Read more
Zusatztext The book begins with an expertly written overview, which will be of value even to specialists in the field ... authoritatively written by two of the leading figures in the field ... contains a wealth of material ... makes pleasant reading. Klappentext This unique book explores the connections between the geometry of mappings and many important areas of modern mathematics such as Harmonic and non-linear Analysis, the theory of Partial Differential Equations, Conformal Geometry and Topology. Much of the book is new. It aims to provide students and researchers in many areas with a comprehensive and up to date account and an overview of the subject as a whole. Zusammenfassung This book provides a survey of recent developments in the field of non-linear analysis and the geometry of mappings.Sobolev mappings, quasiconformal mappings, or deformations, between subsets of Euclidean space, or manifolds or more general geometric objects may arise as the solutions to certain optimisation problems in the calculus of variations or in non-linear elasticity, as the solutions to differential equations (particularly in conformal geometry), as local co-ordinates on a manifold or as geometric realisations of abstract isomorphisms between spaces such as those that arise in dynamical systems (for instance in holomorphic dynamics and Kleinian groups). In each case the regularity and geometric properties of these mappings and related non-linear quantities such as Jacobians, tells something about the problems and the spaces under consideration.The applications studied include aspects of harmonic analysis, elliptic PDE theory, differential geometry, the calculus of variations as well as complex dynamics and other areas. Indeed it is the strong interactions between these areas and the geometry of mappings that underscores and motivates the authors' work. Much recent work is included. Even in the classical setting of the Beltrami equation or measurable Riemann mapping theorem, which plays a central role in holomorphic dynamics, Teichmuller theory and low dimensional topology and geometry, the authors present precise results in the degenerate elliptic setting. The governing equations of non-linear elasticity and quasiconformal geometry are studied intensively in the degenerate elliptic setting, and there are suggestions for potential applications for researchers in other areas. Inhaltsverzeichnis 0: Introduction and Overview 1: Conformal Mappings 2: Stability of the Mobius Group 3: Sobolev Theory and Function Spaces 4: The Liouville Theorem 5: Mappings of Finite Distortion 6: Continuity 7: Compactness 8: Topics from Multilinear Algebra 9: Differential Forms 10: Beltrami Equations 11: Riesz Transforms 12: Integral Estimates 13: The Gehring Lemma 14: The Governing Equations 15: Topological Properties of Mappings of Bounded Distortion 16: Painleve's Theorem in Space 17: Even Dimensions 18: Picard and Montel Theorems in Space 19: Conformal Structures 20: Uniformly Quasiregular Mappings 21: Quasiconformal Groups 22: Analytic Continuation for Beltrami Systems ...
