# Infinitesimally small spheres and conformally invariant metrics

## TL;DR

This paper confirms the conjecture that all isometries of the modulus metric are conformal mappings and explores geometric properties of spheres within this metric space.

## Contribution

It proves that isometries in the modulus metric are conformal and investigates the geometric structure of spheres in this metric space.

## Key findings

- All isometries in the modulus metric are conformal mappings.
- New geometric properties of spheres in the modulus metric space are established.
- Confirmation of a longstanding conjecture from 1991.

## Abstract

The modulus metric (also called the capacity metric) on a domain $D\subset \mathbb{R}^n$ can be defined as $\mu_D(x,y)=\inf\{{\mbox{cap}}\,(D,\gamma)\}$, where ${\mbox{cap}}\,(D,\gamma)$ stands for the capacity of the condenser $(D,\gamma)$ and the infimum is taken over all continua $\gamma\subset D$ containing the points $x$ and $y$. It was conjectured by J. Ferrand, G. Martin and M. Vuorinen in 1991 that every isometry in the modulus metric is a conformal mapping. In this note, we confirm this conjecture and prove new geometric properties of surfaces that are spheres in the metric space $(D,\mu_D)$.