On the hyperbolic distance of n-times punctured spheres

Sugawa, Toshiyuki; Вуоринен, Матти; Zhang, Tanran

doi:10.1007/s11854-020-0112-9

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Logarithmic Möbius Metric1

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2022

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“…We now prove the first part of Theorem 1•15. By using the results from [50] or [49], we could obtain more explicit estimates for the bound c = c(A). However, for brevity, we shall be content with existence of c > 0 only.…”

Section: Logarithmic Möbius Metricmentioning

confidence: 99%

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Conformally invariant complete metrics

Sugawa¹,

Вуоринен²,

Zhang³

2022

Math. Proc. Camb. Phil. Soc.

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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.

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Section: Logarithmic Möbius Metricmentioning

confidence: 99%

“…It is a challenging task, studied in [49,50], to give concrete bounds for the h G distances in domains G whose boundary consists only of isolated points. Since log (1 + x) is a subadditive function on 0 x < +∞, we can easily see that log (1 + m(x, y)) is a distance function on X whenever m(x, y) is a distance function on X [2, 7•42 (1)].…”

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confidence: 99%