# Quasiconformal parametrization of metric surfaces with small dilatation

## TL;DR

This paper proves a conjecture relating quasiconformal parametrizations of metric surfaces to a specific modulus inequality, establishing the optimal bounds and advancing understanding of geometric function theory in metric spaces.

## Contribution

It confirms Rajala's conjecture by showing the existence of a quasiconformal map with optimal modulus bounds for metric surfaces admitting such parametrizations.

## Key findings

- Established the optimal modulus inequality bounds for quasiconformal maps.
- Proved the conjecture that such maps satisfy specific inequalities related to curve families.
- Connected the inequality to a geometric property of planar convex bodies under linear transformations.

## 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 $f: X \rightarrow \Omega$ satisfying the modulus inequality $2\pi^{-1}\text{Mod }\Gamma \leq \text{Mod }f\Gamma \leq 4\pi^{-1}\text{Mod }\Gamma$ 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.