# Logarithmic Potentials and Quasiconformal Flows on the Heisenberg Group

## TL;DR

This paper investigates the construction of quasiconformal mappings on the Heisenberg group with extreme local behavior, establishing criteria related to logarithmic potentials and extending the theory of quasiconformal flows.

## Contribution

It introduces new criteria for when logarithmic potentials induce quasiconformal mappings with specific Jacobian behavior, expanding the understanding of quasiconformal flows on the Heisenberg group.

## Key findings

- Existence of quasiconformal mappings with Jacobian comparable to exponential of logarithmic potentials.
- Conditions under which the canonical and weighted metrics are bi-Lipschitz equivalent.
- Extension of quasiconformal flow theory on the Heisenberg group.

## Abstract

Let $\mathbb{H}$ be the sub-Riemannian Heisenberg group. That $\mathbb{H}$ supports a rich family of quasiconformal mappings was demonstrated by Kor\'{a}nyi and Reimann using the so-called flow method. Here we supply further evidence of the flexible nature of this family, constructing quasiconformal mappings with extreme behavior on small sets. More precisely, we establish criteria to determine when a given logarithmic potential $\Lambda$ on $\mathbb{H}$ is such that there exists a quasiconformal mapping of $\mathbb{H}$ with Jacobian comparable to $e^{2\Lambda}$ (so that the Jaobian is zero or infinity at the same points as $e^{2\Lambda}$). When $\Lambda$ is continuous and meets the criteria, we show the canonical (sub-Riemannian) metric $g_0$ and the weighted metric $g = e^\Lambda g_0$ generate bi-Lipschitz equivalent distance functions. These results rest on an extension to the theory of quasiconformal flows on $\mathbb{H}$ and constructions that adapt the iterative method of Bonk, Heinonen, and Saksman.