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

Mohnke, Klaus

Ann. Math. (2) 154, No. 1, 219-222 (2001).

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)].

Reviewer: Edith Padron (La Laguna)

MSC:

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

Citations:

Zbl 0649.58010; Zbl 0592.53025; Zbl 0618.58021