## Quasisymmetry and rectifiability of quasispheres

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- by Matthew Badger, James T. Gill, Steffen Rohde and Tatiana Toro PDF
- Trans. Amer. Math. Soc.
**366**(2014), 1413-1431 Request permission

## 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 $( \textrm {ess}\sup _{1-t<|x|<1+t} K_f(x)-1 )^{2-\epsilon } dt/t$ 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.## References

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## Additional Information

**Matthew Badger**- Affiliation: Department of Mathematics, Stony Brook University, Stony Brook, New York 11794-3651
- MR Author ID: 962755
- 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
- MR Author ID: 363909
- Email: toro@math.washington.edu
- 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 - © Copyright 2013
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**366**(2014), 1413-1431 - MSC (2010): Primary 30C65; Secondary 28A75, 30C62
- DOI: https://doi.org/10.1090/S0002-9947-2013-05926-0
- MathSciNet review: 3145736