Non-displaceable Lagrangian links in four-manifolds

Abstract

Let $$\omega $$ 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.

Quantitative Heegaard Floer cohomology and the Calabi invariant

“…We can understand holomorphic discs in with boundary on via the ‘tautological correspondence’ between a holomorphic map and a pair of holomorphic maps , where and is a branched covering with all the branch points lying inside the interior of S . The correspondence arises as follows (see also [47, Section 13], [48, Section 3.1] and the references therein). Let be the ‘big diagonal’ comprising all unordered k -tuples of points in at least two of which coincide.…”

“…Remark 4.1). See [48] for a rapid overview and references, [69, Section 5.3] for a ‘monotone’ version closely related to that used here and [14, Proposition 4.34] for a detailed treatment in a general formalism (which would also apply over the Novikov field in the setting of Definition 5.4).…”

“…We briefly discuss the ideas entering into the proof of Theorem 1.13. Following an insight from [48], although some of the individual components are Floer-theoretically trivial in , the link defines a Lagrangian submanifold of the symmetric product which may be nontrivial. A Hamiltonian function determines canonically a function .…”

On two remarkable groups of area-preserving homeomorphisms

“…The aim of the present article is to give an additional insight on that picture. Our approach is based on application of novel Floer-theoretic invariants from [9] (see also [6]), in combination with a soft approach relying on an idea due to Sikorav [10].…”

Exact orbifold fillings of contact manifolds

“…It is worth pointing out straight away that, additionally to the fact that the study of symplectic orbifolds is of independent interest, these singular objects can be very useful in the study of smooth symplectic topology and dynamics. For example, Mak-Smith [MS21] used orbifold Lagrangian Floer theory developed by Cho-Poddar [CP14] to obtain new families of non-displaceable Lagrangian links in symplectic four-manifolds. The idea was further exploited by Polterovich-Shelukhin [PS21] as well as Cristofaro-Gardiner-Humilière-Mak-Seyfaddini-Smith [CGHM + 21] leading to recent breakthroughs on dynamics on surfaces and C 0 symplectic geometry.…”