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Conformally invariant complete metrics

Published online by Cambridge University Press:  30 May 2022

Graduate School of Information Sciences, Tohoku University, Aoba-ku, Sendai 980-8579, Japan. e-mail:
Department of Mathematics and Statistics, University of Turku, FI-20014 Turku, Finland. e-mail:
Department of Mathematics, Soochow University, No.1 Shizi Street, Suzhou 215006, China. e-mail:


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.

Research Article
© The Author(s), 2022. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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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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