The Beltrami Equation : A Geometric Approach The Beltrami Equation: A Geometric Approach will be particularly useful to many specialists in modern geometric analysis quasiconformal mappings and extensions beginning researchers and graduate students with a year’s background in complex variables. This book covers the state of the art in the ongoing study of the Beltrami equation the classical equation that has been studied for more than 100 years. Along with its rich history the Beltrami equation plays a significant role in geometry analysis and physics. The most important feature of this work concerns the unified geometric approach taken based on the modulus method that can be effectively applied to solving many problems in mathematical physics. Beautiful examples illustrate the relationship between mappings with bounded oscillation and those with finite oscillations. Written by authors that are well known specialists in this field this monograph presents recent developments in the theory of Beltrami equations studying a variety of problems like convergence existence uniqueness representation removal of singularities local distortion estimates and boundary behavior of solutions to the Beltrami equations. It contains new types of criteria in the given problems particularly new integral conditions for the existence of regular solutions to the Beltrami equations that turned out to be not only sufficient but also necessary. 1. Introduction 2. Preliminaries 3. The Classical Beltrami Equation ||μ||∞ < 1 4. The Degenerate Case 5. BMO and FMO Quasiconformal Mappings 6. Ring Q Homeomorphisms at Boundary Points 7. Strong Ring Solutions of Beltrami Equations 8. On the Dirichlet Problem for Beltrami Equations 9. On the Beltrami Equations with Two Characteristics 10. Alternating Beltrami Equation References Index.