### Abstract

For a domain D in R^{¯n}, the modulus metric is defined by μ_{D}(x, y) = inf_{γ} 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 language | English |
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Pages (from-to) | 2735-2752 |

Number of pages | 18 |

Journal | Transactions of the American Mathematical Society |

Volume | 372 |

Issue number | 4 |

DOIs | |

State | Published - Aug 15 2019 |

### Keywords

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

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## Cite this

Betsakos, D., & Pouliasis, S. (2019). Isometries for the modulus metric are quasiconformal mappings.

*Transactions of the American Mathematical Society*,*372*(4), 2735-2752. https://doi.org/10.1090/tran/7712