Modulus Characterizations of Bilipschitz Mappings

Qingshan Zhou; Zhiqiang Yang; Antti Rasila; Yuehui He

doi:10.1007/s12220-024-01635-4

Modulus Characterizations of Bilipschitz Mappings

Qingshan Zhou, Zhiqiang Yang, Antti Rasila, Yuehui He^*

^*Corresponding author for this work

Mathematics with Computer Science

Research output: Contribution to journal › Article › peer-review

Abstract

In this paper, we establish six necessary and sufficient conditions for a homeomorphism of Rⁿ onto itself to be strongly quasisymmetric. These conditions are quantitative in terms of conformal moduli of disjoint continua as well as the geometric modulus, which was recently introduced by Tukia and Väisälä. Note that all of them are equivalent to bilipschitz continuity with parameters depending also on two fixed points. As an application, we obtain several quantitative characterizations for a homeomorphism of the Riemann sphere R¯ⁿ onto itself to be strongly quasimöbius.

Original language	English
Article number	155
Journal	Journal of Geometric Analysis
Volume	34
Issue number	6
DOIs	https://doi.org/10.1007/s12220-024-01635-4
State	Published - Jun 2024

Keywords

Bilipschitz mapping
Geometric modulus
Modulus
Ring
Strongly quasimöbius mapping
Strongly quasisymmetric mapping

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10.1007/s12220-024-01635-4

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title = "Modulus Characterizations of Bilipschitz Mappings",

abstract = "In this paper, we establish six necessary and sufficient conditions for a homeomorphism of Rn onto itself to be strongly quasisymmetric. These conditions are quantitative in terms of conformal moduli of disjoint continua as well as the geometric modulus, which was recently introduced by Tukia and V{\"a}is{\"a}l{\"a}. Note that all of them are equivalent to bilipschitz continuity with parameters depending also on two fixed points. As an application, we obtain several quantitative characterizations for a homeomorphism of the Riemann sphere R¯n onto itself to be strongly quasim{\"o}bius.",

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T1 - Modulus Characterizations of Bilipschitz Mappings

AU - Zhou, Qingshan

AU - Yang, Zhiqiang

AU - Rasila, Antti

AU - He, Yuehui

N1 - Publisher Copyright: © Mathematica Josephina, Inc. 2024.

PY - 2024/6

Y1 - 2024/6

N2 - In this paper, we establish six necessary and sufficient conditions for a homeomorphism of Rn onto itself to be strongly quasisymmetric. These conditions are quantitative in terms of conformal moduli of disjoint continua as well as the geometric modulus, which was recently introduced by Tukia and Väisälä. Note that all of them are equivalent to bilipschitz continuity with parameters depending also on two fixed points. As an application, we obtain several quantitative characterizations for a homeomorphism of the Riemann sphere R¯n onto itself to be strongly quasimöbius.

AB - In this paper, we establish six necessary and sufficient conditions for a homeomorphism of Rn onto itself to be strongly quasisymmetric. These conditions are quantitative in terms of conformal moduli of disjoint continua as well as the geometric modulus, which was recently introduced by Tukia and Väisälä. Note that all of them are equivalent to bilipschitz continuity with parameters depending also on two fixed points. As an application, we obtain several quantitative characterizations for a homeomorphism of the Riemann sphere R¯n onto itself to be strongly quasimöbius.

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KW - Geometric modulus

KW - Modulus

KW - Ring

KW - Strongly quasimöbius mapping

KW - Strongly quasisymmetric mapping

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DO - 10.1007/s12220-024-01635-4

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SN - 1050-6926

VL - 34

JO - Journal of Geometric Analysis

JF - Journal of Geometric Analysis

IS - 6

M1 - 155

ER -

Modulus Characterizations of Bilipschitz Mappings

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