Infinitesimally small spheres and conformally invariant metrics

2018/12/11 by Pouliasis, Stamatis, Solynin, Alexander Yu.
#30C65 #30C75 #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.1812.04651

Abstract

The modulus metric (also called the capacity metric) on a domain D⊂ ℝⁿ can be defined as μ_D(x,y)=inf\cap (D,γ)\, where cap (D,γ) stands for the capacity of the condenser (D,γ) and the infimum is taken over all continua γ⊂ 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,μ_D).

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