Rigidity of Quasiconformal Maps on Carnot Groups

Rigidity of Quasiconformal Maps on Carnot Groups

Bibliographic Details
Main Author: Medwid, Mark Edward
Language:English
Published: Bowling Green State University / OhioLINK 2017
Subjects:
Mathematics
quasiconformal mappings
rigidity
Carnot groups
Lie groups
Lie algebras
quasisymmetric mappings
analysis on metric spaces
Online Access:http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1497620176117104
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spelling ndltd-OhioLink-oai-etd.ohiolink.edu-bgsu14976201761171042021-08-03T07:02:54Z Rigidity of Quasiconformal Maps on Carnot Groups Medwid, Mark Edward Mathematics quasiconformal mappings rigidity Carnot groups Lie groups Lie algebras quasisymmetric mappings analysis on metric spaces Quasiconformal mappings were first utilized by Grotzsch in the 1920’s and then later named by Ahlfors in the 1930’s. The conformal mappings one studies in complex analysis are locally angle-preserving: they map infinitesimal balls to infinitesimal balls. Quasiconformal mappings, on the other hand, map infinitesimal balls to infinitesimal ellipsoids of a uniformly bounded eccentricity. The theory of quasiconformal mappings is well-developed and studied. For example, quasiconformal mappings on Euclidean space are almost-everywhere differentiable. A result due to Pansu in 1989 illustrated that quasiconformal mappings on Carnot groups are almost-everywhere (Pansu) differentiable, as well. It is easy to show that a biLipschitz map is quasiconformal but the converse does not hold, in general. There are many instances, however, where globally defined quasiconformal mappings on Carnot groups are biLipschitz. In this paper we show that, under certain conditions, a quasiconformal mapping defined on an open subset of a Carnot group is locally biLipschitz. This result is motivated by rigidity results in geometry (for example, the theorem by Mostow in 1968). Along the way we develop background material on geometric group theory and show its connection to quasiconformal mappings. 2017-08-02 English text Bowling Green State University / OhioLINK http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1497620176117104 http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1497620176117104 unrestricted This thesis or dissertation is protected by copyright: all rights reserved. It may not be copied or redistributed beyond the terms of applicable copyright laws.
collection NDLTD
language English
sources NDLTD
topic Mathematics
quasiconformal mappings
rigidity
Carnot groups
Lie groups
Lie algebras
quasisymmetric mappings
analysis on metric spaces
spellingShingle Mathematics
quasiconformal mappings
rigidity
Carnot groups
Lie groups
Lie algebras
quasisymmetric mappings
analysis on metric spaces
Medwid, Mark Edward
Rigidity of Quasiconformal Maps on Carnot Groups
author Medwid, Mark Edward
author_facet Medwid, Mark Edward
author_sort Medwid, Mark Edward
title Rigidity of Quasiconformal Maps on Carnot Groups
title_short Rigidity of Quasiconformal Maps on Carnot Groups
title_full Rigidity of Quasiconformal Maps on Carnot Groups
title_fullStr Rigidity of Quasiconformal Maps on Carnot Groups
title_full_unstemmed Rigidity of Quasiconformal Maps on Carnot Groups
title_sort rigidity of quasiconformal maps on carnot groups
publisher Bowling Green State University / OhioLINK
publishDate 2017
url http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1497620176117104
work_keys_str_mv AT medwidmarkedward rigidityofquasiconformalmapsoncarnotgroups
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