Synopses & Reviews
Accessible and pedagogical introduction to the theory of harmonic maps, covering recent results and applications.
Review
"...an excellent overview...The reviewer has really enjoyed reading this book." MATH"The presentation of the material is excellent. The book is as self-contained as it can be....The concepts that are introduced are very carefully motivated. Because the book also gives an overview about recent research results (even updated for the second edition), it is not only a pleasure to read, but also a very good reference book for research." Mathematical Reviews
Synopsis
This accessible introduction to harmonic map theory and its analytical aspects, covers recent developments in the regularity theory of weakly harmonic maps. The book begins by introducing these concepts, stressing the interplay between geometry, the role of symmetries and weak solutions. It then presents a guided tour into the theory of completely integrable systems for harmonic maps, followed by two chapters devoted to recent results on the regularity of weak solutions. A presentation of exotic functional spaces from the theory of harmonic analysis is given and these tools are then used for proving regularity results. The importance of conservation laws is stressed and the concept of a Coulomb moving frame is explained in detail. The book ends with further applications and illustrations of Coulomb moving frames to the theory of surfaces.
Synopsis
The author provides an accessible and self-contained introduction to harmonic map theory and its analytical aspects, covering recent developments in the regularity theory of weakly harmonic maps. It introduces the necessary mathematical tools for the study of harmonic maps and presents applications of the methods and theory, bringing in results from recent research on the regularity of weak solutions. The book will be of particular interest to graduate students and researchers in geometry, analysis and partial differential equations. It will also have a wide appeal within the mathematical physics community.
Table of Contents
Preface; Introduction; Acknowledgements; Notations; 1. Geometric and analytic setting; 2. Harmonic maps with symmetries; 3. Compensations and exotic function spaces; 4. Harmonic maps without symmetries; 5. Surfaces with mean curvature in L2; References.