1.1. From the Preface.

In a very simplistic way, a nonlinear differential form is a function, valued in a finite dimensional linear space, whose variables are partial derivatives of a vector field. In a larger context the vector fields are replaced by tensors. This is a discipline in which a cancellation of certain integral terms, due mainly to integration by parts, is the central source of arguments . Sometimes it becomes almost mystical. For example, null Lagrangians pertain to those forms for which the integral is identically zero for all vector fields vanishing at the boundary. From this cancellation phenomenon we will develop some very powerful tools. And this is a field for which most problems motivating interest in the subject cannot be explained at the introductory level. On the other hand , one of the distinctive features of contemporary applied mathematics is its interest in framing the observed world in rigorous mathematical terms. The facts remain but their values change. New results, wider and more unifying, are established, leading to various developments. Tensor fields (a system of functions arranged in a way to satisfy certain transformation laws) become indispensable tools in continuum mechanics, non-linear elasticity, electromagnetics, and so on. Fashions in quasiconformal geometry, as in other fields, can alter. At the present time exterior differential forms become ever more useful , stimulating new partial differential equations which govern an even larger class of mappings of finite distortion.

1.2. Review by: Giovanni Alberti.
Mathematical Reviews MR1678020 (2000g:58002).

These lecture notes are organized into ten chapters: 1. Topics from multilinear algebra, 2. Requisites from analysis and function spaces, 3. Sobolev classes of differential forms, 4. Hodge decompositions, 5. Poincaré inequalities for differential forms, 6. Compensated compactness and integral estimates for Jacobians, 7. Geometric function theory in $\mathbb{R}^{n}$, 8. The Gehring lemma, 9. The limit of mappings with finite distortion, 10. Commutators.

It clearly appears from the index that the subject is presented from the analytical viewpoint: forms, or more precisely their (local) coefficients, are not assumed to be smooth, but belong to distributions or to spaces of summable or weakly differentiable functions (e.g., Luzin, Orlicz, or Sobolev spaces) ...
...
These lecture notes are very well-written and readable. Even if some familiarity with the basic concepts of multilinear algebra and the theory of weakly differentiable functions is probably required, the theory which is needed to proceed is quite effectively summarised in the first chapters. In the following chapters, the author gives a unified presentation of many recent results, many of which, to my knowledge, have never been collected before in lecture form. Every topic is motivated in detail, and accompanied with historical notes, exercises, and separate bibliography (one for almost every chapter). In conclusion, this is an excellent introduction to problems and results that, besides being wonderful pieces of mathematics per se, are also deeply connected to almost all areas of analysis, from the calculus of variations to harmonic analysis, to the regularity theory for PDEs.

2.1. From the Publisher.

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.

2.2. From the Preface.

This book is largely about the geometry of mappings - that. is, functions or deformations between subsets of the Euclidean $n$-space $\mathbb{R}^{n}$ and more generally between manifolds or other geometric objects. Such mappings may be homeomorphisms, diffeomorphisms, branched coverings or more abstract correspondences such as Sobolev mappings, They may arise as the solutions to differential equations, the minima of certain optimisation problems in the calculus of variations, as local coordinates on a manifold or as geometric realisations of abstract isomorphisms between spaces. In each case the regularity and geometric properties of these mappings will tell us something about the problem at hand or the spaces we are investigating.

Of course such a general topic intersects many areas of modern mathematics. Thus we will run into aspects of differential geometry, topology, partial differential equations and harmonic analysis, as well as nonlinear analysis, the calculus of variations and so forth. A good deal of this intersection is surveyed in Chapter 1, in which our aim is to give the reader some appreciation of the diversity of applications and directions in which current research is moving, as well as a glimpse of the substantial body of work which we were unable to cover in any detail here.

This book is essentially a research monograph. We have tried to present a fairly complete account of the most recent developments in these areas as they pertain to the geometry of mappings, and indeed a significant portion of this book was new or recent at the time of writing. However, we do cover and offer new approaches to many aspects of the classical theory as well as devoting a few chapters to foundational material, and we have pitched the level of the book at the competent graduate student.

We wish to express our deep gratitude to the many fellow mathematicians who have contributed in one way or another to this book. In particular, those from the Finnish and Italian Schools with whom we have collaborated and discussed many ideas and whose theorems can be found throughout this book. Also many thanks to Tsukasa Yashiro who created all the pictures for us, and to John Duncan and Volker Meyer who read and commented on a good portion of the manuscript.

