Quasisymmetry and rectifiability of quasispheres

Badger, Matthew; Gill, James; Rohde, Steffen; Toro, Tatiana

doi:10.1090/S0002-9947-2013-05926-0

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.

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