On the mean radius of quasiconformal mappings

TL;DR

This paper introduces a new class of quasiconformal mappings called BIP maps, analyzes their mean radius growth, and demonstrates their asymptotic quasiconformality through the properties of their Zorich transform.

Contribution

The paper defines BIP maps with bounded derivative properties and proves their mean radius function's logarithm is bi-Lipschitz, advancing understanding of quasiconformal map behavior.

Findings

BIP maps form a new subclass of quasiconformal mappings.

The logarithmic mean radius function for BIP maps is bi-Lipschitz.

Asymptotic representations of BIP maps are quasiconformal via Zorich transform analysis.

Abstract

We study the mean radius growth function for quasiconformal mappings. We give a new sub-class of quasiconformal mappings in $\mathbb{R}^n$, for $n\geq 2$, called bounded integrable parameterization mappings, or BIP maps for short. These have the property that the restriction of the Zorich transform to each slice has uniformly bounded derivative in $L^{n/(n-1)}$. For BIP maps, the logarithmic transform of the mean radius function is bi-Lipschitz. We then apply our result to BIP maps with simple infinitesimal spaces to show that the asymptotic representation is indeed quasiconformal by showing that its Zorich transform is a bi-Lipschitz map.