Joint IAS/Princeton/Montreal/Paris/Tel-Aviv Symplectic Geometry Zoominar

In this talk, I will address the (singular) Weinstein conjecture about the existence of (singular) periodic orbits of Reeb vector fields on compact manifolds endowed with singular contact forms. Our motivating examples come from Celestial mechanics (restricted three-body problem) where contact topology techniques were already successful in determining the existence of periodic orbits (Albers-Frauenfelder-Van Koert-Paternain). With the aim of completing this understanding, we deal with the restricted three body example by adding the so-called "infinity set" (via a McGehee regularization). This induces a singularity on the contact structure which permeates the geometry and topology of the problem. Hofer's fine techniques to prove the Weinstein conjecture for overtwisted 3-dimensional contact manifolds can be adapted in this singular set-up under some symmetry assumptions close to the singular set (which also work for the non-compact case). We prove the existence of infinite smooth Reeb periodic orbits on the (compact) critical set of the contact form. This critical set can often be identified with the collision set or set at infinity in the motivating examples from Celestial mechanics. In those examples, escape trajectories can be often compactified as singular periodic orbits. Time permitting, we will end up this talk proving the existence of escape orbits and generalized singular periodic orbits for 3-dimensional singular contact manifolds under some mild assumptions. Our theory benefits in a great manner from the existence of a correspondence (up to reparametrization) between Reeb and Beltrami vector fields (Etnyre and Ghrist) which can be exported to this singular set-up. In particular, Uhlenbeck's genericity results for the eigenfunctions of the Laplacian is a key point of the proof.

The contents of this talk are based on joint works with Cédric Oms and Daniel Peralta-Salas.