Isometries for the modulus metric are quasiconformal mappings

Stamatis Pouliasis, Dimitrios Betsakos

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6 Scopus citations


For a domain D in R¯n, the modulus metric is defined by μD(x, y) = infγ cap(D, γ), where the infimum is taken over all curves γ in D joining x to y, and “cap” denotes the conformal capacity of the condensers. It has been conjectured by J. Ferrand, G. J. Martin, and M. Vuorinen that isometries in the modulus metric are conformal mappings. We prove the conjecture when n = 2. In higher dimensions, we prove that isometries are quasiconformal mappings.

Original languageEnglish
Pages (from-to)2735-2752
Number of pages18
JournalTrans. Amer. Math. Soc.
Issue number4
StatePublished - Aug 15 2019


  • Condenser capacity
  • Conformal mapping
  • Extremal length
  • Modulus metric
  • Quasiconformal mapping
  • Reduced conformal modulus


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