Characteristic Properties of Möbius Transformations and Quasiconformal Mappings Open Access

Abstract

Möbius transformations make up a very special class of conformal mappings. They are differentiable, continuous, and bijective in the entirety of their domain and make up the group of homeomorphisms of the extended complex plane. And with their conformal and homeomorphic properties they preserve the essence of a shape or space, and thus have many applications in physics, engineering, and complex analysis. But while Möbius trans-formations are very beautiful mathematical objects, their rigidity can cause problems when attempting to apply them to more complicated domains and structures. Thus the concept of a quasiconformal or "almost conformal" map was developed. Quasiconformal maps are generalizations of conformal maps and Möbius transformations that are both flexible enough to be applied to more difficult problems and yet have enough structure to be useful and interesting. The purpose of this paper is to gather and prove the key characterizations of Möobius transformations in a clear and succinct manner as well as to generalize some of them to properties for quasiconformal mappings.

Table of Contents

1 Introduction..............................................................................1

1.1 Conformal Mappings .......................................................1

1.2 Möbius Transformations ..................................................4

1.3 Quasiconformal Mappings ...............................................5

1.4 Quasicircles ..................................................................7

2 Characterizations of Möbius Transformations.................................8

2.1 Known Characterizations.................................................8

2.2 Key Results...................................................................11

3 Characterizations of Quasiconformal Mappings and Quasicircles.......14

3.1 Characterizations of Quasiconformal Mappings...................14

3.2 Characterizations of Quasicircles......................................17

3.3 Examples .....................................................................20

4 Open Questions........................................................................24

About this Honors Thesis

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School	Emory College
Department	Mathematics
Degree	BS
Submission	Honors Thesis
Language	English
Research Field	Mathematics
Keyword	Conformal Mobius Quasiconformal Complex Analysis
Committee Chair / Thesis Advisor	Yang, Shanshuang, Emory University
Committee Members	Batterson, Steven L, Emory University Margariti, Roxani, Emory University

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