The Beltrami Equation: A Geometric Approach (Developments in Mathematics, 26)

This book is devoted to the Beltrami equations that play a significant role in Geometry, Analysis and Physics and, in particular, in the study of quasiconformal mappings and their generalizations, Riemann surfaces, Kleinian groups, Teichmuller spaces, Clifford analysis,  meromorphic functions, low dimensional topology, holomorphic motions, complex dynamics,  potential theory, electrostatics, magnetostatics,  hydrodynamics and magneto-hydrodynamics.

The purpose of this book is to present the recent developments in the theory of Beltrami equations; especially those concerning degenerate and alternating Beltrami equations. The authors study a wide circle of problems like convergence, existence, uniqueness, representation, removal of singularities, local distortion estimates and boundary behavior of solutions to the Beltrami equations. The monograph contains a number of 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.

The most important feature of this book concerns the unified  geometric approach based on the modulus method that is effectively applied to solving the mentioned problems. Moreover, it is characteristic for the book application of many new concepts as strong ring solutions, tangent dilatations, weakly flat and strongly accessible boundaries, functions of finite mean oscillations and new integral conditions that make possible to realize a more deep and refined analysis of problems related to the Beltrami equations. Mastering and using these new tools also gives essential advantages for the reader in the research of modern problems in many other domains. Every mathematics graduate library should have a copy of this book.​

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.