Conformally invariant complete metrics

For a domain G in the one-point compactification ¯Rn=Rn∪{∞} of Rn,n⩾2 , we characterise the completeness of the modulus metric μG in terms of a potential-theoretic thickness condition of ∂G, Martio’s M-condition [35]. Next, we prove that ∂G is uniformly perfect if and only if μG admits a minorant in terms of a Möbius invariant metric. Several applications to quasiconformal maps are given.