# Canonical parametrizations of metric discs

## TL;DR

This paper leverages recent advances in the theory of area and energy minimizing discs in metric spaces to establish canonical parametrizations of metric surfaces, providing new proofs and generalizations of key theorems.

## Contribution

It introduces a new approach for parametrizing metric surfaces using minimizers, simplifying proofs of existing theorems and extending their applicability.

## Key findings

- Provides a new proof of Bonk and Kleiner's theorem on quasisymmetric parametrizations.
- Establishes canonical parametrizations for metric surfaces.
- Discusses applications to the geometry of metric surfaces.

## Abstract

We use the recently established existence and regularity of area and energy minimizing discs in metric spaces to obtain canonical parametrizations of metric surfaces. Our approach yields a new and conceptually simple proof of a well-known theorem of Bonk and Kleiner on the existence of quasisymmetric parametrizations of linearly locally connected, Ahlfors $2$-regular metric $2$-spheres. Generalizations and applications to the geometry of such surfaces are described.