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Title:Quasisymmetric spheres constructed over quasidisks
Author(s):Vellis, Vyron
Director of Research:Wu, Jang-Mei
Doctoral Committee Chair(s):Tyson, Jeremy T.
Doctoral Committee Member(s):Wu, Jang-Mei; Merenkov, Sergiy A.; Hinkkanen, Aimo
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
quasisymmetric spheres
distance function
Level sets
chordal property
Abstract:In this thesis we construct concrete examples of quasispheres and quasisymmetric spheres. These examples are double-dome type surfaces in the 3-dimensional 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 2-dimensional 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 chord-arc 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 bi-Lipschitz or quasisymmetric equivalent to the 2-dimensional unit sphere.
Issue Date:2014-09-16
Rights Information:Copyright 2014 Vyron Vellis
Date Available in IDEALS:2014-09-16
Date Deposited:2014-08

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