By Casim Abbas
This booklet presents an advent to symplectic box concept, a brand new and demanding topic that's presently being constructed. the start line of this thought are compactness effects for holomorphic curves proven within the final decade. the writer provides a scientific creation delivering loads of history fabric, a lot of that's scattered during the literature. because the content material grew out of lectures given through the writer, the most objective is to supply an access element into symplectic box concept for non-specialists and for graduate scholars. Extensions of convinced compactness effects, that are believed to be real through the experts yet haven't but been released within the literature intimately, replenish the scope of this monograph.
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Extra resources for An Introduction to Compactness Results in Symplectic Field Theory
54 1 Riemann Surfaces Fig. 20 A good pairs of pants decomposition Since every connected stable surface of signature (g, m, n) can be assembled from 2g − 2 + m + n pairs of pants, its area equals areah (S) = 2π(2g − 2 + m + n). 53 is an important milestone towards a compactness theorem for stable Riemannian surfaces. e. it extends over the punctures. Then we can associate to j a unique hyperbolic metric h with finite area. We now know that S can be decomposed isometrically into 2g − 2 + m + n pairs of pants.
22: If h1 , h2 were two hyperbolic metrics on S inducing the complex structure j then we would get two hyperbolic metrics hd1 , hd2 on the double inducing the same complex structure j d . 22 then asserts that the two metrics on S d are identical, therefore h1 ≡ h2 . Assume now that j is a complex structure on S which is of finite type. Recalling the definition, this means that there is a finite extension i : S → Σ such that i∗ j extends to a complex structure jΣ on Σ . Let p ∈ Σ\S. A suitable neighborhood U ⊂ Σ of p is conformally equivalent to the disk D ⊂ C with the standard complex structure i.
If p : S ∪ S ′ → S d is the projection onto the quotient then we get a chart ψα : p −1 (Uα ) → Br (i) ⊂ H + as follows: ψα p(x) := ϕα (x) ϕα′ (x) if x ∈ Uα if x ∈ Uα′ . With f = ϕα ◦ ϕβ−1 : Br+ (i) → Br+ (i) the transition map between coordinate charts ψα , ψβ is of the form (s, t) → f (s, t) −f (−s, t) if s ≥ 0 if s < 0. In general, this is not a smooth map, but it is if f is holomorphic. In our case, f is a local isometry. If we choose an orientation σ on S, the opposite orientation on S ′ so that f is orientation preserving then it is also holomorphic (see below exercise).