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Title:  Quasisymmetric spheres constructed over quasidisks 
Author(s):  Vellis, Vyron 
Director of Research:  Wu, JangMei 
Doctoral Committee Chair(s):  Tyson, Jeremy T. 
Doctoral Committee Member(s):  Wu, JangMei; Merenkov, Sergiy A.; Hinkkanen, Aimo 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  quasispheres
quasisymmetric spheres distance function Level sets chordal property quasicircles snowflakes 
Abstract:  In this thesis we construct concrete examples of quasispheres and quasisymmetric spheres. These examples are doubledome type surfaces in the 3dimensional Euclidean space over planar Jordan domains. The thesis consists of three parts. Let D be a Jordan domain with boundary C and h a self homeomorphism of the set of non negative real numbers. In the Geometric construction, the surface is the graph of h(dist(x,C)). We examine the properties of the Jordan domains D and of the height functions h ensuring that these surfaces are either quasispheres or quasisymmetric equivalent to the 2dimensional unit sphere. As it turns out, the geometry of the sets of constant distance from C plays a key role in the geometry of these surfaces. The Geometric construction is the motivation of the second part, the study of sets of constant distance from a planar Jordan curve C. We ask what properties of C ensure that these sets are Jordan curves, or uniform quasicircles, or uniform chordarc curves for all sufficiently small distances. Sufficient conditions are given in term of a scaled invariant parameter for measuring the local deviation of subarcs from their chords. The chordal conditions given are sharp. In the third part, we discuss the Analytic construction. In this construction, the level sets of the height of the surface built over a Jordan domain D are the level sets of f for some quasiconformal function f that maps D onto the unit disk. We investigate the properties of f which guarantee that these surfaces are either biLipschitz or quasisymmetric equivalent to the 2dimensional unit sphere. 
Issue Date:  20140916 
URI:  http://hdl.handle.net/2142/50693 
Rights Information:  Copyright 2014 Vyron Vellis 
Date Available in IDEALS:  20140916 
Date Deposited:  201408 
This item appears in the following Collection(s)

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