Pseudoholomorphic curve methods for infinite-dimensional Hamiltonian systems

This thesis is concerned with extending the well-established theory of pseudoholomorphic curve methods for finite-dimensional symplectic manifolds, to the setting of infinite-dimensional symplectic spaces. Our new results are then used to prove the existence of periodic orbits of infinite-dimensional Hamiltonian systems. The systems we consider are non-linear Hamiltonian systems, and a big portion of the material is devoted to the precise description of the type of nonlinearities we allow for, and we show that our results do not hold for other nonlinearities. Examples of systems we consider are nonlinear Hamiltonian differential equations, as well as Hamiltonian particle-field systems. A main problem that arises in generalizing the finite-dimensional results to infinite dimensions, is that of small divisors. This core problem is prevalent in all of our results and it is concisely described how we overcome it.

In the first paper we establish a type of compactness result, where we show the existence of Floer curves in an infinite-dimensional symplectic Hilbert space for admissible Hamiltonians. We then show that these curves give rise to periodic orbits of the Hamiltonian system. We extend this result by coupling the linear theory to a finite-dimensional closed symplectic manifold and prove a cuplength estimate.

In the second paper we extend the latter result for Hamiltonian particle-field systems where the particle is restricted to a closed submanifold of the $n$-torus and the phase space is the cotangent bundle of this submanifold.

The third paper establishes the Fredholm theory, as a step in our program of defining an infinite-dimensional Floer theory. We describe how the small divisor problem permeates the theory and how we overcome it by introducing a modified norm and Sobolev completion related to the small divisors.