[Paper Review] Periodic orbits of magnetic flows for weakly exact unbounded forms and for spherical manifolds

Luca Asselle, Gabriele Benedetti|arXiv (Cornell University)|Dec 1, 2014

Geometric Analysis and Curvature Flows28 references3 citations

TL;DR

This paper establishes the Palais-Smale condition for the action functional of weakly exact magnetic flows with infinite Mañé critical value on contractible loops with bounded, non-zero period, resolving a gap in the existence proof of one closed contractible magnetic geodesic for almost every energy. It further generalizes the method to arbitrary magnetic flows, proving the existence of such geodesics when the base manifold has non-trivial second homotopy group.

ABSTRACT

We show that for weakly exact magnetic flows with infinite Ma\~n\'e critical value the action functional satisfies the Palais-Smale condition on the space of contractible loops with period bounded and bounded away from zero. This fills a gap in the proof of the existence of one closed contractible magnetic geodesic for almost every energy in [Merry, 2010]. The gap has also been fixed independently by Will Merry and it will appear soon in an erratum by the author. The idea used for the weakly exact case is then generalized to arbitrary magnetic flows to study critical sequences for the action 1-form. As a corollary we prove that if the second fundamental group of the base manifold is non-trivial, there exists one closed contractible magnetic geodesic for almost every energy.

Motivation & Objective

• To close a gap in the proof of existence for closed contractible magnetic geodesics in weakly exact magnetic flows with infinite Mañé critical value.
• To establish the Palais-Smale condition for the action functional on contractible loops with bounded, non-zero period.
• To generalize the method to arbitrary magnetic flows to study critical sequences of the action 1-form.
• To prove the existence of at least one closed contractible magnetic geodesic for almost every energy when the second fundamental group of the base manifold is non-trivial.

Proposed method

• Use of the Palais-Smale condition to ensure convergence of critical sequences in the space of contractible loops with period bounded away from zero and bounded above.
• Application of variational methods to the action functional under weakly exact magnetic forms with infinite Mañé critical value.
• Generalization of the approach from weakly exact to arbitrary magnetic flows to analyze critical sequences of the action 1-form.
• Topological argument using the non-triviality of the second fundamental group to guarantee existence of closed geodesics.
• Use of periodicity constraints and energy bounds to control the behavior of loops in the variational setting.
• Leveraging results from symplectic topology and Morse theory to ensure compactness and existence of critical points.

Experimental results

Research questions

• RQ1 Does the action functional for weakly exact magnetic flows with infinite Mañé critical value satisfy the Palais-Smale condition on the space of contractible loops with bounded, non-zero period?
• RQ2 Can the gap in the existence proof of closed contractible magnetic geodesics for almost every energy be resolved via the Palais-Smale condition?
• RQ3 How can the method used for weakly exact forms be extended to arbitrary magnetic flows to study critical sequences of the action 1-form?
• RQ4 Under what topological conditions on the base manifold does a closed contractible magnetic geodesic exist for almost every energy?
• RQ5 What role does the non-triviality of the second fundamental group play in the existence of such geodesics?

Key findings

• The action functional satisfies the Palais-Smale condition on the space of contractible loops with period bounded and bounded away from zero for weakly exact magnetic flows with infinite Mañé critical value.
• The gap in the existence proof of one closed contractible magnetic geodesic for almost every energy is resolved via this condition.
• The method is generalized to arbitrary magnetic flows, enabling analysis of critical sequences for the action 1-form.
• For base manifolds with non-trivial second fundamental group, there exists at least one closed contractible magnetic geodesic for almost every energy level.
• The result confirms the existence of periodic orbits under broad topological and geometric conditions on the magnetic flow.