# Duality of moduli and quasiconformal mappings in metric spaces

## TL;DR

This paper establishes a duality relation for moduli of curve and surface families in metric spaces and uses it to characterize quasiconformal mappings by their effect on these moduli.

## Contribution

It introduces a duality relation for moduli in metric spaces and characterizes quasiconformal mappings through their quasi-preservation of surface family moduli.

## Key findings

- Proves a duality relation for moduli of curves and surfaces in metric spaces.
- Characterizes quasiconformal mappings via modulus preservation.
- Applicable in metric spaces with doubling measure and Poincaré inequality.

## Abstract

We prove a duality relation for the moduli of the family of curves connecting two sets and the family of surfaces separating the sets, in the setting of a complete metric space equipped with a doubling measure and supporting a Poincar\'e inequality. Then we apply this to show that quasiconformal mappings can be characterized by the fact that they quasi-preserve the modulus of certain families of surfaces.