We present a new method to prove transversality for holomorphic curves in symplectic manifolds, and show how it leads to a definition of genus zero Gromov-Witten invariants. The main idea is to introduce additional marked points that are mapped to a symplectic hypersurface of high degree in order to stabilize the domains of holomorphic maps.
We study the coherent orientations of the moduli spaces of holomorphic curves in Symplectic Field Theory, generalizing a construction due to Floer and Hofer. In particular we examine their behavior at multiple closed Reeb orbits under change of the asymptotic direction. The orientations are determined by a certain choice of orientation at each closed Reeb orbit, that is similar to the orientation of the unstable tangent spaces of critical points in finite-dimensional Morse theory.
Bourgeois, Eliashberg, Hofer, Wysocki and Zehnder recently proved a general compactness result for moduli spaces of punctured holomorphic curves arising in symplectic field theory. In this paper we present an alternative proof of this result. The main idea is to determine a priori the levels at which holomorphic curves split, thus reducing the proof to two separate cases: long cylinders of small area, and regions with compact image. The second case requires a generalization of Gromov compactness for holomorphic curves with free boundary.
We use a neck stretching argument for holomorphic curves to produce symplectic disks of small area and Maslov class with boundary on Lagrangian submanifolds of nonpositive curvature. Applications include the proof of Audin's conjecture on the Maslov class of Lagrangian tori in linear symplectic space, the construction of a new symplectic capacity, obstructions to Lagrangian embeddings into uniruled symplectic manifolds, a quantitative version of Arnold's chord conjecture, and estimates on the size of Weinstein neighbourhoods. The main technical ingredient is transversality for the relevant moduli spaces of punctured holomorphic curves with tangency conditions.
Abstract. We prove ( a weak version of) Arnold's Chord Conjecture in [2] using Gromov's "classical" idea in [9] to produce holomorphic disks with boundary on a Lagrangian submanifold.Arnold's Chord Conjecture. In this paper we prove the following theorem which was conjectured by Arnold Theorem 1 will follow as a corollary from the main result of this paper, Theorem 4. In fact it can be applied to a more general situation: Our results include the existence of chords for Legendrians in the standard contact structure on RP 2n−1 proved by Ginzburg and Givental [8,7], although it does not provide their statement of linear growth. They cover results by Abbas [1] and Cieliebak [6] who treat subcases of the problem on the sphere and on boundaries of subcritical Stein manifolds.Lagrangian out of Legendrian embeddings. Consider a closed Legendrian submanifold l ⊂ M 2n−1 in a contact manifold. Given a contact one form α we will construct Lagrangian embeddings of the torus l × S 1 into the symplectization (M × R, d(e s α)) of (M, ξ) and study them. Denote by φ the flow of the Reeb vector field R = R α .Assume l has no Reeb chords of length at most T > 0. Then there is an embeddingα)) = e s ds ∧ dt, (s, t) ∈ R × [0, T ] being the coordinates of the infinite strip. The map is, of course, constructed using the Reeb flow: Φ(x, s, t) := (φ t (x), s).
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