Isometries for the modulus metric in higher dimensions are conformal mappings

Zhang, Xiaohui

doi:10.1007/s11425-018-1670-6

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Application To Quasimeromorphic Maps1

Introduction1

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2022

2024

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Cited by 2 publications

(2 citation statements)

References 23 publications

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“…Very recently this result was strengthened by Pouliasis and Yu. Solynin [44] and independently by Zhang [56]: μ-isometries are conformal in all dimensions n 2 .…”

Section: Application To Quasimeromorphic Mapsmentioning

confidence: 93%

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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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“…Very recently this result was strengthened by Pouliasis and Yu. Solynin [44] and independently by Zhang [56]: μ-isometries are conformal in all dimensions n 2 .…”

Section: Application To Quasimeromorphic Mapsmentioning

confidence: 93%

“…THEOREM A ( [8,44,56]). A homeomorphism f : G → G , where G and G are domains in R n , n 2, is an isometry between (G, μ G ) and (G , μ G ) if and only if f is conformal.…”

Section: Introductionmentioning

confidence: 99%

Conformally invariant complete metrics

Sugawa¹,

Вуоринен²,

Zhang³

2022

Math. Proc. Camb. Phil. Soc.

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Conformally Invariant Metrics and Lack of Hölder Continuity

Kargar,

Rainio

2024

Bull. Malays. Math. Sci. Soc.

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The modulus metric between two points in a subdomain of $$\mathbb {R}^n, n\ge 2,$$ R n , n ≥ 2 , is defined in terms of moduli of curve families joining the boundary of the domain with a continuum connecting the two points. This metric is one of the conformally invariant hyperbolic-type metrics that have become a standard tool in geometric function theory. We prove that the modulus metric is not Hölder continuous with respect to the hyperbolic metric.

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Isometries for the modulus metric in higher dimensions are conformal mappings

Cited by 2 publications

References 23 publications

Conformally invariant complete metrics

Conformally invariant complete metrics

Conformally Invariant Metrics and Lack of Hölder Continuity

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