Symplectically knotted cubes

While by a result of McDuff the space of symplectic embeddings of a closed 4-ball into an open 4-ball is connected, the situation for embeddings of cubes C4=D2×D2 is very different. For instance, for the open ball B4 of capacity 1, there exists an explicit decreasing sequence c1,c2,…→1/3 such that for c<ck there are at least k symplectic embeddings of the closed cube C4(c) of capacity c into B4 that are not isotopic. Furthermore, there are infinitely many non-isotopic symplectic embeddings of C4(1/3) into B4.


A similar result holds for several other targets, like the open 4-cube, the complex projective plane, the product of two equal 2-spheres, or a monotone product of such manifolds and any closed monotone toric symplectic manifold.


The proof uses exotic Lagrangian tori.


This is joint work with Joé Brendel and Grisha Mikhalkin.



Université de Neuchâtel


Felix Schlenk