Some non-collarable slices of Lagrangian surfaces

TL;DR

This paper introduces the concept of collarable slices of Lagrangian submanifolds, providing examples of non-collarable slices in ^2, advancing understanding of Lagrangian-contact hypersurface interactions.

Contribution

It defines collarable slices of Lagrangian submanifolds and presents explicit examples of non-collarable slices in ^2, highlighting new phenomena in symplectic topology.

Findings

Defined collarable slices of Lagrangian submanifolds.

Constructed explicit examples of non-collarable Lagrangian slices.

Showed these slices are transverse to contact hypersurfaces.

Abstract

In this note we define the notion of collarable slices of Lagrangian submanifolds. Those 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 $\mathbb{C}^2$ transverse to $S^3$ whose slices are non-collarable.