Quasiconformal Jordan Domains
Ikonen, T. (2021). Quasiconformal Jordan Domains. Analysis and Geometry in Metric Spaces, 9(1), 167-185. https://doi.org/10.1515/agms-2020-0127
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Analysis and Geometry in Metric SpacesAuthors
Date
2021Copyright
© 2021 Toni Ikonen, published by De Gruyter.
We extend the classical Carathéodory extension theorem to quasiconformal Jordan domains (Y,dY). We say that a metric space (Y,dY) is a quasiconformal Jordan domain if the completion Y of (Y,dY) has finite Hausdor 2-measure, the boundary ∂Y=Y\Y is homeomorphic to S1, and there exists a homeo-morphism φ:D→(Y,dY) that is quasiconformal in the geometric sense. We show that φ has a continuous, monotone, and surjective extension Φ:D→Y. This result is best possible in this generality. In addition, we find a necessary and suffcient condition for Φ to be a quasiconformal homeomorphism. We provide suffcient conditions for the restriction of Φ to S1 being a quasisymmetry and to ∂Y being bi-Lipschitz equivalent to a quasicircle in the plane.
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Walter de Gruyter GmbHISSN Search the Publication Forum
2299-3274Keywords
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https://converis.jyu.fi/converis/portal/detail/Publication/101837523
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Research Council of FinlandFunding program(s)
Academy Project, AoFAdditional information about funding
The author was supported by the Academy of Finland, project number 308659 and by the Vilho, Yrjö and Kalle Väisälä Foundation.License
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