Non-displaceable Lagrangian links in four-manifolds

Abstract

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