Mathematics > Complex Variables
[Submitted on 11 Dec 2018]
Title:Infinitesimally small spheres and conformally invariant metrics
View PDFAbstract: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 $(D,\mu_D)$.
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