A remark on deformation of Gromov non-squeezing

Abstract

Let $R,r$ be as in the classical Gromov non-squeezing theorem, and let $ϵ=(\pi R^2-\pi r^2)/\pi r^2$. We first conjecture that the Gromov non-squeezing phenomenon persists for deformations of the symplectic form on the range $C^0$ (w.r.t. the standard metric) ϵ-nearby to the standard symplectic form. We prove this in some special cases, in particular when the dimension is four and when $R<\sqrt{2}r$. Given such a perturbation, we can no longer compactify the range and hence the classical Gromov argument breaks down. Our main method consists of a certain trap idea for holomorphic curves, analogous to traps in dynamical systems.