Harmonic Maps, Conservation Laws and Moving Frames

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.

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.