Holomorphic disks and the Chord Conjecture

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.

In this paper we prove the following theorem which was conjectured by Arnold [2]:

THEOREM 1 (Arnold’s Chord Conjecture). For every closed Legendrian submanifold in $S^{2n-1}$ with the standard contact structure and any contact form for this structure, there is a Reeb chord, i.e. an integral curve of the Reeb vector field which begins and ends on the Legendrian submanifold.

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:

THEOREM 2. Let $(M,\xi)$ be a contact manifold which arises as smooth boundary of a compact subcritical Stein manifold (see [4] for a definition). Then for any closed Legendrian submanifold and any contact one-form corresponding to $\xi$ there is a Reeb chord.

Our results include the existence of chords for Legendrians in the standard contact structure on $\mathbb{R} P^{2n-1}$ 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 $l \subset M^{2n-1}$ in a contact manifold. Given a contact one-form $\alpha$ we will construct Lagrangian embeddings of the torus $l \times S^1$ into the symplectization $(M \times \mathbb{R},d(e^s\alpha))$ of $(M,\xi)$ and study them. Denote by $\phi$ the flow of the Reeb vector field $R = R_\alpha$.

DOI: 10.2307/3062116

Authors

Klaus Mohnke