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| Format: | Preprint |
| Veröffentlicht: |
2023
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| Online-Zugang: | https://arxiv.org/abs/2311.18510 |
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| _version_ | 1866909047847911424 |
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| author | Oh, Yong-Geun Yu, Seungook |
| author_facet | Oh, Yong-Geun Yu, Seungook |
| contents | In the present paper, we formulate a contact analogue on the one-jet bundle $J^1B$ of Weinstein's observation which reads the classical action functional on the cotangent bundle is a generating function of any Hamiltonian isotope of the zero section. We do this by identifying the correct action functional which is defined on the space of Hamiltonian-translated (piecewise smooth) horizontal curves of the contact distribution, which we call the Carnot path space. Then we give a canonical construction of the Legendrian generating function which is the Legendrian counterpart of Laudenbach-Sikorav's canonical construction of the generating function of Hamiltonian isotope of the zero section on the cotangent bundle which utilizes a finite dimensional approximation of the action functional.
Motivated by this construction, we develop a Floer theoretic construction of spectral invariants for the Legendrian submanifolds in the sequel [OY] which is the contact analog to the construction given in [Oh97, Oh99] for the Lagrangian submanifolds in the cotangent bundle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_18510 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Contact action functional, calculus of variation and canonical generating function of Legendrian submanifolds Oh, Yong-Geun Yu, Seungook Symplectic Geometry 53D10, 37J51 In the present paper, we formulate a contact analogue on the one-jet bundle $J^1B$ of Weinstein's observation which reads the classical action functional on the cotangent bundle is a generating function of any Hamiltonian isotope of the zero section. We do this by identifying the correct action functional which is defined on the space of Hamiltonian-translated (piecewise smooth) horizontal curves of the contact distribution, which we call the Carnot path space. Then we give a canonical construction of the Legendrian generating function which is the Legendrian counterpart of Laudenbach-Sikorav's canonical construction of the generating function of Hamiltonian isotope of the zero section on the cotangent bundle which utilizes a finite dimensional approximation of the action functional. Motivated by this construction, we develop a Floer theoretic construction of spectral invariants for the Legendrian submanifolds in the sequel [OY] which is the contact analog to the construction given in [Oh97, Oh99] for the Lagrangian submanifolds in the cotangent bundle. |
| title | Contact action functional, calculus of variation and canonical generating function of Legendrian submanifolds |
| topic | Symplectic Geometry 53D10, 37J51 |
| url | https://arxiv.org/abs/2311.18510 |