Lectures on Quasiconformal Mappings, second edition
These lectures develop the theory of planar quasiconformal mapping. Each linear mapping L of the plane onto itself maps the unit circle to an ellipse. The ratio of its axes (major/minor) is called the dilatation of L. Roughly, a quasiconformal mapping is a homeomorphism such that the dilatation of its derivative (computed pointwise) is bounded. A more technical definition is needed for non-differentiable maps: either the differentiability is relaxed to absolute continuity on almost all lines (analytic definition) or the differential properties are replaced by global conditions of geometric nature. The comparison of these two approaches is one of the first achievements of the lectures. Further results and topics include estimates of moduli of special condensers with an excursion to elliptic and modular functions, the sharp Hölder estimate (Mori's theorem), analysis of mappings of quadruplets of points, boundary behaviour, quasiconformal reflection, the solution of the Beltrami equation and the Calderón-Zygmund inequality. The lectures conclude with a treatment of Teichmüller spaces (including the Bers embedding and the Teichmüller curve).
The book is based on Ahlfors' course that was given at Harvard University in 1964. The first edition appeared in 1966. The new second edition contains three supplementary chapters demonstrating the efficiency of methods of quasiconformal mappings in various branches of analysis and geometry. A supplementary chapter is written by Earle and Kra. They begin with a brief survey of the theory of quasiconformal mappings with emphasis on issues mentioned in the lectures and their new developments. Most of the chapter is, however, devoted to the theory of Teichmüller spaces and their connections to Kleinian groups. A chapter by Shishikura presents applications of quasiconformal theory in complex dynamics. The third appendix, by Hubbard, shows how tools like quasiconformal mappings, the measurable Riemann mapping theorem and quasi-Fuchsian groups have been combined with some new methods to obtain Thurston's very deep theory on hyperbolization of irreducible 3-manifolds. The lectures and supplements constitute a very efficient way of learning some complicated theories with numerous applications.