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.