Infinitesimally Small Spheres and Conformally Invariant Metrics

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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).

Original languageEnglish
JournalJournal d'Analyse Mathematique
StateAccepted/In press - 2021


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