Abstract
The modulus metric (also called the capacity metric) on a domain D ⊂ ℝn can be defined as μD(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, γD).
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Pouliasis, S., Solynin, A.Y. Infinitesimally small spheres and conformally invariant metrics. JAMA 143, 179–205 (2021). https://doi.org/10.1007/s11854-021-0152-9
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DOI: https://doi.org/10.1007/s11854-021-0152-9