Conformally invariant complete metrics

Abstract

For a domain G in the one-point compactification $\overline{\mathbb{R}}^n = {\mathbb{R}}^n \cup \{ \infty\}$ of ${\mathbb{R}}^n, n \geqslant 2$ , we characterise the completeness of the modulus metric $\mu_G$ in terms of a potential-theoretic thickness condition of $\partial G\,,$ Martio’s M-condition [35]. 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.

On Harnack inequality and harmonic Schwarz lemma

“…It is important to clarify that the theorem presented herein diverges from Theorem 5.2 in [23]. Specifically, our theorem assumes that f G is a A-uniform domain with a connected boundary, while Sugawa et al [23] regarded ∂ f G as uniformly perfect. The connectedness of ∂ f G is decisive in the following theorem, as demonstrated in Remark 3.5.…”

“…The connectedness of ∂ f G is decisive in the following theorem, as demonstrated in Remark 3.5. Conversely, in the proof of Theorem 5.2, Sugawa et al [23] employ the definition of the modulus metric μ G to establish an upper bound, whereas we utilize a general upper bound derived from Lemma 10.6(2) of [10] for y ∈ B n (x, sd G (x)). Moreover, the constant C(s) obtained here is more generality than the constant obtained by Sugawa et al in [23].…”