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Logarithmic Potentials and Quasiconformal Flows on the Heisenberg Group

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posted on 19.10.2016, 00:00 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

19/10/2016

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