Some non-collarable slices of Lagrangian surfaces

In this note, we define the notion of collarable slices of Lagrangian submanifolds. These are slices of Lagrangian submanifolds which can be isotoped through Lagrangian submanifolds to a cylinder over a Legendrian embedding near a contact hypersurface. Such a notion arises naturally when studying intersections of Lagrangian submanifolds with contact hypersurfaces. We then give two explicit examples of Lagrangian disks in ℂ2 transverse to S3 whose slices are non-collarable.