By Casim Abbas
This booklet presents an advent to symplectic box concept, a brand new and demanding topic that is at present being built. the place to begin of this conception are compactness effects for holomorphic curves validated within the final decade. the writer provides a scientific creation delivering loads of history fabric, a lot of that's scattered in the course of the literature. because the content material grew out of lectures given via the writer, the most target is to supply an access element into symplectic box concept for non-specialists and for graduate scholars. Extensions of sure compactness effects, that are believed to be real by way of the experts yet haven't but been released within the literature intimately, fill up the scope of this monograph.
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Extra resources for An Introduction to Compactness Results in Symplectic Field Theory
The case where ∂S = ∅ then simply follows from taking the double of S. 65. The statement can be refined as follows: The corresponding geodesics γ1 and γ2 agree as point sets if and only if there are nonzero integers k1 , k2 such that α k1 and α k2 are freely homotopic. Assume that γ1 (R) = γ2 (R). Then there are nonzero integers k1 , k2 such that γ1k1 and γ2k2 are freely homotopic. Then so are α k1 and α k2 . If α k1 and α k2 are freely homotopic then we proceed as in the uniqueness proof above and lift the homotopy to the universal cover.
Then the map ϕ : R2 −→ H (t, s) −→ expγ (t) s i γ˙ (t) c is a diffeomorphism. We have ϕ ∗ gH + ∂ ∂ , ∂s ∂s ϕ ∗ gH + ≡ 1, ∂ ∂ , ∂t ∂t = c2 cosh2 s and ϕ ∗ gH + ∂ ∂ , ∂s ∂t ≡ 0. See the following remark for the computations. 34 It is sufficient to do the calculation for the geodesic γ (t) = iet because any other geodesic can be written as φ ◦ γ for suitable φ ∈ Conf(H). Find all conformal isomorphisms φ such that φ(i) = γ (t) = iet . We obtain φ(z) = (az + b)/(cz + d) with b = −et/2 cos θ a = et/2 sin θ, and c = e−t/2 cos θ, d = e−t/2 sin θ.
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. 17): (1) α(t ˜ + 1) = α(t) ˜ ±1 ˜ (2) α(t ˜ + 1) = e α(t). Our first objective is to show that case (1) cannot occur.