We introduce canonical antisymmetric quasiconformal maps, which minimize the quasiconformality constant among maps sending the unit circle to a given quasicircle. As an application we prove Astala’s conjecture that the Hausdorff dimension of a k-quasicircle is at most 1+k2.
This research was partially supported by the NSF grant DMS-9706875 as well as by the EU RTN CODY and Swiss NSF.
"Dimension of quasicircles." Acta Math. 205 (1) 189 - 197, 2010. https://doi.org/10.1007/s11511-010-0053-8