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Title:  Quasiconformal mappings on planar surfaces 
Author(s):  Ackermann, Colleen Teresa 
Director of Research:  Hinkkanen, Aimo; Tyson, Jeremy 
Doctoral Committee Chair(s):  Wu, JangMei 
Doctoral Committee Member(s):  Nikolaev, Igor 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  quasiconformal mapping
Grushin plane 
Abstract:  This thesis discusses three different projects concerning quasiconformal mappings on planar surfaces. In the first two projects we show that a priori weaker conditions still suffice to prove quasiconformality. The geometric definition states that an orientationpreserving homeomorphism f:U → f(U) is quasiconformal if there exists K ≥ 1 such that for all Q, Q ⊂ U the ratio of the modulus of f(Q) to the modulus of Q is bounded above by K. We show that for the subclass of homeomorphisms that preserves the set of vertical lines, it suffices to just consider squares with sides parallel to the coordinate axes and at fortyfive degree angles to the coordinate axes. Another more recent sufficient condition for quasiconformality discovered by Hubbard in 2006, requires that the skews of triangles be only distorted by a bounded amount. Haïssinsky, Hinkkanen and I proved that if there exists a constant K such that (f(T)) ≤ K for all equilateral triangles T, then f is quasiconformal. Furthermore, this condition is also sufficient for mappings between finitedimensional Hilbert spaces. In the last project we study quasiconformal mappings on a generalized class of Grushin planes. We define quasisymmetries between these Grushin planes and the complex plane, and use them to find a Grushin Beltrami equation and state an analytic definition of quasisymmetry on the Grushin plane. Finally we look at a previously discovered class of conformal mappings on the Grushin plane, and show that these conformal mappings agree with our analytic definition of quasisymmetry in the conformal case. 
Issue Date:  20160422 
Type:  Thesis 
URI:  http://hdl.handle.net/2142/90613 
Rights Information:  Copyright 2016 Colleen Ackermann 
Date Available in IDEALS:  20160707 
Date Deposited:  201605 
This item appears in the following Collection(s)

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois