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

Abstract

Motivated by Poincaré's orbits going to infinity in the (restricted) three-body problem [see H. Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste, Vol. 3 (Gauthier-Villars, 1899) and A. Chenciner, Poincaré and the three-body problem, in Henri Poincaré, 1912-2012 (Birkhäuser, Basel, 2015), pp. 51-149], we investigate the generic existence of heteroclinic-like orbits in a neighborhood of the critical set of a b-contact form. This is done by using a singular counterpart [R. Cardona, E. Miranda and D. Peralta-Salas, Euler flows and singular geometric structures, Philos. Trans. R. Soc. A 377(2158) (2019) 20190034] of Etnyre-Ghrist's contact/Beltrami correspondence [J. Etnyre and R. Ghrist, Contact topology and hydrodynamics: I. Beltrami fields and the Seifert conjecture, Nonlinearity 13(2) (2000) 441-458], and genericity results concerning eigenfunctions of the Laplacian established by Uhlenbeck [Generic properties of eigenfunctions, Amer. J. Math. 98(4) (1976) 1059-1078]. 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 four escape orbits. Generalized singular periodic orbits are trajectories of the vector field whose α-and ω-limit sets intersect the critical surface. These results are a first step towards proving the singular Weinstein conjecture. © 2022 World Scientific Publishing Company.