By Casim Abbas

This ebook offers an creation to symplectic box conception, a brand new and critical topic that's at present being constructed. the place to begin of this idea are compactness effects for holomorphic curves demonstrated within the final decade. the writer provides a scientific advent delivering loads of heritage fabric, a lot of that's scattered in the course of the literature. because the content material grew out of lectures given via the writer, the most objective is to supply an access aspect into symplectic box thought for non-specialists and for graduate scholars. Extensions of sure 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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**Additional info for An Introduction to Compactness Results in Symplectic Field Theory**

**Example text**

Then we have the following commutative diagram where all the maps are local isometries: (V˜ , gH + ) −−−−→ (H + , gH + ) ⏐ ⏐ ⏐ ⏐ π (V˜ , gH + ) ⏐ ⏐ q π˜ ψ A := (V˜ /DV˜ (π), ψ ∗ h) −−−−→ (V \{p}, h) −−−−→ (S, h) We have H := q∗ π1 (V˜ ) ⊂ π1 (A) ≈ Z, and H is as a non-trivial subgroup of an infinite cyclic group also infinite cyclic. Then DV˜ (π) is isomorphic to N (H )/H where N (H ) denotes the normalizer of H in π1 (A)4 which then equals π1 (A). Hence there are two possibilities for the 4 Let H be a subgroup of a group G.

55 and let π : H → S ∗ be the (Riemannian) universal covering map. 2 Riemann Surfaces and Hyperbolic Geometry 41 Fig. 14 A hyperbolic flare and filling the gap between two strips component of π −1 (S) ⊂ H. Since π is a local isometry S˜ is a connected domain in H with piecewise geodesic boundary, and all interior angles between geodesic boundary arcs are ≤π . 57 A closed subset C ⊂ H is convex if and only if it is connected and locally convex. (You should try to prove this as an exercise. First show that it is sufficient to prove the lemma for the Euclidean plane instead of the hyperbolic plane using the correspondence by the hyperbolic plane and the upper half plane model, see Sect.

Then the isometry T : z → aa z (with = log(a /a) to be consistent with our previous notation) maps δ onto itself, and it satisfies T (γ (t)) = γ (t). We obtain a hyperbolic surface (see Fig. 6) C by identifying γ (t) with γ (t). 2 Riemann Surfaces and Hyperbolic Geometry 29 Fig. 5 Constructing a hyperbolic cylinder Fig. 6 Hyperbolic cylinder a closed geodesic in C of length . We will identify C with H/Γ , where Γ = {T k | k ∈ Z} ⊂ I. Another description of the hyperbolic cylinder can be obtained by using Fermi coordinates.