Quasisymmetry and rectifiability of quasispheres
  We obtain Dini conditions with "exponent 2" that guarantee that an asymptotically conformal quasisphere is rectifiable. In particular, we show that for any e>0 integrability of   (esssup_{1-t < |x| < 1+t} K_f(x)-1)^{2-e} 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. 
Matthew Badger, James T. Gill, Steffen Rohde, Tatiana Toro
math.CA math.CV math.MG