Both authors would like to acknowledge the support they received from the US National Science Foundation and the NZ Marsden Fund.

While this book is dedicated to our families, there are two others we must acknowledge. These are our teachers, Bogdan Bojarski and Fred Gehring, who pioneered much of the theory presented here. In particular, Fred brought us together in Ann Arbor from either end of the world to do mathematics, and throughout our careers he and his wife Lois have been unfailingly supportive. Thanks!

2.3. Review by: Mario Bonk.
Mathematical Reviews MR1859913 (2003c:30001).

In its title the term "geometric function theory" has to be understood from a modern point of view as the study of quasiconformal, quasiregular and related mappings whose theory evolved from the classical theory of analytic functions. The second part of the title, "non-linear analysis", emphasizes that the subject is treated from an analytic perspective similar in spirit to Reshetnyak's approach.

If we compare this book with Reshetnyak's book [Space mappings with bounded distortion (1989)], then we see how the subject has advanced in the past two decades. One of the main points is a considerable increase in tools from harmonic analysis and PDEs which allow the investigation of more subtle problems. Iwaniec and Martin make full use of this recent progress.

Classical subjects are not neglected, though. ... Apart from these classical topics, the authors are particularly interested in questions whose investigation has only recently become possible. ...
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Many discussions in the book are based on very recent research literature, and the authors tried to include some of the latest developments. A certain disadvantage of this is that some of the results presented are already superseded. As the reviewer learned from the authors, the book was written in a remarkably short period of time. At some places this shows, and some errors slipped in (for example: Gaussian curvature is not preserved under conformal change of the metric (p. 1); (Paul) du Bois-Reymond is one person (p. 406)).

Every reader interested in the modern viewpoint of geometric function theory will find a wide range of appealing topics in this book. It is particularly useful to see how some highly sophisticated machinery of real and harmonic analysis is developed and employed. This book will become a standard reference for the field.

2.4. Comment by: Tadeusz Iwaniec.
Rev. Mat. Iberoam. 28 (3) (2012), 681-722.

From the very beginning of my mathematical life I fell in love with logic and later as a young scholar with geometry and harmonic analysis. There are so many captivating topics in geometric analysis. I was especially fascinated by the foundation of Geometric Function Theory, the mysteries in the Calculus of Variations and Nonlinear Partial Differential Equations. Nowadays, these fields are essential in material science and nonlinear elasticity, which are critical in modern technology and many engineering problems. Myriad practical problems of nonlinear elasticity and numerous elegant conjectures are very appealing to me.

I and my amazingly imaginative colleague Gaven Martin have presented Geometric Function Theory in all dimensions in our book Geometric function theory and non-linear analysis. Well, we did not make a fortune with this book, nor did we become famous. But I have heard someone say, "Hey, I have read your book". How satisfying!

3.1. From the Publisher.

The "measurable Riemann Mapping Theorem" (or the existence theorem for quasiconformal mappings) has found a central role in a diverse variety of areas such as holomorphic dynamics, Teichmuller theory, low dimensional topology and geometry, and the planar theory of PDEs. Anticipating the needs of future researchers, the authors give an account of the "state of the art" as it pertains to this theorem, that is, to the existence and uniqueness theory of the planar Beltrami equation, and various properties of the solutions to this equation. The classical theory concerns itself with the uniformly elliptic case (quasiconformal mappings). Here the authors develop the theory in the more general framework of mappings of finite distortion and the associated degenerate elliptic equations.

3.2. From the Abstract.

The "measurable Riemann Mapping Theorem" (or the existence theorem for quasiconformal mappings) has found a central role in a diverse variety of areas such as holomorphic dynamics, Teichmüller theory, low dimensional topology and geometry, and the planar theory of PDEs. Anticipating the needs of future researchers we give an account of the "state of the art" as it pertains to this theorem, that is to the existence and uniqueness theory of the planar Beltrami equation, and various properties of the solutions to this equation. The classical theory concerns itself with the uniformly elliptic case (quasiconformal mappings). Here we develop the theory in the more general framework of mappings of finite distortion and the associated degenerate elliptic equations.

