The Beurling-Ahlfors extension and Conformal welding

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Title: The Beurling-Ahlfors extension and Conformal welding
Author: Karlsson, Ville
Contributor: University of Helsinki, Faculty of Science
Publisher: Helsingin yliopisto
Date: 2020
Language: eng
Thesis level: master's thesis
Discipline: Matematiikka
Abstract: This thesis introduces the theory of conformal welding and shows that every quasisymmetric function is a welding homeomorphism. Conformal weldings appear naturally in Teichmüller theory when we consider the conditions under which two Riemann surfaces can be joined together. In the second chapter we investigate the properties of quasisymmetric and quasiconformal functions and derive various inequalities for them. Later in the chapter we define the so-called Beurlin-Ahlfors extension to quasisymmetric functions and show that this function is in fact quasiconformal. In the third chapter, we begin by introducing the fundamentals of Sobolev spaces and present some theorems that allow us to study the properties of compositions of quasiconformal functions. We prove Stoilow's factorization theorem and use it to show that the solution to Beltrami's equation can naturally be normalized so that the solution is unique. Using Stoilow's factorization and a few other lemmas and theorems we derived earlier, we finally show the main result of the thesis that is, we show that every quasisymmetric function is a welding homeomorphism. In the last chapter, we answer the question of whether perhaps all increasing homeomorphisms are welding homeomorphisms. The answer is no and this is shown by the counterexample invented by Bishop.
Subject: Sobolev spaces
Conformal welding
Beurling-Ahlfors extenstion

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