Logarithmic Potentials and Quasiconformal Flows on the Heisenberg Group

In a first contribution to the quasiconformal Jacobian problem outside the Euclidean setting, we show that if the total variation of the measure associated to a quasi-logarithmic potential on the Heisenberg group is sufficiently small, then there is a quasiconformal mapping of the Heisenberg group, whose Jacobian is almost everywhere comparable to the exponential of twice the quasi-logarithmic potential. This is analogous to work of Bonk, Heinonen and Saksman in the Euclidean spaces. As a precursor we extend the flow method of Koranyi and Reimann for generating quasiconformal mappings of the Heisenberg group, and as an application we show that a family of metric spaces conformally equivalent to the sub-Riemannian Heisenberg group are in fact bi-Lipschitz equivalent to the Heisenberg group.