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The author presents 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. The book begins by introducing these concepts, stressing the interplay between geometry, the role of symmetries and weak solutions. The reader is then presented with 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 self-contained 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.

Harmonic maps. --- Riemannian manifolds. --- Manifolds, Riemannian --- Riemannian space --- Space, Riemannian --- Geometry, Differential --- Manifolds (Mathematics) --- Maps, Harmonic --- Mappings (Mathematics)

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This book provides a broad yet comprehensive introduction to the analysis of harmonic maps and their heat flows. The first part of the book contains many important theorems on the regularity of minimizing harmonic maps by Schoen-Uhlenbeck, stationary harmonic maps between Riemannian manifolds in higher dimensions by Evans and Bethuel, and weakly harmonic maps from Riemannian surfaces by Helein, as well as on the structure of a singular set of minimizing harmonic maps and stationary harmonic maps by Simon and Lin. The second part of the book contains a systematic coverage of heat flow of harmon

Harmonic maps --- Heat equation --- Riemannian manifolds --- Manifolds, Riemannian --- Riemannian space --- Space, Riemannian --- Geometry, Differential --- Manifolds (Mathematics) --- Diffusion equation --- Heat flow equation --- Differential equations, Parabolic --- Maps, Harmonic --- Mappings (Mathematics)

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Differential geometry. Global analysis --- Diffeomorphisms. --- Riemannian manifolds. --- Difféomorphismes --- Riemann, Variétés de --- 51 <082.1> --- Mathematics--Series --- Difféomorphismes --- Riemann, Variétés de --- Diffeomorphisms --- Riemannian manifolds --- Manifolds, Riemannian --- Riemannian space --- Space, Riemannian --- Geometry, Differential --- Manifolds (Mathematics) --- Differential topology

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Differential geometry. Global analysis --- Variational inequalities (Mathematics) --- Riemannian manifolds. --- Inégalités variationnelles. --- Riemann, Variétés de. --- Riemannian manifolds --- Inequalities, Variational (Mathematics) --- Calculus of variations --- Differential inequalities --- Manifolds, Riemannian --- Riemannian space --- Space, Riemannian --- Geometry, Differential --- Manifolds (Mathematics)

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Differential geometry. Global analysis --- Riemannian manifolds. --- Riemann, Variétés de. --- Geodesics (Mathematics) --- Géodésiques (mathématiques) --- Riemannian manifolds --- Geometry, Differential --- Global analysis (Mathematics) --- Mathematics --- Manifolds, Riemannian --- Riemannian space --- Space, Riemannian --- Manifolds (Mathematics)