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.
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
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Tadeusz Iwaniec is a John Raymond French Distinguished Professor of Mathematics at Syracuse University. Gaven John Martin is a Professor of Mathematics and James Cook Fellow of Royal Society, New Zealand.
