A note on Lagrangian intersections and Legendrian Cobordism

TL;DR

This paper explores how Legendrian cobordisms influence intersections with pre-Lagrangian submanifolds in contact manifolds, revealing that certain intersection properties are preserved under cobordism in the hypertight setting.

Contribution

It demonstrates that intersection properties with pre-Lagrangian submanifolds are preserved under Legendrian cobordisms in hypertight contact manifolds.

Findings

If $ ext{Lambda}$ intersects a pre-Lagrangian $P$, then $ ext{Lambda'}$ also intersects $P$ under cobordism.

The result holds in the hypertight setting with Floer homology considerations.

Shows invariance of intersection properties under Legendrian cobordism.

Abstract

Let $\Lambda, \Lambda'$ be a pair of closed Legendrian submanifolds in a closed contact manifold $(Y, \xi = Ker(\alpha))$ related by a Legendrian cobordism $W\subset (\mathbb{C}\times Y, \tilde{\xi}=Ker(-y dx +\alpha))$. In this note, we show that in the hypertight setting, if $\Lambda$ intersects a closed, weakly exact or monotone pre-Lagrangian $P\subset Y$ for reasons of Floer homology, then so does $\Lambda'$.