## Harmonic Maps, Conservation Laws and Moving FramesThis 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. |

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### Contents

Harmonic maps with symmetry | 49 |

Compensations and exotic function spaces | 114 |

Harmonic maps without symmetry | 165 |

Surfaces with mean curvature in L2 | 221 |

254 | |

263 | |

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### Common terms and phrases

belongs boundary bounded chapter choose coefficients compact conformal immersion conformal transformation conservation laws consider constant mean curvature construct continuous converges Coulomb frame covariant critical points deduce define definition denote diffeomorphism differential Dirichlet integral domain dx dy energy equation equivalent estimate Euclidean example fact frame field geometric Hardy space hence Hl(M,Af holomorphic image manifold implies inequality invariant isometric isometric embedding lemma Lie algebra Lipschitz loop group Lorentz spaces maps with values mean curvature metric g minimizing Moreover moving frame Noether harmonic Noether's theorem norm Notice obtain open subset orthonormal basis orthonormal frame Proof of theorem prove Remark Riemannian manifold Riemannian surface satisfies second fundamental form sequence simply connected smooth SO(n sphere Step stress-energy tensor submanifold suppose symmetry taking values tangent tensor theory variational problem vector field weak solutions weak topology weakly harmonic maps Wente's write yields

### Popular passages

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Page 255 - R. Coifman, P.-L. Lions. Y. Meyer, S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appi.

Page 257 - L. Nirenberg, On functions of bounded mean oscillations, Comm. Pure Appl. Math. 14 (1961), 415-426.

Page 257 - I. Rodnianski. On the global regularity of wave maps in the critical Sobolev norm, IMRN 13 (2001), 655-677.