Estimating Reeb chords using microlocal sheaf theory
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866915453205479424 |
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| author | Li, Wenyuan |
| author_facet | Li, Wenyuan |
| contents | We show that for a closed Legendrian submanifold in a 1-jet bundle, if there is a sheaf with compact support, perfect stalk and singular support on that Legendrian, then (1) the number of Reeb chords has a lower bound by half of the sum of Betti numbers of the Legendrian; (2) the number of Reeb chords between the original Legendrian and its Hamiltonian pushoff has a lower bound in terms of Betti numbers when the oscillation norm of the Hamiltonian is small comparing with the length of Reeb chords. In the proof we develop a duality exact triangle and use the persistence structure (which comes from the action filtration) of microlocal sheaves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2106_04079 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Estimating Reeb chords using microlocal sheaf theory Li, Wenyuan Symplectic Geometry Geometric Topology We show that for a closed Legendrian submanifold in a 1-jet bundle, if there is a sheaf with compact support, perfect stalk and singular support on that Legendrian, then (1) the number of Reeb chords has a lower bound by half of the sum of Betti numbers of the Legendrian; (2) the number of Reeb chords between the original Legendrian and its Hamiltonian pushoff has a lower bound in terms of Betti numbers when the oscillation norm of the Hamiltonian is small comparing with the length of Reeb chords. In the proof we develop a duality exact triangle and use the persistence structure (which comes from the action filtration) of microlocal sheaves. |
| title | Estimating Reeb chords using microlocal sheaf theory |
| topic | Symplectic Geometry Geometric Topology |
| url | https://arxiv.org/abs/2106.04079 |