Low-action holomorphic curves and invariant sets
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916306512510976 |
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| author | Cristofaro-Gardiner, Dan Prasad, Rohil |
| author_facet | Cristofaro-Gardiner, Dan Prasad, Rohil |
| contents | We prove a compactness theorem for sequences of low-action punctured holomorphic curves of controlled topology, in any dimension, without imposing the typical assumption of uniformly bounded Hofer energy. In the limit, we extract a family of closed Reeb-invariant subsets. Then, we prove new structural results for the U-map in ECH and PFH, implying that such sequences exist in abundance in low-dimensional symplectic dynamics. We obtain applications to symplectic dynamics and the geometry of surfaces. First, we prove generalizations to higher genus surfaces and three-manifolds of the celebrated Le Calvez-Yoccoz theorem. Second, we show that for any closed Riemannian or Finsler surface a dense set of points have geodesics passing through them that visit different sections of the surface. Third, we prove a version of Ginzburg-Gürel's "crossing energy bound" for punctured holomorphic curves, of arbitrary topology, in symplectizations of any dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_14445 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Low-action holomorphic curves and invariant sets Cristofaro-Gardiner, Dan Prasad, Rohil Symplectic Geometry Dynamical Systems We prove a compactness theorem for sequences of low-action punctured holomorphic curves of controlled topology, in any dimension, without imposing the typical assumption of uniformly bounded Hofer energy. In the limit, we extract a family of closed Reeb-invariant subsets. Then, we prove new structural results for the U-map in ECH and PFH, implying that such sequences exist in abundance in low-dimensional symplectic dynamics. We obtain applications to symplectic dynamics and the geometry of surfaces. First, we prove generalizations to higher genus surfaces and three-manifolds of the celebrated Le Calvez-Yoccoz theorem. Second, we show that for any closed Riemannian or Finsler surface a dense set of points have geodesics passing through them that visit different sections of the surface. Third, we prove a version of Ginzburg-Gürel's "crossing energy bound" for punctured holomorphic curves, of arbitrary topology, in symplectizations of any dimension. |
| title | Low-action holomorphic curves and invariant sets |
| topic | Symplectic Geometry Dynamical Systems |
| url | https://arxiv.org/abs/2401.14445 |