A note on exact Lagrangian cobordisms with disconnected Legendrian ends

TL;DR

This paper presents specific examples of exact Lagrangian cobordisms with disconnected Legendrian ends, highlighting limitations in composition and properties of primitives on ends, using explicit Lagrangian immersions.

Contribution

It provides explicit constructions of Lagrangian immersions illustrating key properties and limitations of exact Lagrangian cobordisms with disconnected ends.

Findings

Two Lagrangian submanifolds cannot be composed exactly.

An example of an exact Lagrangian cobordism with non-constant primitive on the negative end.

Positive end is a stabilization, negative end admits augmentations.

Abstract

We provide in this note two relevant examples of Lagrangian cobordisms. The first one gives an example of two exact Lagrangian submanifolds which cannot be composed in an exact fashion. The second one is an example of an exact Lagrangian cobordism on which all primitive of the Liouville form is not constant on the negative end and such that the positive end is a stabilisation whereas the negative end admits augmentations. These examples emphasise point (i) of the definition of exact Lagrangian cobordisms in [6]. In order to provide such examples we construct Lagrangian immersions with single double points using an explicit model and interpret such Lagrangians as cobordisms from the Hopf link.