Download An Introduction to Compactness Results in Symplectic Field by Casim Abbas PDF

By Casim Abbas

This booklet offers an creation to symplectic box conception, a brand new and demanding topic that is presently being built. the place to begin of this thought are compactness effects for holomorphic curves demonstrated within the final decade. the writer offers a scientific advent 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 objective is to supply an access element into symplectic box idea for non-specialists and for graduate scholars. Extensions of yes compactness effects, that are believed to be precise 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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Extra resources for An Introduction to Compactness Results in Symplectic Field Theory

Example text

We obtain φ(z) = (az + b)/(cz + d) with a = et/2 sin θ, b = −et/2 cos θ and c = e−t/2 cos θ, d = e−t/2 sin θ. Then we obtain a geodesic δ(s) = φ γ (s) = and we compute ˙ = δ(0) ies+t sin θ − et cos θ ies cos θ + sin θ −iet . (cos θ − i sin θ )2 Fermi coordinates near the geodesic γ are then given by ϕ : R2 −→ H (t, s) −→ expiet −set . The curve δ0 : s → expiet (−set ) is the geodesic for which δ0 (0) = iet and δ˙0 (0) = −et . If we choose θ = 3π/4 so that e2iθ = −i then δ ≡ δ0 . Simplifying, we then obtain ϕ(t, s) = et + ies+t = −et tanh s + iet sech s 1 − ies 26 1 Riemann Surfaces and Dϕ(t, s) = et − tanh s sech s − sech2 s .

Z) = eℓ/m z. Consider now the curves α˜ and φ(α). ˜ We claim that they intersect. e. ϕ : (s, t) −→ expγ˜ (t) s i γ˜ ′ (t) ℓ = eℓt (− tanh s + i sech s) after a similar calculation as on page 25. In these coordinates, the transformation φ is given by ϕ −1 ◦ φ ◦ ϕ (s, t) = s, t + 1 . m If we define then α˜ = (α1 , α2 ) := ϕ −1 ◦ α˜ β˜ := ϕ −1 ◦ φ ◦ α˜ = α1 , α2 + 1 . m From α(t ˜ + 1) = Γα (α(t)) ˜ we conclude that α1 (τ + 1) = α1 (τ ) and α2 (τ + 1) = α2 (τ ) + 1 and therefore lim α2 (τ ) = ±∞.

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. Let C : H + → H +, C(z) = −¯z which is an orientation reversing isometry. We define now a chart near the point ι(x) ∈ S ′ by ϕα′ ⊃ Uα′ → Br− (i) := z ∈ H + | Re(z) ≤ 0, |z − i| < r ϕα′ z′ := C ϕα ι−1 z′ .

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