Quasiconformal parametrization of metric surfaces with small dilatation

Abstract

We verify a conjecture of Rajala: if $(X,d)$ is a metric surface of locally finite Hausdorff 2-measure admitting some (geometrically) quasiconformal parametrization by a simply connected domain $\Omega \subset {\mathbb{R}}^2$, then there exists a quasiconformal mapping $f:X\to \Omega$ satisfying the modulus inequality $2{\pi }^{-1}\text{Mod }\Gamma \le \text{Mod }f\Gamma \le 4{\pi }^{-1}\text{Mod }\Gamma$ for all curve families $\Gamma$ in $X$. This inequality is the best possible. Our proof is based on an inequality for the area of a planar convex body under a linear transformation which attains its Banach-Mazur distance to the Euclidean unit ball.

arXiv comment7 pages, 2 figures, to appear in Indiana Univ. Math. J