An Introduction to Compactness Results in Symplectic Field by Casim Abbas

By Casim Abbas

This ebook offers an advent to symplectic box idea, a brand new and significant 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 offering 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 target is to supply an access element into symplectic box conception for non-specialists and for graduate scholars. Extensions of yes compactness effects, that are believed to be actual by way of the experts yet haven't but been released within the literature intimately, replenish the scope of this monograph.

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Any lift α˜ : [0, 1] → H of the loop α satisfies α(0) ˜ = α(1). 28. If we view α and α˜ as curves defined on R such that α(0) = α(1) then we have for all t ˜ . α(t ˜ + 1) = Γα α(t) So there is φ ∈ Conf(H) such that φΓα φ −1 either equals P ± : z → z ± 1 or T : z → e z for some > 0. Replacing the universal cover π with the universal ˜ we may assume that Γα equals one of cover π ◦ φ −1 and replacing α˜ with φ(α) these standard isometries. Hence we have to consider the following two cases (see Fig.

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. 8). The metric gH + on the upper half plane induces a hyperbolic metric on any hexagon. With respect to this metric hexagons always have finite area. 3, [74] pp. 83–85). Consider now a hexagon G ⊂ H with sides a1 , b1 , a2 , b2 , a3 , b3 parameterized on the unit interval [0, 1]. Let now G := {x + iy ∈ C | x − iy ∈ G} be a copy of G in the negative half-plane H − .

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).

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