Pseudo-holomorphic triangles and the median quasi-state
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914719046041600 |
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| author | Haim-Kislev, Pazit Khanevsky, Michael Kislev, Asaf Rosen, Daniel |
| author_facet | Haim-Kislev, Pazit Khanevsky, Michael Kislev, Asaf Rosen, Daniel |
| contents | We prove that the minimal area of a holomorphic triangle whose boundary lies on the union of any three Lagrangian submanifolds is bounded from above by the Lagrangian spectral norm of any pair taken out of the three. We show a relation between this result and the median quasi-state on the 2-sphere. The median quasi-state gives rise to a measure of Poisson non-commutativity of any pair of functions. We answer a question of Entov--Polterovich--Zapolsky by giving a sharp upper bound on the ratio between this measure and the uniform norm of the Poisson bracket of the pair. The sharpness of this bound also implies a lower bound on the defect of the Calabi quasi-morphism. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_12248 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Pseudo-holomorphic triangles and the median quasi-state Haim-Kislev, Pazit Khanevsky, Michael Kislev, Asaf Rosen, Daniel Symplectic Geometry 53Dxx (Primary) 37Jxx (Secondary) We prove that the minimal area of a holomorphic triangle whose boundary lies on the union of any three Lagrangian submanifolds is bounded from above by the Lagrangian spectral norm of any pair taken out of the three. We show a relation between this result and the median quasi-state on the 2-sphere. The median quasi-state gives rise to a measure of Poisson non-commutativity of any pair of functions. We answer a question of Entov--Polterovich--Zapolsky by giving a sharp upper bound on the ratio between this measure and the uniform norm of the Poisson bracket of the pair. The sharpness of this bound also implies a lower bound on the defect of the Calabi quasi-morphism. |
| title | Pseudo-holomorphic triangles and the median quasi-state |
| topic | Symplectic Geometry 53Dxx (Primary) 37Jxx (Secondary) |
| url | https://arxiv.org/abs/2403.12248 |