Resumen de The geometry and topology of steady euler flows, integrability and singular geometric structures
Robert Cardona
n this thesis, we make a deep investigation of the geometry and dynamics of several objects (singular or not) appearing in nature. The main goal is to study rigidity versus flexibility dynamical behavior of the objects considered. In particular, we inspect normal forms, h-principles, classifications, and existence theorems. These concern a series of objects which are either close or far away from what we call “integrable situations" in the sense of Frobenius theorem and the existence of first integrals. Such dynamical systems arise in the context of symplectic and contact geometry (and their singular counterparts), as well as in the Euler equations on Riemannian manifolds.
As integral objects, we consider integrable systems appearing in symplectic manifolds but also on singular symplectic manifolds. Singularities show up naturally on these phase spaces by considering spaces with cylindrical ends and studying b-symplectic forms as initiated by Guillemin-Miranda-Pires. Other types of singularities are folded structures originally considered by Martinet and then by Cannas da Silva, Guillemin, Woodward for geometrical purposes. We give classification results of steady Euler flows which admit a Morse-Bott first integral using techniques coming from the symplectic world, and study obstructions arising from the ambient topology.
Our analysis includes the existence of action-angle coordinates on folded symplectic manifolds, and a correspondence between the recently introduced b-contact forms and Beltrami fields on b-manifolds. As examples of systems that are “far from integral" we consider the case of contact manifolds and their close allies in the study of Euler flows (Beltrami vector fields). This gives us the possibility to extend the h-principles from the contact realm to that of Beltrami vector fields. This last observation enables us to consider universality properties, as introduced by Tao, of steady Euler flows by analyzing those of high-dimensional Reeb flows in contact geometry. In the same spirit, we address the construction of steady Euler flows in dimension 3 which simulate a universal Turing machine, using tools coming from symbolic dynamics. In particular, these solutions have undecidable trajectories. In all these discussions, a key role is played by different classes of vector fields such as geodesible, Beltrami, and Eulerisable fields. We set up the study of the relations between such classes in higher odd-dimensions, showing that new phenomena arise as soon as one leaves the realm of three-dimensional manifolds. For these high dimensional Euler flows (or more generally, flows admitting a strongly adapted one-form), we show that they satisfy the periodic orbit conjecture, which was known to be satisfied with the weaker assumption of geodesibility.
