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Conformally invariant complete metrics

Part of: Geometric function theory

Published online by Cambridge University Press:  30 May 2022

TOSHIYUKI SUGAWA ,
MATTI VUORINEN  and
TANRAN ZHANG
TOSHIYUKI SUGAWA
Affiliation:
Graduate School of Information Sciences, Tohoku University, Aoba-ku, Sendai 980-8579, Japan. e-mail: [email protected]
MATTI VUORINEN
Affiliation:
Department of Mathematics and Statistics, University of Turku, FI-20014 Turku, Finland. e-mail: [email protected]
TANRAN ZHANG
Affiliation:
Department of Mathematics, Soochow University, No.1 Shizi Street, Suzhou 215006, China. e-mail: [email protected]
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Abstract

For a domain G in the one-point compactification $\overline{\mathbb{R}}^n = {\mathbb{R}}^n \cup \{ \infty\}$ of ${\mathbb{R}}^n, n \geqslant 2$ , we characterise the completeness of the modulus metric $\mu_G$ in terms of a potential-theoretic thickness condition of $\partial G\,,$ Martio’s M-condition [ 35 ]. Next, we prove that $\partial G$ is uniformly perfect if and only if $\mu_G$ admits a minorant in terms of a Möbius invariant metric. Several applications to quasiconformal maps are given.

MSC classification

Primary: 30C35: General theory of conformal mappings
Secondary: 30C55: General theory of univalent and multivalent functions
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 174 , Issue 2 , March 2023 , pp. 273 - 300
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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Footnotes

†

The authors were supported in part by JSPS KAKENHI Grant Number JP17H02847 and NSF of the Higher Education Institutions of Jiangsu Province, China, Grant Number 17KJB110015, and NSFC Grant Number 12001391.

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