On the singular Weinstein conjecture and the existence of escape orbits for $b$-Beltrami fields
| Autor: | Miranda, Eva, Oms, Cédric, Peralta-Salas, Daniel |
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| Rok vydání: | 2020 |
| Předmět: | |
| Zdroj: | Communications in Contemporary Mathematics (2021) 2150076 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1142/S0219199721500760 |
| Popis: | Motivated by Poincar\'e's orbits going to infinity in the (restricted) three-body (see [26] and [6]), we investigate the generic existence of heteroclinic-like orbits in a neighbourhood of the critical set of a $b$-contact form. This is done by using the singular counterpart [3] of Etnyre--Ghrist's contact/Beltrami correspondence [9], and genericity results concerning eigenfunctions of the Laplacian established by Uhlenbeck [29]. Specifically, we analyze the $b$-Beltrami vector fields on $b$-manifolds of dimension $3$ and prove that for a generic asymptotically exact $b$-metric they exhibit escape orbits. We also show that a generic asymptotically symmetric $b$-Beltrami vector field on an asymptotically flat $b$-manifold has a generalized singular periodic orbit and at least $4$ escape orbits. Generalized singular periodic orbits are trajectories of the vector field whose $\alpha$- and $\omega$-limit sets intersect the critical surface. These results are a first step towards proving the singular Weinstein conjecture. Comment: 18 pages, 2 figures, minor changes |
| Databáze: | arXiv |
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