Intrinsic metrics in ring domains

TL;DR

This paper investigates three hyperbolic-type metrics within annular ring domains, establishing sharp inequalities and a new M"obius-invariant lower bound for conformal capacity using Euclidean midpoint rotation.

Contribution

It introduces the Euclidean midpoint rotation method to derive bounds for hyperbolic metrics and proves a novel M"obius-invariant lower bound for conformal capacity.

Findings

Sharp inequalities for the metrics are established.

The Euclidean midpoint rotation effectively bounds the metrics.

A new M"obius-invariant lower bound for conformal capacity is proved.

Abstract

Three hyperbolic type metrics including the triangular ratio metric, the $j^*$-metric and the M\"obius metric are studied in an annular ring. The Euclidean midpoint rotation is introduced as a method to create upper and lower bounds for these metrics, and their sharp inequalities are found. A new M\"obius-invariant lower bound is proved for the conformal capacity of a general ring domain by using a symmetric quantity defined with the M\"obius metric.