  For a domain $G$ in the one-point compactification $\overline{\mathbb{R}}^n =
\mathbb{R}^n \cup \{ \infty\}$ of $\mathbb{R}^n, n \ge 2$, we characterize the
completeness of the modulus metric $\mu_G$ in terms of a potential-theoretic
thickness condition of $\partial G\,,$ Martio's $M$-condition. Next, we prove
that $\partial G$ is uniformly perfect if and only if $\mu_G$ admits a minorant
in terms of a M\"obius invariant metric. Several applications to quasiconformal
maps are given.
