Planar Quasiconformal Mappings: Fundamental Properties and Characterizations

Quasiconformal (QC) mappings generalize conformal mappings. Since their introduction in the 1930s, QC mappings have become a versatile tool in various fields of mathematics, ranging from PDEs to holomorphic dynamics. This thesis is an exposition of the five most widespread descriptions of QC mappings in the plane, as well as the most valuable properties thereof. We present a proof of the equivalence of the three main definitions: the metric, analytic, and geometric. Two additional characterizations are discussed in detail. The first is the partial identification of QC mappings with quasisymmetric mappings. This is done via conformal invariants. Once this identification is obtained, we use it to demonstrate that QC maps form a pseudogroup. We also use quasisymmetries to obtain the compactness properties of certain families of QC maps. Further, we demonstrate, using complex variables, several analytic properties, such as the change of variables and area formulae. We present a proof of the Measurable Riemann Mapping Theorem, which identifies quasiconformal mappings as the solutions of the Beltrami's equation - this is the fifth characterization. It is the interplay between the alternative characterizations that is arguably the most prominent feature of QC mappings. For this reason, an emphasis is put on highlighting the relationships between various descriptions and approaches to proofs