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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Quasisymmetry and rectifiability of quasispheres


Authors: Matthew Badger, James T. Gill, Steffen Rohde and Tatiana Toro
Journal: Trans. Amer. Math. Soc. 366 (2014), 1413-1431
MSC (2010): Primary 30C65; Secondary 28A75, 30C62
Published electronically: September 12, 2013
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Abstract: We obtain Dini conditions that guarantee that an asymptotically conformal quasisphere is rectifiable. In particular, we show that for any $ \epsilon >0$ integrability of $ ( {\rm ess}\sup _{1-t<\vert x\vert<1+t} K_f(x)-1 )^{2-\epsilon } dt/t$ implies that the image of the unit sphere under a global quasiconformal homeomorphism $ f$ is rectifiable. We also establish estimates for the weak quasisymmetry constant of a global $ K$-quasiconformal map in neighborhoods with maximal dilatation close to 1.


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Additional Information

Matthew Badger
Affiliation: Department of Mathematics, Stony Brook University, Stony Brook, New York 11794-3651
Email: badger@math.sunysb.edu

James T. Gill
Affiliation: Department of Mathematics and Computer Science, Saint Louis University, St. Louis, Missouri 63103
Email: jgill5@slu.edu

Steffen Rohde
Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195-4350
Email: rohde@math.washington.edu

Tatiana Toro
Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195-4350
Email: toro@math.washington.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05926-0
PII: S 0002-9947(2013)05926-0
Keywords: Quasisymmetry, quasisphere, asymptotically conformal, rectifiable, Hausdorff measure, Reifenberg flat, linear approximation property, Jones $\beta$-number, modulus
Received by editor(s): January 18, 2012
Published electronically: September 12, 2013
Additional Notes: The first author was partially supported by NSF grant #0838212
The second author was partially supported by NSF grant #1004721
The third author was partially supported by NSF grant #0800968
The fourth author was partially supported by NSF grant #0856687
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.