By Casim Abbas

This ebook offers an creation to symplectic box idea, a brand new and critical topic that's presently being built. the start line of this conception are compactness effects for holomorphic curves demonstrated within the final decade. the writer provides a scientific advent supplying loads of heritage fabric, a lot of that's scattered through the literature. because the content material grew out of lectures given by way of the writer, the most goal 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 real by means of the experts yet haven't but been released within the literature intimately, fill up the scope of this monograph.

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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 .

Assume that S is a surface equipped with a hyperbolic metric h such that all components of ∂S are geodesics. Let S be another copy of S (equipped with the same metric h), and let ι : S → S be the map3 x → x. We define the doubled surface S d by S d := S ∪i S = S ∪ S / x ∼ ι(x) | x ∈ ∂S . Let ϕα : S ⊃ Uα → Br+ (i) ⊂ H + be a local coordinate chart near a boundary point x where Br+ (i) := {z ∈ H + | Re(z) ≥ 0, |z − i| < r}. 15 we can assume that ϕα is an isometry if we equip H + with the usual hyperbolic metric.

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 .