Remarks on the oscillation energy of Legendrian isotopies

Exact Lagrangians in four‐dimensional symplectisations

“…In this article, we show that closed exact Lagrangians exist also in four‐dimensional symplectisations of overtwisted contact three‐manifolds. In particular, this answers [2, Question 1] in recent work by Cant.…”

“…The symplectisation of a contact manifold is an open symplectic manifold, and, further, due to its concave cylindrical end, it does not have bounded geometry at infinity in the sense of [22]. Certain symplectisations can still be embedded in symplectic manifolds that have bounded geometry at infinity, but this is not the case for symplectisations of overtwisted contact manifolds; see, for example, work [14] by Eliashberg for closed contact manifolds and more recent work by Cant [2] for open contact manifolds. Hence, there is no reason to expect that Floer homology is well behaved in these symplectisations.…”