By Casim Abbas

This ebook offers an advent to symplectic box idea, a brand new and critical topic that's presently being constructed. the start line of this conception are compactness effects for holomorphic curves confirmed within the final decade. the writer provides a scientific advent delivering loads of historical past fabric, a lot of that is scattered during the literature. because the content material grew out of lectures given by way of the writer, the most target is to supply an access element into symplectic box idea for non-specialists and for graduate scholars. Extensions of sure compactness effects, that are believed to be real through 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

Example text

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. Then we obtain a geodesic δ(s) = φ γ (s) = ies+t sin θ − et cos θ ies cos θ + sin θ and we compute ˙ = δ(0) −iet . (cos θ − i sin θ )2 Fermi coordinates near the geodesic γ are then given by ϕ : R2 −→ H (t, s) −→ expiet −set .

This does not change the fact that we are working with a minimal sequence. After passing to a suitable subsequence, for each j the points pj,n will converge to points pj , and the geodesic segments will converge as well to a geodesic segments between the points pj and pj +1 . Denote the resulting curve between p1 and pm by δ. Since δ minimizes the length among all curves from p1 to pm it must be a geodesic. In particular, it must be smooth. Moreover, it must intersect Γ0 and Γ1 orthogonally. In order to see this, use Fermi coordinates near Γ0 and Γ0 .

If a component Γ of the boundary contains a geodesic arc of length then we attach to it a strip {(x, y) ∈ [0, ∞) × [0, ]} equipped with the hyperbolic metric ds 2 = dx 2 + cosh2 x dy 2 (compare with the Fermi coordinates discussed earlier). After attaching these ‘Fermi strips’ we attach at each vertex p with interior angle α ≤ π an infinite circle sector {p} ∪ ((0, ∞) × [0, π − α]) equipped with the hyperbolic metric ds 2 = dρ 2 + sinh2 ρ dσ 2 , where 0 < ρ < ∞, 0 ≤ σ ≤ π − α. Verify as an exercise that the surface S ∗ obtained this way carries a complete metric.