We recount aspects of this classical theory for the uninitiated, and then develop the more general theory. Much of this is either new at the time of writing, or provides a new approach and new insights into the theory. Indeed, it is the substantial recent advances in non-linear harmonic analysis, Sobolev theory and geometric function theory that motivated our approach here. The concept of a principal solution and its fundamental role in understanding the natural domain of definition of a given Beltrami operator is emphasised in our investigations. We believe our results shed considerable new light on the theory of planar quasiconformal mappings and have the potential for wide applications, some of which we discuss.

3.3. Review by: Alexander Vasil'ev.
Mathematical Reviews MR2377904 (2008m:30019).

The Beltrami equation has been at the heart of the geometric and analytic interplay between PDEs and function theory since the classical 1938 paper by C B Morrey, Jr in 1938. ...
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The survey part of the work under review gives a nice historical overview of the distinguished history of the Beltrami equation. Recently, much interest turned to a weaker solution to this equation ...
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The main starting point of the monograph is represented by two papers, one by G David (1988) and the other by M A Brakalova and J A Jenkins (1998) ... The authors synthesise and extend these two approaches to study the degenerate Beltrami equation and discuss existence, regularity and uniqueness results which become close to optimal. The authors place all classical results in a modern setting and discuss future development and applications of the theory of the planar Beltrami equation. This short monograph is highly recommended for a wide audience of analysts.

4.1. From the Publisher.

This book explores the most recent developments in the theory of planar quasiconformal mappings with a particular focus on the interactions with partial differential equations and nonlinear analysis. It gives a thorough and modern approach to the classical theory and presents important and compelling applications across a spectrum of mathematics: dynamical systems, singular integral operators, inverse problems, the geometry of mappings, and the calculus of variations. It also gives an account of recent advances in harmonic analysis and their applications in the geometric theory of mappings.

The book explains that the existence, regularity, and singular set structures for second-order divergence-type equations - the most important class of PDEs in applications - are determined by the mathematics underpinning the geometry, structure, and dimension of fractal sets; moduli spaces of Riemann surfaces; and conformal dynamical systems. These topics are inextricably linked by the theory of quasiconformal mappings. Further, the interplay between them allows the authors to extend classical results to more general settings for wider applicability, providing new and often optimal answers to questions of existence, regularity, and geometric properties of solutions to nonlinear systems in both elliptic and degenerate elliptic settings.

4.2. From the Preface.

This book presents the most recent developments in the theory of planar quasiconformal mappings and their wide-ranging applications in partial differential equations and nonlinear analysis, conformal geometry, holornorphic dynamical systems, singular integral operators, inverse problems, the geometry of mappings and, more generally, the calculus of variations. It is a simply amazing fact that the mathematics that underpins the geometry, structure and dimension of such concepts as Julia sets and limit sets of Kleinien groups, the spaces of moduli of Riemann surfaces, conformal dynamical systems and so forth is the very same as t hat which underpins existence , regularity, singular set structure and so forth for precisely t he most import ant class of differential equations one meets in physical applications, namely, second-order divergence-type equations. All these subjects are inextricably linked in two dimensions by the theory of quasiconformal mappings.

There have been profound developments in the three or four decades since the publication of Lars Ahlfors' beautiful little book and the classical text of Olli Lehto and Kalle Virtanen. Indeed, whole subjects have blossomed, conformal and holomorphic dynamics, holomorphic motions, nonlinear partial differential equations and connections with the calculus of variations, to name just a few.

This book gives a fairly comprehensive account of the modern theory, but for those planning to present a semester course in the theory of quasiconformal mappings and their applications in modern complex analysis, the contents of Chapters 3 and 5, with selected applications chosen from Chapters 12, 13 and some of the later chapters, should provide ample material at an easy pace. Further, the material in Chapter 4 presents a reasonable and self-contained introduction to harmonic analysis and the theory of singular integral operators in two dimensions.

The latter parts of the book present perhaps the most recent advances in the area. Indeed, more than a few results and proofs in this monograph are new. These chapters also serve to illustrate the wide applicability of the ideas and techniques developed in the earlier part of the book.

