Reeb chords of Lagrangian slices (Q6601735)

This paper introduces the notion of Lagrangian slices in a given contact manifold \(Y\), which are submanifolds \(\Lambda\) that are not necessarily Legendrian, but come from transversal intersection of some non-cylindrical Lagrangian embedding of the product \(\Lambda\times (1-\varepsilon, 1+\varepsilon)\) with \(Y\) in its sympletization \(Y\times\mathbb{R}\). The author studies some basic properties of Lagrangian slices with relations to existence of Reeb chords, one of which is a Viterbo-type result: if \(Y\) admits a filling by subcritical Weinstein manifold and \(\Lambda\) has an exact Lagrangian filling inside, then \(\Lambda\) admits a Reeb chord in \(Y\).\N\NI am not completely convinced by this paper that the notion of Lagrangian slices is interesting enough for people to study, at least before any sorts of Floer theory can be established on this type of submanifolds. One question that might be interesting to me is, how effective this kind of generalization of Legendrian submanifolds is for us to study flexibility properties of say, Weinstein fillings? I hope the author can reach some results concerning these type of questions in the future work.\N\NFor the entire collection see [Zbl 1515.53004].