By Casim Abbas

ISBN-10: 3642315429

ISBN-13: 9783642315428

ISBN-10: 3642315437

ISBN-13: 9783642315435

This publication presents an creation to symplectic box conception, a brand new and demanding topic that's presently being built. the start line of this idea are compactness effects for holomorphic curves confirmed within the final decade. the writer provides a scientific creation offering loads of historical past fabric, a lot of that is scattered in the course of 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 idea 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, stock up the scope of this monograph.

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**Extra info for An Introduction to Compactness Results in Symplectic Field Theory**

**Example text**

E. there is a homotopy h(s, t) with h(0, t) = δ(t), h(1, t) = δ (t), h(0, t) ∈ Γ0 and h(1, t) ∈ Γ1 then δ ≡ δ . Moreover, if δ is a shortest geodesic arc connecting two other boundary components Γ0 and Γ1 where Γ1 = Γ1 then δ and δ do not intersect. 59 If all curves connecting Γ0 with Γ1 are homotopic then δ is indeed unique. This is the case if S is a surface diffeomorphic to a pair of pants. 15 42 1 Riemann Surfaces Fig. 15 Two different shortest geodesics indicates that uniqueness will not hold if there are several homotopy classes of paths connecting Γ0 and Γ1 .

Assume that S → Σ is a finite extension, and let p ∈ Σ\S be a puncture. Then there exist an open neighborhood U ⊂ Σ of p and an isometric embedding ϕ : C −→ U \{p}, h , where C is a cusp C := z ∈ H | Im(z) ≥ K > 0 /{z ∼ z + 1} equipped with the hyperbolic metric induced from H. 2 Riemann Surfaces and Hyperbolic Geometry 45 Proof By doubling the surface we may assume without loss of generality that S has no boundary and that π : (H, gH + , i) −→ (S, h, j ) is the universal covering, where π is a local isometry and holomorphic.

4 Annuli We discuss now Riemann surfaces A which are diffeomorphic to (0, 1) × S 1 (open annuli or cylinders). 40 (Hyperbolic cylinders) Consider the geodesic δ(t) = iet in H. Let now γ and γ be geodesics intersecting δ orthogonally at the points ia and ia , respectively, where a < a (see Fig. 5). Parameterize γ and γ with unit speed and with orientations as indicated in Fig. 5. 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).

### An Introduction to Compactness Results in Symplectic Field Theory by Casim Abbas

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