Infinitesimally small spheres and conformally invariant metrics

Abstract

The modulus metric (also called the capacity metric) on a domain $D\subset {\mathbb{R}}^n$ can be defined as $μ_D(x,y)=\inf\{\mbox{cap}\,(D,γ)\}$, where ${\mbox{cap}}\,(D,γ)$ 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 $(D,{\mu }_D)$.