By Casim Abbas
This publication offers an creation to symplectic box conception, a brand new and significant topic that's at present being constructed. the place to begin of this conception are compactness effects for holomorphic curves verified within the final decade. the writer offers a scientific advent delivering loads of history fabric, a lot of that's scattered through the literature. because the content material grew out of lectures given through the writer, the most objective is to supply an access aspect into symplectic box thought for non-specialists and for graduate scholars. Extensions of convinced compactness effects, that are believed to be precise by way of the experts yet haven't but been released within the literature intimately, refill the scope of this monograph.
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Extra info for An Introduction to Compactness Results in Symplectic Field Theory
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 (, , ), 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 . 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 . (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.
An Introduction to Compactness Results in Symplectic Field Theory by Casim Abbas