Conformally invariant complete metrics

Abstract

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 Möbius invariant metric. Several applications to quasiconformal maps are given.

28 pages

Cited by in corpus (1)

• On Harnack inequality and harmonic Schwarz lemma