Cédric Oms defended his PhD thesis Global Hamiltonian Dynamics on Singular Symplectic Manifolds (pdf), supervised by Professor Eva Miranda, on October 2, 2020, within the UPC doctoral program in Applied Mathematics. Currently, he is a post-doctoral student at the Unité de Mathématiques Pures et Appliquées at the École Normale Supérieure in Lyon.
Thesis Summary
This thesis studies the Reeb and Hamiltonian dynamics on singular symplectic and contact manifolds. Those structures are motivated by singularities coming from classical mechanics and fluid dynamics.
The branch of symplectic and contact geometry emerged as a set-up for the study of classical Hamiltonian systems, as for instance celestial mechanics. The equations of motion of the Hamiltonian system can be geometrically interpreted as the flow of the Hamiltonian vector field associated to the smooth energy function H, called Hamiltonian, on a symplectic manifold (W, ω). The holy grail in Hamiltonian systems is to establish existence of periodic orbits. Contact manifolds appear as level-sets of H and the Hamiltonian dynamics can be described intrinsically using the Reeb vector field. Whenever there are singularities in the Hamiltonian system, the approach of classical symplectic and contact topology fails. However many Hamiltonian systems coming from fluid dynamics or celestial mechanics do admit singularities and hence it is crucial to understand the dynamics whenever there are singularities in the geometric structure.
In the contact realm, the singularities considered in this thesis consist of a generalization of contact structures where the nonintegrability condition fails on a hypersurface called the critical hypersurface. Those structures are called b-contact structures. In particular they arise from hypersurfaces in b-symplectic manifolds that have been studied extensively in the past. Formerly, this odd-dimensional counterpart to b-symplectic geometry has been neglected in the existing vast literature.
The first chapter of this thesis, the local geometry of those manifolds is examined using the language of Jacobi manifolds, which provides an adequate set-up and leads to understanding the geometric structure on the critical hypersurface. Topological obstructions to the existence of those structures are studied and the topology of b^m-contact manifolds is related to the existence of convex contact hypersurfaces. The results concerning the topology and geometry of b-contact manifolds can be found in [2].
The next chapter delves into the dynamical properties of the Reeb vector field associated to a given b^m-contact form. The relation of those structures to celestial mechanics underlines the relevance for existence results of periodic orbits of the Hamiltonian vector field in the b^m-symplectic setting and Reeb vector fields for b^m-contact manifolds. In this light, it is proven that in dimension 3, there are always infinitely many periodic Reeb orbits on the critical surface. However, explicit examples without periodic orbits away from the critical set are given. In contrast to contact topology, it is shown that there exist traps for this vector field. We prove that the well-known Weinstein conjecture holds for compact b-contact manifolds that satisfy some additional conditions.
The mentioned results shed new light towards a singular version of that conjecture. Finally, the obtained results are applied to the particular case of the restricted planar circular three body problem, where the results from previous chapters imply that after the McGehee change, there are infinitely many non-trivial periodic orbits at the manifold at infinity for positive energy values. The results on the dynamics of the b-Reeb vector field can be found in [2]. See also the continuation [3], where a semi-local version of the singular Weinstein conjecture is proved.
Highlighted publications
References
[1] Miranda, Eva, and Cédric Oms. “The singular Weinstein conjecture.” Advances in Mathematics 389 (2021): 107925.
[2] Miranda, Eva, and Cédric Oms. “The geometry and topology of contact structures with singularities.” arXiv preprint arXiv:1806.05638 (2018).
[3] Miranda, Eva, Cédric Oms, and Daniel Peralta-Salas. “On the singular Weinstein conjecture and the existence of escape orbits for b Beltrami fields.” Communications in contemporary mathematics (2021).