By Frédéric Hélein
This obtainable creation to harmonic map thought and its analytical points, covers fresh advancements within the regularity concept of weakly harmonic maps. The ebook starts by way of introducing those ideas, stressing the interaction among geometry, the function of symmetries and vulnerable suggestions. It then provides a guided travel into the speculation of thoroughly integrable platforms for harmonic maps, by means of chapters dedicated to fresh effects at the regularity of susceptible strategies. A presentation of "exotic" useful areas from the speculation of harmonic research is given and those instruments are then used for proving regularity effects. the significance of conservation legislation is under pressure and the idea that of a "Coulomb relocating body" is defined intimately. The publication ends with extra purposes and illustrations of Coulomb relocating frames to the speculation of surfaces.
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This available creation to harmonic map idea and its analytical points, covers contemporary advancements within the regularity idea of weakly harmonic maps. The publication starts off via introducing those suggestions, stressing the interaction among geometry, the function of symmetries and susceptible suggestions. It then offers a guided journey into the idea of thoroughly integrable structures for harmonic maps, by way of chapters dedicated to contemporary effects at the regularity of susceptible recommendations.
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Extra info for Harmonic maps, conservation laws, and moving frames
The first is obtained in the case where the symmetry group acts on the image manifold, and the second is connected to the symmetries of the domain. 1 Symmetries on N We start with a simple example, that of harmonic maps from an open set Ω of Rm , taking values in the sphere S 2 ⊂ R3 . 24) where |du|2 = 3 m i=1 α=1 ∂ui ∂xα 2 = 2e(u), and m ∆u = ∂2u . 25) or equivalently, m ∂ α ∂x α=1 u× ∂u ∂xα m = ∂u ∂u × α + u × ∆u = 0 . 26). Its interest is that it singles out a vector field j = (j 1 , . . , j m ) on Ω, given by jα = u × ∂u ∂xα which is divergence-free.
54) The interest of this theorem is that the group of conformal transformations of a Riemannian surface is very big. 3 Conservation laws for harmonic maps 29 and every holomorphic (or anti-holomorphic) diffeomorphism over ω induces a conformal transformation. Thus, for every holomorphic function F : ω −→ C, the vector field X = (F ) ∂ ∂ + (F ) ∂x ∂y is a conformal Killing field. In these coordinates (Sαβ ) = |du|2 2 = 1 2 1 0 0 1 |ux |2 u x , uy − − φ φ φ φ |du|2 = ∂u ∂x u x , uy |uy |2 , where 2 + ∂u ∂y 2 , and φ= ∂u ∂x 2 − ∂u ∂y 2 − 2i ∂u ∂u , ∂x ∂y .
Still in the case m = 2, we have ∆e2v g (φ) = e−2v ∆g φ . 9) Therefore, for m = 2, every function which is harmonic over (M, g) will also be so over (M, e2v g). 11) and where λ= ∂T i ∂T j 1 αβ g (x)gij (T (x)) α β . 2 The Dirichlet integral, and the set of harmonic functions over an open subset of a Riemannian surface, depend only on the conformal structure of this surface. This phenomenon, characteristic of dimension 2, has many consequences, among them the following, which is very useful: first recall that according to the theorem below, locally all conformal structures are equivalent.
Harmonic maps, conservation laws, and moving frames by Frédéric Hélein