Non-displaceable Lagrangian links in four-manifolds

TL;DR

This paper constructs specific Lagrangian links in certain symplectic four-manifolds, demonstrating that some non-displaceable links can be composed of displaceable components, challenging previous assumptions about Lagrangian displaceability.

Contribution

It introduces families of Lagrangian links in symplectic 4-manifolds where individual components are displaceable but the links themselves are non-displaceable, revealing new phenomena in symplectic topology.

Findings

Existence of displaceable Lagrangian tori forming non-displaceable links

Construction of such links in product symplectic manifolds

Insights into the displaceability properties of Lagrangian links

Abstract

Let $\omega$ denote an area form on $S^2$. Consider the closed symplectic 4-manifold $M=(S^2\times S^2, A\omega \oplus a \omega)$ with $0<a<A$. We show that there are families of displaceable Lagrangian tori $L_{0,x},\, L_{1,x} \subset M$, for $x \in [0,1]$, such that the two-component link $L_{0,x} \cup L_{1,x}$ is non-displaceable for each $x$.