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arXiv · 2010.00564

On the singular Weinstein conjecture and the existence of escape orbits for $b$-Beltrami fields

Eva Miranda·Cédric Oms·Daniel Peralta-Salas
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Abstract

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.

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Eva Miranda, Cédric Oms, Daniel Peralta-Salas. 2020-10-01. On the singular Weinstein conjecture and the existence of escape orbits for $b$-Beltrami fields. https://doi.org/10.1142/s0219199721500760

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