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 (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.

arXiv comment18 pages, 2 figures, minor changes

References in corpus (11)

• Symplectic and Poisson geometry on b-manifolds
• The contact geometry of the restricted 3-body problem
• Oscillatory orbits in the restricted elliptic planar three body problem
• Global surfaces of section in the planar restricted 3-body problem
• Examples of integrable and non-integrable systems on singular symplectic manifolds
• Action-angle variables and a KAM theorem for b-Poisson manifolds
• Universality of Euler flows and flexibility of Reeb embeddings
• The geometry of E-manifolds
• Chazy-Type Asymptotics and Hyperbolic Scattering for the $n$-Body Problem
• Euler flows and singular geometric structures
• Contact structures and Beltrami fields on the torus and the sphere

Cited by in corpus (1)

• Classical KMS Functionals and Phase Transitions in Poisson Geometry