Intrinsic metrics under conformal and quasiregular mappings

TL;DR

This paper investigates how six intrinsic metrics distort under conformal and quasiregular mappings in simple domains, providing improved inequalities and new sharp Schwarz lemmas for these transformations.

Contribution

It introduces refined bounds for metric distortion, improves existing inequalities, and develops sharp Schwarz lemmas for quasiregular and quasiconformal mappings.

Findings

Improved inequalities between hyperbolic and intrinsic metrics.

Bounded conformal distortion of the triangular ratio metric.

New sharp Schwarz lemmas for quasiregular mappings.

Abstract

The distortion of six different intrinsic metrics and quasi-metrics under conformal and quasiregular mappings is studied in a few simple domains $G\subsetneq\mathbb{R}^n$. The already known inequalities between the hyperbolic metric and these intrinsic metrics for points $x,y$ in the unit ball $\mathbb{B}^n$ are improved by limiting the absolute values of the points $x,y$ and the new results are then used to study the conformal distortion of the intrinsic metrics. For the triangular ratio metric between two points $x,y\in\mathbb{B}^n$, the conformal distortion is bounded in terms of the hyperbolic midpoint and the hyperbolic distance of $x,y$. Furthermore, quasiregular and quasiconformal mappings are studied, and new sharp versions of the Schwarz lemma are introduced.