By Casim Abbas

ISBN-10: 3642315429

ISBN-13: 9783642315428

ISBN-10: 3642315437

ISBN-13: 9783642315435

This e-book offers an creation to symplectic box thought, a brand new and significant topic that is presently being constructed. the start line of this thought are compactness effects for holomorphic curves confirmed within the final decade. the writer offers a scientific creation offering 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 goal is to supply an access aspect into symplectic box concept for non-specialists and for graduate scholars. Extensions of definite compactness effects, that are believed to be real by means of the experts yet haven't but been released within the literature intimately, replenish the scope of this monograph.

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**Sample text**

48 The surface Y together with its complex structure and its hyperbolic metric is called a pair of pants (see Fig. 10). 49 According to the above definition a pair of pants is a surface constructed by gluing together two hexagons (as described above) together with the obvious hyperbolic metric and complex structure. Later we will show that any surface diffeomorphic to a sphere with three punctures and equipped with a hyperbolic metric of finite area can be obtained by the above construction. If we repeat the same identification with the closures of the two hexagons then we obtain a compact topological space Y .

0m , m ∈ Z\{−1, 0, +1} Let α, ˜ Γα (z) = e z, γ , γ˜ as before. We will first prove the following claim: Claim Assume α is simple. Let σ be a covering transformation of the universal cover π : H → S such that σ (α(R)) ˜ = α(R). ˜ Then σ α(R) ˜ ∩ α(R) ˜ = ∅. We will assume that, σ (α(R)) ˜ ∩ α(R) ˜ = ∅ and we will show that this either implies that α is not simple or that σ (α(R)) ˜ = α(R). ˜ Let t0 , s ∈ R such that σ α(t ˜ 0 + s) = α(t ˜ 0 ). 50 1 Riemann Surfaces Fig. 18 Proving the claim on page 49 The covering transformation Γα is uniquely determined by the equation Γα (α(t)) ˜ = α(t ˜ + 1) or Γα−k (α(t ˜ + k)) = α(t) ˜ for any k ∈ Z.

39. 43? We will explicitly determine its conformal class. The cylinder Sˆ can be described alternatively as Sˆ = H ++ /{z ∼ 2z} where H ++ := {z ∈ H + | Re(z) > 0} which we equip with the standard complex structure i and the metric gH ++ = Im(z)−2 geucl and which is not complete. The cylinder Sˆ inherits a metric and a complex structure from H ++ because the transformation z → 2z is at the same time holomorphic and an isometry. The projection ˆ The transformaH ++ → Sˆ is a covering map, and H ++ is the universal cover of S.

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

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