It is our pleasure to acknowledge the wide-ranging support we have had from a number of places that has made this book possible. First, we have all been partly supported by the Academy of Finland, the Marsden Fund of New Zealand and the National Science Foundation of the United States at one time or another. We all shared Research in Peace fellowships at Institute Mittag-Leffler (Sweden) where the first real progress toward a book was made. Of course our home institutions - Massey University (New Zealand), Syracuse University (United States) and the University of Helsinki (Finland) - have all hosted and supported us as a group at various times.

There are many people to thank as well. In particular, Pekka Koskela let us use his notes on quasisymmetrlc functions (a good part of Chapter 3), Stanislaw Smirnov let us present his unpublished proof of the dimension bounds for quasicircles and Laszlo Lempert communicated Chirka's proof of the $\lambda$-lemma to us while we were at Oberwolfach. There are also the people who read various drafts of the book and made substantial and valuable comments. These include Tomasz Adamowicz, Samuel Dillon, Daniel Faraco, Peter Haïssinsky, Jarmo Jääskelfäinen, Matti Lessas, Martti Nikunen, Jani Onninen, Lessi Päivärinta, Istvan Prause, Eero Saksman, Carlo Sbordone, Ignacio Uriarte-Tuero and Antti Vähäkangas. We would also like to thank the team at Princeton University Press - Kathleen Cioffi, Carol Dean, Lucy Day Hobor and Vickie Kearn - who skilfully guided us through the production process and whose considerable efforts improved this book.

Finally, during the writing of this book the "quasi-world" was saddened by the premature death of one of its leading figures, Juha Heinonen. We wish to record here the deep respect we have for Juha and the contributions he made. He was an inspiration to all of us.

4.3. Review by: Olli Martio.
Mathematical Reviews MR2472875 (2010j:30040).

This book belongs to the long line of books on plane quasiconformal mappings, the best known examples being [O Lehto and K I Virtanen, Quasiconformal mappings in the plane, 1973; L V Ahlfors, Lectures on quasiconformal mappings, 1966]. The emphasis of the book is clearly on the analytical aspects and not so much on the geometrical theory where the main method is the modulus of a path family [see J Väisälä, Lectures on $n$-dimensional quasiconformal mappings, 1971; O Martio et al, Moduli in modern mapping theory]. The aim is to present the latest achievements of the analytical theory and their consequences to plane PDE theory and nonlinear analysis in general. Here the authors are successful, although it requires 676 pages. Another aim is to present the optimal assumptions on the existence and regularity of quasiconformal mappings and related PDEs. These are not so apparent in the classical set up but the study of very weak solutions and quasiconformal mappings with finite dilatation opens new possibilities and applications. Much of the recent work has been done in this area where the authors have been the key persons. Many proofs in the book are new.

The book consists of 21 chapters and one appendix. Although complex notation is used rather sparingly, it plays an important role in the study of Cauchy-Riemann operators and the Beurling transformation. Thus it takes some time for the reader with purely real analytic background to grasp the fundamental ideas typical in the plane case. Chapters 2 and 3 consist of an outline of 80 pages on (now) classical theory on quasiconformal plane mappings; the authors return to these questions, such as the Hölder continuity, in subsequent chapters. Various $L^{p}$ estimates (for maximal functions, Riesz transform, Hilbert transform, Beurling transform, etc.) play an essential role throughout the book. These estimates fill Chapter 4 and many subsequent sections. Everywhere the intention is to get the best possible results; when not successful, then the failure is mentioned.

Because of the rather extensive scope of the book it is only possible to highlight some topics. Plane first-order systems and the corresponding differential operators and their homotopy classes get a full treatment. ...

4.4. Review by: Canadian Mathematical Society.
Canadian Mathematical Society Notes 41 (8) (2009), 7.

This book explores the recent developments in the theory of planar quasiconformal mappings with a particular focus on the interactions with partial differential equations and nonlinear analysis. There are profound applications in such wide ranging areas as holomorphic dynamical systems, singular integral operators, inverse problems, geometry of maps, and more generally, calculus of variations - all of these are presented in the book. The theory of quasiconformal mappings and their applications in complex analysis is discussed in the first few chapters. There is a chapter entitled 'complex potentials' which presents a good self-contained introduction to harmonic analysis and the theory of singular integral operators in two dimensions. The later chapters are devoted to recent advances in special areas such as Beltrami equations and operators, and aspects of calculus of variations. The book will be useful to research mathematicians looking for powerful modern techniques in diverse applications.