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

Hardcover, xiii + 301 pages, NOT ex-library. Clean and bright throughout. Some internal creasing. Issued without a dust jacket. - A treatment of Beltrami equations, with a focus on a unified geometric approach based on the modulus method. This monograph serves both as a reference work and pedagogical text for graduate-level readers engaged in research on quasiconformal mappings, complex analysis, and nonlinear elliptic partial differential equations. It is especially relevant for contemporary mathematical investigations involving generalized and degenerate structures, boundary value problems, and geometric analysis. At the heart of the book is the Beltrami equation "f sub z-bar equals mu of z times f sub z", a fundamental nonlinear PDE that generalizes the Cauchy-Riemann equations. This equation plays a central role in the theory of quasiconformal mappings, which in turn are deeply connected to Teichmüller theory, Riemann surfaces, low-dimensional topology, and various areas in mathematical physics. The authors present a geometric method founded on modulus and capacity theory, allowing for new insights into problems such as existence, uniqueness, convergence, distortion, and boundary behavior of solutions under general, degenerate, or singular conditions. The book is structured into ten chapters that progressively develop the analytic and geometric theory of Beltrami equations. After an introductory chapter that contextualizes the equation within geometry, analysis, and applications, the authors establish foundational material in Chapter 2, covering BMO and FMO function spaces, Sobolev classes, modulus, capacity, convergence theorems, and integral conditions. These preliminaries support later rigorous treatments of classical and degenerate Beltrami equations. Chapters 3 & 4 examine the classical equation "essential supremum of the absolute value of mu is strictly less than one" and its degenerate forms. The degenerate case, which includes singularities and sets where "the absolute value of mu is equal to one" is especially relevant for modern research, as it captures behaviors arising in limiting configurations and critical geometric flows. Through new integral conditions, the authors unify and extend many existing results, establishing criteria that are both necessary and sufficient for solvability in degenerate domains. Chapters 5 & 6 study BMO- and FMO-quasiconformal mappings and ring Q-homeomorphisms at boundary points. The development of distortion estimates, existence theorems, and removability results reflects a significant methodological advancement, particularly the refinement of local and boundary regularity conditions. These tools are essential in modern complex dynamics and potential theory, especially in settings where traditional smoothness assumptions are too restrictive. Chapter 7 introduces the notion of strong ring solutions, including factorization techniques and integral characterizations that allow for a robust treatment of Beltrami equations with measurable coefficients. This facilitates deeper engagement with the non-linear and non-uniform elliptic structures present in various applied contexts. Chapter 8 addresses the Dirichlet problem under generalized Beltrami conditions. This discussion includes new integral criteria for existence, regularity of homeomorphic solutions, and techniques for extending mappings to the boundary - a topic of importance in the study of Riemann surfaces and variational methods. Chapters 9 & 10 explore more recent developments: Beltrami equations with two characteristics and alternating Beltrami equations. These extensions are crucial in studying branched and folded mappings, which have applications in image processing, geometric function theory, and the modeling of material deformations. The detailed classifications and constructions of BF-maps (branched folded maps) and cusp structures provide a novel toolkit for researchers analyzing discontinuous or multi-valued phenomena.…