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Logarithmic Potentials and Quasiconformal Flows on the Heisenberg Group

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posted on 2016-10-19, 00:00 authored by Alex D. Austin
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.

History

Advisor

Dumas, David

Department

Mathematics, Statistics, and Computer Science

Degree Grantor

University of Illinois at Chicago

Degree Level

  • Doctoral

Committee Member

Tyson, Jeremy T. Baudoin, Fabrice Csornyei, Marianna Whyte, Kevin

Submitted date

2016-08

Language

  • en

Issue date

2016-10-19

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