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Quasisymmetric spheres over Jordan domains


Authors: Vyron Vellis and Jang-Mei Wu
Journal: Trans. Amer. Math. Soc. 368 (2016), 5727-5751
MSC (2010): Primary 30C65; Secondary 30C62
DOI: https://doi.org/10.1090/tran/6634
Published electronically: October 20, 2015
MathSciNet review: 3458397
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Abstract: Let $ \Omega $ be a planar Jordan domain. We consider double-dome-like surfaces $ \Sigma $ defined by graphs of functions of $ \operatorname {dist}(\cdot ,\partial \Omega )$ over $ \Omega $. The goal is to find the right conditions on the geometry of the base $ \Omega $ and the growth of the height so that $ \Sigma $ is a quasisphere or is quasisymmetric to $ \mathbb{S}^2$. An internal uniform chord-arc condition on the constant distance sets to $ \partial \Omega $, coupled with a mild growth condition on the height, gives a close-to-sharp answer. Our method also produces new examples of quasispheres in $ \mathbb{R}^n$, for any $ n\ge 3$.


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Additional Information

Vyron Vellis
Affiliation: Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, Illinois 61820
Address at time of publication: Department of Mathematics and Statistics, P. O. Box 35, University of Jyväskylä, FI-40014, Finland
Email: vellis1@illinois.edu, vyron.v.vellis@jyu.fi

Jang-Mei Wu
Affiliation: Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, Illinois 61801
Email: jmwu@illinois.edu

DOI: https://doi.org/10.1090/tran/6634
Keywords: Quasispheres, quasisymmetric spheres, double-dome-like surfaces, level chord-arc property
Received by editor(s): July 17, 2014
Published electronically: October 20, 2015
Additional Notes: This research was supported in part by the NSF grant DMS-1001669.
Article copyright: © Copyright 2015 American Mathematical Society

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