Memoirs of the American Mathematical Society 2008; 92 pp; softcover Volume: 191 ISBN10: 0821840452 ISBN13: 9780821840450 List Price: US$64 Individual Members: US$38.40 Institutional Members: US$51.20 Order Code: MEMO/191/893
 The "measurable Riemann Mapping Theorem" (or the existence theorem for quasiconformal mappings) has found a central rôle in a diverse variety of areas such as holomorphic dynamics, Teichmüller theory, low dimensional topology and geometry, and the planar theory of PDEs. Anticipating the needs of future researchers, the authors give an account of the "state of the art" as it pertains to this theorem, that is, to the existence and uniqueness theory of the planar Beltrami equation, and various properties of the solutions to this equation. The classical theory concerns itself with the uniformly elliptic case (quasiconformal mappings). Here the authors develop the theory in the more general framework of mappings of finite distortion and the associated degenerate elliptic equations. The authors recount aspects of this classical theory for the uninitiated, and then develop the more general theory. Much of this is either new at the time of writing, or provides a new approach and new insights into the theory. Indeed, it is the substantial recent advances in nonlinear harmonic analysis, Sobolev theory and geometric function theory that motivated their approach here. The concept of a principal solution and its fundamental role in understanding the natural domain of definition of a given Beltrami operator is emphasized in their investigations. The authors believe their results shed considerable new light on the theory of planar quasiconformal mappings and have the potential for wide applications, some of which they discuss. Table of Contents  Introduction
 Quasiconformal mappings
 Partial differential equations
 Mappings of finite distortion
 Hardy spaces and BMO
 The principal solution
 Solutions for integrable distortion
 Some technical results
 Bibliography
