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

TL;DR

This paper studies the existence of escape orbits and periodic trajectories in $b$-Beltrami fields on $b$-manifolds, providing new generic results that advance understanding of the singular Weinstein conjecture.

Contribution

It establishes the generic existence of escape orbits and singular periodic orbits in $b$-Beltrami fields, advancing the singular Weinstein conjecture.

Findings

Generic $b$-Beltrami fields have escape orbits.

Existence of generalized singular periodic orbits under certain conditions.

Results contribute to the proof of the singular Weinstein conjecture.

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.