A proof of Gromov's non-squeezing theorem
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916541669310464 |
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| author | Faisal, Shah |
| author_facet | Faisal, Shah |
| contents | 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 $(\mathbb{CP}^1\times T^{2n-2}, ω_{\mathrm{FS}}\oplus ω_{\mathrm{std}})$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_18462 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A proof of Gromov's non-squeezing theorem Faisal, Shah Symplectic Geometry Differential Geometry 53Dxx, 32Q65 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 $(\mathbb{CP}^1\times T^{2n-2}, ω_{\mathrm{FS}}\oplus ω_{\mathrm{std}})$ 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. |
| title | A proof of Gromov's non-squeezing theorem |
| topic | Symplectic Geometry Differential Geometry 53Dxx, 32Q65 |
| url | https://arxiv.org/abs/2412.18462 |