2023/11/10 by Moreno, Agustin
#Dynamical Systems (math.DS) #FOS: Mathematics #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.2311.06174
In this article, we extend the methods from arXiv:2011.06568, where the five dimensional analogue of the three dimensional finite energy foliations introduced by Hofer--Wysocki--Zehnder was identified, to the case where there the underlying (IP) contact 5-fold admits a 6-dimensional (IP) symplectic filling. We show that the filling induces a moduli space of pseudo-holomorphic curves which is itself a symplectic filling of the standard 3-sphere, and hence symplectomorphic to the 4-dimensional ball. We further show that whenever the contact form on the 5-fold is strictly convex, then this moduli space is a strictly convex domain, so that the induced Reeb dynamics at the boundary is in particular dynamically convex. For the circular restricted three-body problem, this implies that whenever the spatial dynamics (near one of the heavy masses) is strictly convex, the holomorphic shadow of arXiv:2011.06568 is dynamically convex; this is shown to hold for near-integrable cases close to the Kepler problem, and with mass ratio either zero or sufficiently close to 1.