Quantitative tightness for three-dimensional contact manifolds: a sub-Riemannian approach
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866915621857394688 |
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| author | Agrachev, Andrei A. Baranzini, Stefano Bellini, Eugenio Rizzi, Luca |
| author_facet | Agrachev, Andrei A. Baranzini, Stefano Bellini, Eugenio Rizzi, Luca |
| contents | Through the use of sub-Riemannian metrics we provide quantitative estimates for the maximal tight neighbourhood of a Reeb orbit on a three-dimensional contact manifold. Under appropriate geometric conditions we show how to construct closed curves which are boundaries of overtwisted disks. We introduce the concept of \emph{contact} Jacobi curve, and prove lower bounds of the so-called tightness radius (from a Reeb orbit) in terms of Schwarzian derivative bounds. We compare these results with the corresponding ones from [Etnyre, Komendarczyk, Massot - Invent. Math. 2012 and Trans. Amer. Math. Soc. 2016], and we show that our estimates are sharp for classical model structures. We also prove similar, but non-sharp, estimates in terms of sub-Riemannian canonical curvature bounds. We apply our results to K-contact sub-Riemannian manifolds. In this setting, we prove a contact analogue of the celebrated Cartan--Hadamard theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_00770 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantitative tightness for three-dimensional contact manifolds: a sub-Riemannian approach Agrachev, Andrei A. Baranzini, Stefano Bellini, Eugenio Rizzi, Luca Differential Geometry Metric Geometry 53C17, 57K33 Through the use of sub-Riemannian metrics we provide quantitative estimates for the maximal tight neighbourhood of a Reeb orbit on a three-dimensional contact manifold. Under appropriate geometric conditions we show how to construct closed curves which are boundaries of overtwisted disks. We introduce the concept of \emph{contact} Jacobi curve, and prove lower bounds of the so-called tightness radius (from a Reeb orbit) in terms of Schwarzian derivative bounds. We compare these results with the corresponding ones from [Etnyre, Komendarczyk, Massot - Invent. Math. 2012 and Trans. Amer. Math. Soc. 2016], and we show that our estimates are sharp for classical model structures. We also prove similar, but non-sharp, estimates in terms of sub-Riemannian canonical curvature bounds. We apply our results to K-contact sub-Riemannian manifolds. In this setting, we prove a contact analogue of the celebrated Cartan--Hadamard theorem. |
| title | Quantitative tightness for three-dimensional contact manifolds: a sub-Riemannian approach |
| topic | Differential Geometry Metric Geometry 53C17, 57K33 |
| url | https://arxiv.org/abs/2407.00770 |