On isometries of conformally invariant metric

TL;DR

This paper proves that isometries in a conformally invariant metric are quasiconformal in general domains and Möbius transformations in punctured space, resolving a specific conjecture.

Contribution

It establishes the nature of isometries in conformally invariant metrics, confirming they are quasiconformal generally and Möbius in punctured space, thus solving a known conjecture.

Findings

Isometries are quasiconformal in general domains.

In punctured space, isometries are Möbius transformations.

Conjecture by Ferrand, Martin, and Vuorinen is confirmed in this case.

Abstract

We prove that isometries in a conformally invariant metric of a general domain are quasiconformal. In the particular case of the punctured space, we prove that isometries in this metric are Mobius, thus resolving a conjecture of Ferrand, Martin and Vuorinen [FMV, p. 200] in this particular case.