Symplectic folding and non-isotopic polydisks

Our main result demonstrates the existence of different Hamiltonian isotopy classes of symplectically embedded polydisks inside a 4-ball, and by the same argument also in the complex projective plane. Furthermore, we find exactly how large the ball can be before the embeddings become isotopic; the optimal isotopy is a version of symplectic folding. Before stating the result precisely we fix some notation. We work in R with coordinates (x1, y1, x2, y2) and standard symplectic form ω = ∑ i dxi ∧ dyi. We denote by B(a) the open ball of capacity a, that is