Conformally invariant complete metrics

Abstract

<p>For a domain <em>G</em> in the one-point compactification ¯R<em>n</em>=R<em>n</em>∪{∞} of R<em>n</em>,<em>n</em>⩾2 , we characterise the completeness of the modulus metric <em>μ</em><em>G</em> in terms of a potential-theoretic thickness condition of ∂<em>G</em>, Martio’s <em>M</em>-condition [<strong><a href="https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/conformally-invariant-complete-metrics/17D6309D53B5787EAF16DFBF5F5011C8#r35">35</a></strong>]. Next, we prove that ∂<em>G</em> is uniformly perfect if and only if <em>μ</em><em>G</em> admits a minorant in terms of a Möbius invariant metric. Several applications to quasiconformal maps are given.</p>