Mathematics Ph.D. Dissertations

Rigidity of Quasiconformal Maps on Carnot Groups

Date of Award

2017

Document Type

Dissertation

Degree Name

Doctor of Philosophy (Ph.D.)

Department

Mathematics

First Advisor

Xiangdong Xie (Advisor)

Second Advisor

Alexander Tarnovsky (Other)

Third Advisor

Mihai Staic (Committee Member)

Fourth Advisor

Juan Bes (Committee Member)

Abstract

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.

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