A remark on deformation of Gromov non-squeezing

2023/02/27 by Savelyev, Yasha · 1 citation
#Differential Geometry (math.DG) #FOS: Mathematics #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.2302.13985

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 < √ 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.

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• On Gromov Width under C⁰ Deformations

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