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
Published inAnalysis and Geometry in Metric Spaces
© 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.
PublisherWalter de Gruyter GmbH
Publication in research information system
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Related funder(s)Academy of Finland
Funding program(s)Academy Project, AoF
Additional information about fundingThe author was supported by the Academy of Finland, project number 308659 and by the Vilho, Yrjö and Kalle Väisälä Foundation.
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