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

24 February 2017 adminTopology

By Frédéric Hélein

This available creation to harmonic map concept and its analytical points, covers contemporary advancements within the regularity concept of weakly harmonic maps. The publication starts off through introducing those strategies, stressing the interaction among geometry, the position of symmetries and vulnerable options. It then offers a guided journey into the idea of thoroughly integrable structures for harmonic maps, via chapters dedicated to contemporary effects at the regularity of susceptible suggestions. A presentation of "exotic" practical 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 extra functions and illustrations of Coulomb relocating frames to the speculation of surfaces.

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Additional resources for Harmonic maps, conservation laws, and moving frames

Sample text

Hence, we notice the invariance of the Dirichlet integral under the action of the diffeomorphism group Diff(Ω) = {Φ ∈ C 1 (Ω, Ω)| Φ is invertible and Φ−1 ∈ C 1 (Ω, Ω)} on the pairs (g, u). We expect to find conservation laws. Nevertheless, two remarkable differences should be pointed out, when compared to the classical setting of Noether’s theorem (like the one we saw before). First, the group Diff(Ω) is infinite dimensional. Next, the Lagrangian of the harmonic map is not exactly invariant under this group action, to the extent that we need to change the metric g, and thus the Lagrangian L, at the same time as we change the map u, in order to obtain the invariance.

The first thing to check is whether Hg1 (M, N ) = ∅. Some obstructions may show up. 6 M = B 2 = {(x, y) ∈ R2 | x2 + y 2 < 1}, N = S 1 = {(x, y) ∈ R2 | x2 + y 2 = 1} = ∂B 2 , and g(x, y) = (x, y). Then, there is no finite energy extension of g in Hg1 (B 2 , S 1 ). The reason is a topological obstruction. For each C 1 map g : S 1 −→ S 1 (in 1 fact it would suffice to suppose that g ∈ H 2 (S 1 )), the following quantity is called the topological degree (or winding number): deg(g) = 1 2π ∂B 2 g 1 dg 2 − g 2 dg 1 .

E. for any vector fields X, Y , and for any vector Z in Ty N , d(hy (X, Y ))(Z) = hy (∇Z X, Y ) + hy (X, ∇Z Y ). e. for any vector fields X, Y , ∇X Y − ∇Y X − [X, Y ] = 0 . ∇ is called the Levi-Civita connection. Let (y 1 , . . , y n ) be a local coordinate system on N , and hij (y) the coefficients of the metric h in these coordinates. 12) are the Christoffel symbols. Let u : M −→ N be a smooth map. 1 u is a harmonic map from (M, g) to (N , h) if and only if u satisfies at each point x in M the equation ∆g ui + g αβ (x)Γijk (u(x)) ∂uj ∂uk = 0.

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