By Casim Abbas

ISBN-10: 3642315437

ISBN-13: 9783642315435

This booklet offers an advent to symplectic box idea, a brand new and critical topic that's at present being constructed. the place to begin of this idea are compactness effects for holomorphic curves verified within the final decade. the writer offers a scientific creation supplying loads of heritage fabric, a lot of that is scattered in the course of the literature. because the content material grew out of lectures given by means of the writer, the most objective is to supply an access aspect into symplectic box conception for non-specialists and for graduate scholars. Extensions of convinced compactness effects, that are believed to be real by way of the experts yet haven't but been released within the literature intimately, refill the scope of this monograph.

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**Extra info for An Introduction to Compactness Results in Symplectic Field Theory**

**Example text**

53. We know already that a stable surface is diffeomorphic to a surface obtained by gluing together pairs of pants. 48. e. the lengths of the boundary geodesics of the pants together with the twist parameters, determine the surface up to isometry. 20 the gluing data (also called Fenchel–Nielsen parameters) then also determines the complex structures of finite type on the surface S. Summarizing our efforts so far, we have managed to describe the space J (S)finite / Diff+ (S) by a finite set of real parameters (although the correspondence between the Riemann moduli space and the Fenchel–Nielsen parameters is not oneto-one if we view the twist parameters as real numbers vs.

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 . Then the shortest curve between Γ0 and δ(ε) is the geodesic arc intersecting Γ0 orthogonally and passing through δ(ε). The curve δ has no self-intersections. Indeed, if we had 0 < t0 < t1 < 1 with δ(t0 ) = δ(t1 ) then the continuous curve δ|[0,t0 ] ∪ δ|[t1 ,1] would connect Γ0 with Γ1 and it is shorter than δ, a contradiction.

E. can be extended over the punctures. 63 a neighborhood of a puncture is isometric to a standard cusp. e. the complex structure extends. 65 (Straightening closed curves) Let S be a stable surface, and let h be a hyperbolic metric of finite area. If ∂S ̸= ∅, we assume that ∂S consists of closed geodesics. Assume that α : [0, 1] → S\∂S, α(0) = α(1), is a closed curve that V˜ is a connected component of the set π −1 (V \{p}) which is closed in H, hence V˜ is itself closed in H. 6 Then the free homotopy class of α contains a closed geodesic γ which is unique up to reparametrization and which is either disjoint from ∂S or contained in ∂S.

### An Introduction to Compactness Results in Symplectic Field Theory by Casim Abbas

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