Symplectic folding and nonisotopic polydisks

Abstract

Let $P_1$ be a polydisk and $P_2=\varphi \left(P_1\right)$ where $\varphi$ is a certain symplectic fold. We determine sharp lower bounds on the size of a ball containing the support of a symplectomorphism mapping $P_1$ to $P_2$. Optimal symplectomorphisms are the folds themselves. As a result, we construct symplectically nonisotopic polydisks in balls and in the complex projective plane.