Stretch maps on the affine-additive group

TL;DR

This paper introduces linear and radial stretch maps within the affine-additive group, proving they minimize mean quasiconformal distortion using modulus of curve families and minimal stretching properties.

Contribution

It defines new stretch maps in the affine-additive group and proves their optimality in minimizing quasiconformal distortion.

Findings

Stretch maps are minimizers of the mean quasiconformal distortion functional.

The proof employs modulus of curve families and the minimal stretching property.

The approach is compatible with the geometric settings of the maps.

Abstract

We define linear and radial stretch maps in the affine-additive group, and prove that they are minimizers of the mean quasiconformal distortion functional. For the proofs we use a method based on the notion of modulus of a curve family and the minimal stretching property (MSP) of the afore-mentioned maps. MSP relies on certain given curve families compatible with the respective geometric settings of the strech maps.