Dimension of quasicircles

TL;DR

This paper introduces canonical antisymmetric quasiconformal maps that minimize quasiconformality and proves Astala's conjecture that the Hausdorff dimension of a k-quasicircle is at most 1+k^2, advancing geometric function theory.

Contribution

It establishes the existence of canonical maps minimizing quasiconformality and proves a key conjecture relating quasicircle dimensions to quasiconformality constants.

Findings

Proof of Astala's conjecture on quasicircle dimensions

Introduction of canonical antisymmetric quasiconformal maps

Maximal Hausdorff dimension bound for k-quasicircles

Abstract

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 $1+k^2$.