Download PDF by Frédéric Hélein: Harmonic maps, conservation laws, and moving frames

By Frédéric Hélein

ISBN-10: 0511016603

ISBN-13: 9780511016608

ISBN-10: 0521811600

ISBN-13: 9780521811606

This available creation to harmonic map concept and its analytical features, covers contemporary advancements within the regularity concept of weakly harmonic maps. The booklet starts by way of introducing those ideas, stressing the interaction among geometry, the function of symmetries and susceptible strategies. It then provides a guided journey into the idea of thoroughly integrable platforms for harmonic maps, via chapters dedicated to fresh effects at the regularity of vulnerable 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 concept that of a "Coulomb relocating body" is defined intimately. The booklet ends with additional purposes and illustrations of Coulomb relocating frames to the speculation of surfaces.

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2. 2 The Dirichlet integral, and the set of harmonic maps on an open subset of a Riemannian surface, depend only on the conformal structure. 3, we can always suppose that we have locally conformal coordinates (x, y) ∈ R2 on (M, g). 13) becomes ∂ 2 ui ∂ 2 ui + + Γijk (u) 2 ∂x ∂y 2 ∂uj ∂uk ∂uj ∂uk + ∂x ∂x ∂y ∂y = 0. 13), but an alternative one where we think of N as a submanifold of a Euclidean space. In fact, thanks to the Nash–Moser theorem ([123], [102], [77]), we know that, provided h is C 3 , it is always possible to isometrically embed (N , h) into a vector space RN , with the Euclidean scalar product .

In the case where m = 2, this method no longer works, but Luc Lemaire proved a similar result which is valid for every simply connected open set [106]. 44) which we used in Case A to deduce a conservation law in the case where there exists a Killing vector field. Writing Sβα = g αγ Sγβ , this identity states that, for any vector field X, 2Sαβ ∇ ∂ ∂xβ X α = −(LX g αβ )Sαβ . 37) implies that ∇ ∂α Sβα = 0 for ∂x all β, and thus ∇ ∂ ∂xβ (Sαβ X α ) = Sαβ ∇ ∂ ∂xβ Xα + ∇ 1 = − (LX g αβ )Sαβ . 52) Geometric and analytic setting 28 We see that for the vector field Y = Sαβ X α ∂x∂ β to be covariant divergencefree, it is necessary and sufficient that (LX g αβ )Sαβ = 0 .

1 Let X be a Lipschitz tangent vector field on N , which is an infinitesimal symmetry for L. 29) or equivalently, using the coordinates (x1 , . . , xm ) on Ω such that dµ = ρ(x)dx1 . . dxm , m ∂ α ∂x α=1 ρ(x)X i (u) ∂L (x, u, du) ∂Aiα = 0. 2 (i) In the case where L is the Lagrangian of the harmonic map, this result was first obtained in [134]. (ii) The vector field J defined over Ω by J α = ρ(x) ∂L (x, u, du)X i (u) ∂Aiα is often called the Noether current by physicists. 30) are called conservation laws.

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Harmonic maps, conservation laws, and moving frames by Frédéric Hélein

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