Some non-collarable slices of Lagrangian surfaces (Q2918790)

Let \(Y\times\mathbb{R}\) be the symplectization of a contact manifold \(Y\) and \(L\subset Y\times\mathbb{R}\) a Lagrangian submanifold. The author calls a transverse intersection \(L\cap Y\times\left\{t\right\}\) collarable if it can be isotoped through Lagrangian submanifolds to a cylinder over a Legendrian embedding near \(Y\times\left\{t\right\}\). Such a notion arises naturally (via the trivialization through the Liouville vector field) when studying intersections of Lagrangian submanifolds with contact hypersurfaces in sympectic manifolds.NEWLINENEWLINEThe paper under review gives two explicit examples of Lagrangian disks in \(\mathbb C^{2}\) transverse to \(S^{3}\) whose slices are non-collarable. The one example is a Lagrangian cobordism T from the trivial Legendrian knot \(K_0\subset S^3\) to the empty set with one global maximum \(m\) for the `distance from the origin' function such that, for any sufficiently small \(\delta > 0\), the slice \(T \cap S^{3}_{m-\delta}\) is non-collarable. The other example yields Lagrangian disks in \(D^4\) transverse to \(S^3\) whose intersections with \(S^3\) are non-collarable.