Isometries for the modulus metric are quasiconformal mappings

Abstract

For a domain D D in R ¯ n \bar {\mathbb R}^n , the modulus metric is defined by μ D ( x , y ) = inf γ cap ( D , γ ) \mu _D(x,y)=\inf _\gamma \textrm {cap}(D,\gamma ) , where the infimum is taken over all curves γ \gamma in D D joining x x to y 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 n=2 . In higher dimensions, we prove that isometries are quasiconformal mappings.