Quasisymmetric spheres constructed over quasidisks

摘要

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.