An Introduction to Compactness Results in Symplectic Field by Casim Abbas

By Casim Abbas

This ebook presents an advent to symplectic box idea, a brand new and significant topic that is at present being built. the place to begin of this thought are compactness effects for holomorphic curves validated within the final decade. the writer offers a scientific creation 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 goal is to supply an access aspect into symplectic box concept for non-specialists and for graduate scholars. Extensions of sure compactness effects, that are believed to be precise via the experts yet haven't but been released within the literature intimately, replenish the scope of this monograph.

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After choosing isothermal coordinates near the maximum x0 of β, assume that β(x0 ) > 0. Then eβ(x0 ) − 1 > 0 and also eβ − 1 > 0 on a sufficiently small ball B centered at x0 . Since κ ≤ 0 we obtain β ≥ 0 on B. The maximum principle [29] implies now that the function β cannot have an interior 24 1 Riemann Surfaces maximum on B unless it is constant. Because x0 was an absolute maximum for β we get β|B ≡ β(x0 ). The set x0 ∈ S | β(x0 ) = sup β(z) > 0 z∈S is closed and also open by our reasoning above.

Let δ, δ be two shortest homotopic geodesics connecting Γ0 with Γ1 . Using a similar argument as before, δ and δ do not intersect, and there is a geodesic quadrilateral Q in S with sides δ, δ and two subarcs of Γ0 and Γ1 , respectively. 56 Q is isometric to a right angled convex polygon in H with four vertices, but such a polygon does not exist. 60 Let Q ⊂ H be a geodesic polygon with k vertices and interior angles α1 , . . , αk . Then − Q i dz ∧ d z¯ = 2y 2 k αi + (2 − k)π. 61 Let S be a surface diffeomorphic to a nondegenerate pair of pants equipped with a hyperbolic metric h so that ∂S consists of closed geodesics.

R The first two assertions are trivial, the third is not much better. In fact, identifying R2 with C the exponential map yields a conformal transformation from the strip (0, π) × R to H. Now (0, r) × S 1 = (0, r) × R/2πZ is conformally equivalent 2 to (0, π) × R/ 2πr Z via (s, t) → πr (s, t), and the exponential map transforms the (0, r) × S 1 ≈ H/Γ with = 2π 2 r Z action on the second factor of (0, π) × R into a T2π 2 /r action on the hyperbolic plane H. 46 Let A be an annulus such that A\∂A is conformally equivalent to (0, r) × S 1 for some 0 < r ≤ +∞.

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