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Geometric function theory and nonlinear analysis. (English) Zbl 1045.30011
Oxford Mathematical Monographs. Oxford: Oxford University Press (ISBN 0-19-850929-4/hbk). xv, 552 p. (2001).
The theory of quasiconformal and quasiregular mappings in the Euclidean space $$\mathbb R^n, n\geq 2,$$ emerged from the works of H. Grötzsch, O. Teichmüller, and L. Ahlfors for $$n=2$$ and of M. A. Lavrent’ev, F. W. Gehring, J. Väisälä, Yu. G. Reshetnyak for $$n \geq 3.$$ This history of the classical theory is well recorded in the books by L. V. Ahlfors [Lectures on quasiconformal mappings (1987; Zbl 0605.30002)], O. Lehto and K. I. Virtanen, [Quasiconformal mappings in the plane (1973; Zbl 0267.30016)], J. Väisälä [Lectures on $$n$$-dimensional quasiconformal mappings (1971; Zbl 0221.30031)], Yu. G. Reshetnyak [Space mappings with bounded distortion (1989; Zbl 0667.30018); und Stability theorems in geometry and analysis (1994; Zbl 0925.53005)] M. Vuorinen [Conformal geometry and quasiregular mappings (1988; Zbl 0646.30025)], J. Heinonen, . Kilpeläinen and O. Martio, [Nonlinear potential theory of degenerate elliptic equations (1993; Zbl 0780.31001)] S. Rickman [Quasiregular mappings (1993; Zbl 0816.30017)] G. D. Anderson, M. K. Vamanamurthy and M. Vuorinen [Conformal invariants, inequalities, and quasiconformal maps (1997; Zbl 0885.30012)] (listed here roughly in the order of publication).
The book under review, authored by T. Iwaniec and G. Martin, demonstrates that the area has been developing vigorously during the past decade. The authors, themselves leading experts in this area, invite the reader to join an exciting journey through the multitude of results which they present here. The emphasis is on the authors’ work during the past decade which has strong connections to nonlinear PDEs. The reader of this research monograph is supposed to be well prepared, at least on the level of “the competent graduate student” as the authors put it. The book is organized into twenty-three chapters as follows.
1. Introduction and overview, 2. Conformal mappings, 3. Stability of the Möbius group, 4. Sobolev theory and function spaces, 5. The Liouville theorem, 6. Mappings with finite distortion, 7. Continuity, 8. Compactness, 9. Topics from multilinear algebra, 10. Differential forms, 11. Beltrami equations, 12. Riesz transforms, 13. Integral estimates, 14. The Gehring lemma, 15. The governing equations, 16. Topological properties of mappings of bounded distortion, 17. Painlevé’s theorem in space, 18. Even dimensions, 19. Picard and Montel theorems in space, 20. Conformal structures, 21. Uniformly quasiregular mappings, 22. Quasiconformal groups, 23. Analytic continuation for Beltrami systems.
For graduate student readers it would have been helpful if the authors had included exercises with solutions to deepen and underline some key ideas of the text. On the other hand, the lack of exercises may not be a problem since most readers are likely to be specialists in the area of mathematical analysis.
The contents of the book is highly original, because other books do not deal with these topics and also because much of the material was unpublished at the time of the writing of the book. A novel theme is the study of maps with finite distortion. The book is very useful for specialists in the area of multidimensional geometric function theory because it brings under one roof results scattered in journals. The authors display brilliant expertise in this technically very difficult area, much of which is due to the authors and their collaborators. The book addresses many new topics which look very promising, and I think that this book will have an important role in the future development of several areas of mathematical analysis such as nonlinear PDEs, iteration of quasiregular maps, and maps of finite distortion. This book is likely to remain a landmark of the quasiregular mapping theory for many years to come.

##### MSC:
 30C65 Quasiconformal mappings in $$\mathbb{R}^n$$, other generalizations 30-02 Research exposition (monographs, survey articles) pertaining to functions of a complex variable
##### MathOverflow Questions:
Quasiconformal maps in arbitrary dimensions
##### Keywords:
quasiconformal maps; quasiregular maps