals

Published: Mar 1, 2004

Keywords: Apollonian metric; hyperbolic metric; quasihyperbolic metric; Möbius maps; 30F45; 30C65

References