Proof of Hofer-Wysocki-Zehnder's two or infinity conjecture
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866917618740363264 |
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| author | Cristofaro-Gardiner, Dan Hryniewicz, Umberto Hutchings, Michael Liu, Hui |
| author_facet | Cristofaro-Gardiner, Dan Hryniewicz, Umberto Hutchings, Michael Liu, Hui |
| contents | We prove that every Reeb flow on a closed connected three-manifold has either two or infinitely many simple periodic orbits, assuming that the associated contact structure has torsion first Chern class. As a special case, we prove a conjecture of Hofer-Wysocki-Zehnder published in 2003 asserting that a smooth and autonomous Hamiltonian flow on $\mathbb{R}^4$ has either two or infinitely many simple periodic orbits on any regular compact connected energy level that is transverse to the radial vector field. Other corollaries settle some old problems about Finsler metrics: we show that every Finsler metric on $S^2$ has either two or infinitely many prime closed geodesics; and we show that a Finsler metric on $S^2$ with at least one closed geodesic that is not irrationally elliptic must have infinitely many prime closed geodesics. The novelty of our work is that we do not make any nondegeneracy hypotheses. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_07636 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Proof of Hofer-Wysocki-Zehnder's two or infinity conjecture Cristofaro-Gardiner, Dan Hryniewicz, Umberto Hutchings, Michael Liu, Hui Symplectic Geometry Dynamical Systems We prove that every Reeb flow on a closed connected three-manifold has either two or infinitely many simple periodic orbits, assuming that the associated contact structure has torsion first Chern class. As a special case, we prove a conjecture of Hofer-Wysocki-Zehnder published in 2003 asserting that a smooth and autonomous Hamiltonian flow on $\mathbb{R}^4$ has either two or infinitely many simple periodic orbits on any regular compact connected energy level that is transverse to the radial vector field. Other corollaries settle some old problems about Finsler metrics: we show that every Finsler metric on $S^2$ has either two or infinitely many prime closed geodesics; and we show that a Finsler metric on $S^2$ with at least one closed geodesic that is not irrationally elliptic must have infinitely many prime closed geodesics. The novelty of our work is that we do not make any nondegeneracy hypotheses. |
| title | Proof of Hofer-Wysocki-Zehnder's two or infinity conjecture |
| topic | Symplectic Geometry Dynamical Systems |
| url | https://arxiv.org/abs/2310.07636 |