关于Gromov非挤压变形的注记
A remark on deformation of Gromov non-squeezing
AI总结：
本文研究了Gromov非挤压现象在辛形式微小变形下的持续性，通过特定方法证明了在某些情况下该现象仍然成立。
摘要：
设$R,r$如经典的Gromov非挤压定理所示，令$ε= (πR^{2} - πr^{2})/ πr^{2} $。我们首先猜测Gromov非挤压现象在范围$C^{0}$（相对于标准度量）$ε$附近的标准辛形式变形下仍然成立。我们证明了某些特殊情况，特别是当维度为四且$R < \sqrt{2} r$时。给定这样的扰动，我们不能再对范围进行紧致化，因此经典的Gromov论证失效。我们的主要方法由某种针对全纯曲线的陷阱思想组成，类似于动力系统中的陷阱。
英文摘要：
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.
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