A remark on deformation of Gromov non-squeezing
Yasha Savelyev
TL;DR
The work investigates whether Gromov non-squeezing persists under small $C^0$-deformations of the ambient symplectic form on $M = S^2 \times \mathbb{R}^{2n-2}$, introducing the notion of a $J$-holomorphic trap to address noncompact ranges. It establishes the conjecture in the four-dimensional case and for the regime $R < \sqrt{2}\,r$, using two regularization frameworks—Banach-analytic and polyfold—to prove the existence of a 1D cobordism of moduli spaces of class-$A$ $J_t$-holomorphic curves and deriving a contradiction with monotonicity. The core idea is that, despite the lack of range compactification, the trap argument enforces compactness of the relevant moduli spaces, enabling a Gromov-type non-squeezing obstruction to persist under $C^0$-close deformations. The paper thus extends non-squeezing phenomena to near-identity symplectic perturbations and introduces holomorphic-trap techniques that may apply beyond the current setting.
Abstract
Let $R,r$ be as in the classical Gromov non-squeezing theorem, and let $ε= (πR ^{2} - πr ^{2})/ πr ^{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.