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https://hdl.handle.net/2142/72527
Description
Title
Approximable Quasidisks
Author(s)
Koh, Ngin-Tee
Issue Date
2009
Doctoral Committee Chair(s)
Miles, Joseph
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Abstract
Here we provide a simple outline of the contents of this dissertation.
Chapter 1 gives a brief account of some developments in the theory of quasiconformal mappings as well as its important role in complex analysis and other branches of mathematics.
In Chapter 2, we introduce some basic concepts about the theory of quasiconformal mappings.
Chapter 3 lists some distortion theorems. Other estimates are derived as needed. Finally we apply these results to construct a quasiconformal reflection with bounded partial derivatives near the boundary.
Chapter 4 provides univalence and quasiconformal extension criteria for an arbitrary quasidisk.
In Chapter 5, we prove that a quasidisk with an analytic boundary is approximable. This technical undertaking involves several lemmas. The main idea is to construct a family of quasiconformal reflections whose partial derivatives converge uniformly near the boundary of the quasidisk. Our boundary assumptions, which make this possible, also allow for a choice of reflections which are anti-conformal in a neighborhood of the boundary. Without this last feature, which is crucial to establishing Proposition 5.7, it is unlikely that 2.2 in Definition 2.23 would be satisfied.
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