Quasisymmetry and rectifiability of quasispheres

TL;DR

This paper establishes Dini conditions with exponent 2 that ensure asymptotically conformal quasispheres are rectifiable, linking integrability conditions of the dilatation to geometric rectifiability.

Contribution

It introduces new Dini conditions with exponent 2 that guarantee rectifiability of quasispheres under asymptotic conformality, and provides estimates for weak quasisymmetry constants.

Findings

Integrability of (esssup_{1-t < |x| < 1+t} K_f(x)-1)^{2-e} dt/t implies rectifiability.

Estimates for weak quasisymmetry constants near maximal dilatation close to 1.

Conditions under which asymptotically conformal quasispheres are rectifiable.

Abstract

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.