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Holomorphic disks and the chord conjecture. (English) Zbl 1007.53065

The author proves a weak version of Arnold’s Chord Conjecture [V. I. Arnold, Russ. Math. Surv. 41, No. 6, 1–21 (1986); translation from Usp. Mat. Nauk 41, No. 6, 3–8 (1986; Zbl 0618.58021)]. In fact, he obtains that for every closed Legendrian submanifold in the sphere \(S^{2n-1}\) with its standard contact structure and any contact form for this structure, there is an integral curve of the Reeb vector field which begins and ends on the Legendrian submanifold. The idea of the proof is based on Gromov’s method to produce holomorphic disks with boundary on a Lagrangian submanifold [M. Gromov, Invent. Math. 82, 307–347 (1985; Zbl 0592.53025)].

MSC:

53D35 Global theory of symplectic and contact manifolds
53D12 Lagrangian submanifolds; Maslov index
32Q35 Complex manifolds as subdomains of Euclidean space
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