Infinitesimally small spheres and conformally invariant metrics

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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).

Original languageEnglish
Pages (from-to)179-205
Number of pages27
JournalJournal d'Analyse Mathematique
Volume143
Issue number1
DOIs
StatePublished - Jun 2021

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