Conformally invariant complete metrics

Important geometric or analytic properties of domains in the Euclidean space $mathbb{R}^n$ or its one-point compactification (the Mobius space) $overline{mathbb{R}}^n$ $(nge 2)$ are often characterized by comparison inequalities between two intrinsic metrics on a domain. For instance, a proper subdomain $G$ of $mathbb{R}^n$ is {it uniform} if and only if the quasihyperbolic metric $k_G$ is bounded by a constant multiple of the distance-ratio metric $j_G.$ Motivated by this idea we first characterize the completeness of the modulus metric of a proper subdomain $G$ of $overline{mathbb{R}}^n$ in terms of Martios $M$-condition. Next, we prove that if the boundary is uniformly perfect, then the modulus metric is minorized by a constant multiple of a Mobius invariant metric which yields a new characterization of uniform perfectness of the boundary of a domain. Further, in the planar case, we obtain a new characterization of uniform domains. In contrast to the above cases, where the boundary has no isolated points, we study planar domains whose complements are finite sets and establish new upper bounds for the hyperbolic distance between two points in terms of a logarithmic Mobius metric. We apply our results to prove Holder continuity with respect to the Ferrand metric for quasiregular mappings of a domain in the Mobius space into a domain with uniformly perfect boundary.