Some non-collarable slices of Lagrangian surfaces

Abstract

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 C 2 transverse to S 3 whose slices are non-collarable.

Topologically distinct Lagrangian and symplectic fillings

“…Our main geometric tool for constructing Lagrangian fillings, embodied by the theorem below, was first announced by Ekholm, Honda, and Kálmán [11]. The first part of the theorem was proven by Chantraine [6], and the last two parts are formulated as in [5]; see also Rizell's work [35]. 6,11,35]).…”

“…The first part of the theorem was proven by Chantraine [6], and the last two parts are formulated as in [5]; see also Rizell's work [35]. 6,11,35]). If two oriented Legendrian links Λ − and Λ + in the standard contact R 3 are related by any of the following three moves, then there exists an oriented exact Lagrangian cobordism Λ − ≺ L Λ + .…”

“…The first realization of this principle was Eliashberg's proof that any symplectically aspherical filling of the standard contact S 3 is diffeomorphic to B 4 [13]; see also [27,30] for similar results in higher dimensions and [26,31,32,43,45] in low dimensions. In the relative setting, the original principle was first realized by a theorem of Chantraine [6]: if L is an orientable Lagrangian filling of an oriented Legendrian link in the boundary of a Stein surface, then (1.1) tb(Λ) = −χ(L).…”

A note on exact Lagrangian cobordisms with disconnected Legendrian ends

“…In order to construct the explicit examples, we uses elementary cobordisms as defined in [2] together with an additional elementary Lagrangian immersion provided by the following theorem:…”

“…In Section 3 we illustrate explicitly the necessity of (3) of the definition of exact Lagrangian cobordism and prove Proposition 1.1. We also provide, similar to [6,Question 8.10], an example of a non-decomposable Lagrangian cobordism as defined in [2].…”