Quasisymmetric spheres over Jordan domains
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- by Vyron Vellis and Jang-Mei Wu PDF
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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 $n\ge 3$.References
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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
- MR Author ID: 184770
- Email: jmwu@illinois.edu
- 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.
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 5727-5751
- MSC (2010): Primary 30C65; Secondary 30C62
- DOI: https://doi.org/10.1090/tran/6634
- MathSciNet review: 3458397