Conformally invariant complete metrics

TL;DR

This paper characterizes when the modulus metric is complete in domains of Euclidean space using potential theory and explores conditions under which the boundary's geometric properties influence metric invariance, with applications to quasiconformal maps.

Contribution

It provides a new characterization of metric completeness via potential-theoretic boundary conditions and links boundary geometric properties to metric invariance under Möbius transformations.

Findings

Completeness of the modulus metric is characterized by Martio's M-condition.

Boundary uniform perfectness is equivalent to the existence of a Möbius invariant minorant.

Applications to quasiconformal maps demonstrate the theoretical results.

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 M\"obius invariant metric. Several applications to quasiconformal maps are given.