By Casim Abbas

This publication presents an advent to symplectic box idea, a brand new and demanding topic that is presently being constructed. the start line of this thought are compactness effects for holomorphic curves proven within the final decade. the writer offers a scientific creation delivering loads of history fabric, a lot of that is scattered through the literature. because the content material grew out of lectures given by means of the writer, the most goal is to supply an access element into symplectic box thought for non-specialists and for graduate scholars. Extensions of yes compactness effects, that are believed to be real via the experts yet haven't but been released within the literature intimately, fill up the scope of this monograph.

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

**Example text**

Then A is isometrically isomorphic to one of the following models: (1) C/Z with the metric induced by the Euclidean metric on C, (2) the parabolic cylinder or exactly one of the hyperbolic cylinders H/Γℓ with the induced metric from the hyperbolic plane. Another description of the open annuli is given by the following: We identify S 1 with R/2πZ, and we equip R × S 1 (and all cylinders contained therein) with the standard complex structure i. Then we have the following conformal equivalences: R × S 1 ≈ C/Z, and (0, ∞) × S 1 ≈ H/ P k | k ∈ Z 2π 2 .

C) 32 1 Riemann Surfaces Fig. 8 Hexagons in H. 5 Hexagons in the Upper Half Plane and Pairs of Pants Our aim is now to build stable hyperbolic surfaces from simple building blocks, the so-called pairs of pants. In the next section we will show that this process is reversible, any stable hyperbolic surface can be cut up into these elementary pieces. e. if p, q ∈ G then the geodesic arc connecting p with q is also contained in G. If one or more of the sides bk is replaced by a point on the real line or the point {∞} then G is called a degenerate hexagon (see Fig.

Hence a generator α of DV˜ (π) is conjugate by an isometry φ either to P ± : z → z ± 1 or to Tℓ : z → eℓ z. Since (P ± )k , Tℓk ̸= Id for any k ∈ Z, α cannot be of finite order. Using the isometry φ, the semi-closed annulus A is then isometric to the quotient φ(V˜ )/G where G is the infinite cyclic group generated by either P + or Tℓ with the induced metric from gH + . Moreover the set φ(V˜ ) is closed5 in H, simply connected and invariant under the group action. 16 shows what φ(V˜ ) typically looks like (the hyperbolic area of the fundamental domains indicated in the figure then equals the area of the annulus A).