Hyperbolic Lambert Quadrilaterals and Quasiconformal Mappings

TL;DR

This paper establishes precise bounds for hyperbolic distances in Lambert quadrilaterals within the unit disk and examines their behavior under quasiconformal mappings, providing sharp results in both contexts.

Contribution

It introduces sharp bounds for hyperbolic distances in Lambert quadrilaterals and analyzes their images under quasiconformal mappings, extending understanding of geometric distortions.

Findings

Sharp bounds for hyperbolic distances in Lambert quadrilaterals

Results on the images of quadrilaterals under quasiconformal maps

Optimal bounds demonstrating the extremal cases

Abstract

We prove sharp bounds for the product and the sum of two hyperbolic distances between the opposite sides of hyperbolic Lambert quadrilaterals in the unit disk. Furthermore, we study the images of Lambert quadrilaterals under quasiconformal mappings from the unit disk onto itself and obtain sharp results in this case, too.