Conformally Invariant Metrics and Lack of Hölder Continuity

The modulus metric between two points in a subdomain of Rⁿ, n≥ 2, R n , n ≥ 2 , is defined in terms of moduli of curve families joining the boundary of the domain with a continuum connecting the two points. This metric is one of the conformally invariant hyperbolic-type metrics that have become a standard tool in geometric function theory. We prove that the modulus metric is not Hölder continuous with respect to the hyperbolic metric.