A note on Lagrangian cobordisms between Legendrian submanifolds of ℝ²ⁿ⁺¹

Abstract

We study the relation of an embedded Lagrangian cobordism between two closed, orientable Legendrian submanifolds of R 2n+1 . More precisely, we investigate the behavior of the Thurston-Bennequin number and (linearized) Legendrian contact homology under this relation. The result about the Thurston-Bennequin number can be considered as a generalization of the result of Chantraine which holds when n = 1. In addition, we provide a few constructions of Lagrangian cobordisms and prove that there are infinitely many pairs of exact Lagrangian cobordant and not pairwise Legendrian isotopic Legendrian n-tori in R 2n+1 .

A note on the front spinning construction

“…T 2k+1 for k > j, see the proof of Proposition 1.6 in [16] or Section 8.1 in [12]. It is easy to see that tb(T 2k+1 ) = 2k − 1.…”

“…Here we prove Proposition 1.3 by mimicking the proof of Proposition 1.5 from [16]. Given two closed, orientable Legendrian submanifolds Λ − , Λ + ⊂ R 2n+1 such that…”

“…Note that from the proof of Proposition 1.4 in [16] it follows that there are exact Lagrangian cobordisms L…”

“…From the fact that Legendrian isotopy implies that there exists an exact Lagrangian cylinder, see Proposition 1.4 in [16], it follows that H geom Λ is a Legendrian invariant. Note that Formula 4.3 holds for every exact Lagrangian filling L Λ + of Λ + obtained by gluing the positive end of an exact Lagrangian filling L Λ − to the negative end of L.…”

“…We now recall that tb(Λ) = (−1) n/2+1 χ(L Λ ) i fn is even; (−1) (n−2)(n−1)/2+1 χ(L Λ ) if n is odd, (5.2) see [15,Remark 3.5]. Formulas (5.1) and (5.2) imply that…”