Contact non-squeezing at large scale via generating functions
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866908318938693632 |
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| author | Fraser, Maia Sandon, Sheila Zhang, Bingyu |
| author_facet | Fraser, Maia Sandon, Sheila Zhang, Bingyu |
| contents | Using SFT techniques, Eliashberg, Kim and Polterovich (2006) proved that if $πR_2^2 \leq K \leq πR_1^2$ for some integer $K$ then there is no contact squeezing in $\mathbb{R}^{2n} \times S^1$ of the prequantization of the ball of radius $R_1$ into the prequantization of the ball of radius $R_2$. This result was extended to the case of balls of radius $R_1$ and $R_2$ with $1 \leq πR_2^2 \leq πR_1^2$ by Chiu (2017) and the first author (2016), using respectively microlocal sheaves and SFT. In the present article we recover this general contact non-squeezing theorem using generating functions, a classical method based on finite dimensional Morse theory. More precisely, we develop an equivariant version, with respect to a certain action of a finite cyclic group, of the generating function homology for domains of $\mathbb{R}^{2n} \times S^1$ defined by the second author (2011). A key role in the construction is played by translated chains of contactomorphisms, a generalization of translated points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_11993 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Contact non-squeezing at large scale via generating functions Fraser, Maia Sandon, Sheila Zhang, Bingyu Symplectic Geometry Geometric Topology Using SFT techniques, Eliashberg, Kim and Polterovich (2006) proved that if $πR_2^2 \leq K \leq πR_1^2$ for some integer $K$ then there is no contact squeezing in $\mathbb{R}^{2n} \times S^1$ of the prequantization of the ball of radius $R_1$ into the prequantization of the ball of radius $R_2$. This result was extended to the case of balls of radius $R_1$ and $R_2$ with $1 \leq πR_2^2 \leq πR_1^2$ by Chiu (2017) and the first author (2016), using respectively microlocal sheaves and SFT. In the present article we recover this general contact non-squeezing theorem using generating functions, a classical method based on finite dimensional Morse theory. More precisely, we develop an equivariant version, with respect to a certain action of a finite cyclic group, of the generating function homology for domains of $\mathbb{R}^{2n} \times S^1$ defined by the second author (2011). A key role in the construction is played by translated chains of contactomorphisms, a generalization of translated points. |
| title | Contact non-squeezing at large scale via generating functions |
| topic | Symplectic Geometry Geometric Topology |
| url | https://arxiv.org/abs/2310.11993 |