Holomorphic Disks and the Chord Conjecture

Abstract

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).

Asymptotically linear Hamiltonian systems with Lagrangian boundary conditions

“…Arnold conjectured that, under some conditions, the Lagrangian intersection number has a lower bound estimated by the sum of all Beti numbers of the Lagrangian submanifold in the nondegenerate case; this sum is in turn estimated by the cup-length of the Lagrangian submanifold (see for example [Conley and Zehnder 1984;Hofer 1988;Floer 1988Floer , 1989Oh 1995;Ono 1996;Chekanov 1996Chekanov , 1998Liu 2005a]). For the Arnold chord conjecture, we mention [Arnold 1986;Mohnke 2001]. The multiplicity of the fixed energy problem (1-1) was studied in [Guo and Liu 2007].…”

Displacement energy of unit disk cotangent bundles

“…The proof is same as that of Theorem 4 in [14] and is omitted. Notice that we do not need a condition on π 1 (which is assumed in Theorem 4 [14]), since we are working on an exact symplectic manifold.…”

“…Our proof of Corollary 1.11 is completely different from the arguments in [16], and makes use of the following lemma, based on arguments of Mohnke [14]. Recall that for any contact manifold (Y, λ), its symplectization is Y × R >0 endowed with a 1-form λ(z, r) := rλ(z) (z ∈ Z, r ∈ R >0 ).…”

“…Notice that we do not need a condition on π 1 (which is assumed in Theorem 4 [14]), since we are working on an exact symplectic manifold. Now we prove Corollary 1.11 from Theorem 1.1 and Lemma 1.13:…”

Untitled

“…Arnold conjectured the existence of characteristic chords on the three dimensional sphere for any contact form inducing the standard contact structure and for any Legendrian knot [10]. After a partial result by the author in [3] this conjecture was finally confirmed by K. Mohnke in [25]. It is natural to ask the existence question for characteristic chords not only for M = S 3 , but also for general contact manifolds.…”