A proof of Gromov's non-squeezing theorem

TL;DR

This paper provides two elementary proofs of the compactness of the moduli space of pseudo-holomorphic spheres, which is crucial for Gromov's non-squeezing theorem, avoiding complex bubbling analysis.

Contribution

It introduces simplified, elementary proofs of the key compactness result in Gromov's non-squeezing theorem, bypassing bubbling analysis and Gromov's removable singularity theorem.

Findings

Two proofs of the moduli space compactness are established.

The proofs use reparametrization and gradient bounds via mean value inequality or Gromov-Schwarz lemma.

The approach simplifies the original proof of Gromov's non-squeezing theorem.

Abstract

The original proof of the Gromov's non-squeezing theorem [Gro85] is based on pseudo-holomorphic curves. The central ingredient is the compactness of the moduli space of pseudo-holomorphic spheres in the symplectic manifold $(\mathbb{CP}^1\times T^{2n-2}, \omega_{\mathrm{FS}}\oplus \omega_{\mathrm{std}})$ representing the homology class $[\mathbb{CP}^1\times\{\operatorname{pt}\}]$. In this article, we give two proofs of this compactness. The fact that the moduli space carries the minimal positive symplectic area is essential to our proofs. The main idea is to reparametrize the curves to distribute the symplectic area evenly and then apply either the mean value inequality for pseudo-holomorphic curves or the Gromov-Schwarz lemma to obtain a uniform bound on the gradient. Our arguments avoid bubbling analysis and Gromov's removable singularity theorem, which makes our proof of Gromov's non-squeezing theorem more elementary.