Principal Investigator: Ely Kerman
This proposal is comprised of three projects which concern the relation between various invariants of a symplectic manifold and the periodic orbits of the Hamiltonian flows which it supports. Recent work by Kerman shows that one of these invariants, the Hofer-Zehnder capacity, is finite for tubular neighborhoods of certain symplectic submanifolds. Using a decomposition theorem of Biran, this implies several new kinds of symplectic intersection phenomena for compact Kahler manifolds. The goal of the first project is to study these new intersection results which suggest that many basic symplectic properties of a compact Kahler manifold are determined by the Biran decompositions it admits. The second project is a joint effort with V.L. Ginzburg and B. Gurel. It involves the construction of a generalized version of Hamiltonian Floer homology in which periodic orbits in different homotopy classes are allowed to interact via a generalized Floer differential that counts perturbed holomorphic curves with punctures. The construction is motivated by the Symplectic Field Theory of Eliashberg, Givental and Hofer. The resulting theory should also have a rich algebraic structure, as well as a variety of applications including new calculations of the Hofer-Zehnder capacity for weakly-exact symplectic manifolds. The third project is a program to prove a conjecture which asserts the existence of periodic orbits on all level sets near a nondegenerate symplectic critical submanifold of a Hamiltonian. This is a generalization of some similar conjectures of Arnold which concern periodic orbits of a charged particle moving in a magnetic field. The first step is to construct a Floer-type invariant for the underlying variational principle. Once it is rigorously defined, this should quickly lead to many new existence results. It is also hoped that this invariant can be used to augment Symplectic Field Theory by allowing one to split a symplectic manifold along certain hypersurfaces which are not of contact type.
Hamiltonian flows are used to model many important physical systems in which energy is conserved. Such systems include planets and satellites moving under their mutual gravitational attraction, a charged particle moving in an electro-magnetic field, and the flow of an incompressible ideal fluid. These motions are often quite complex and one way to begin to understand their global behavior is to look for repeating patterns, i.e., periodic orbits. While most Hamiltonian flows have many periodic orbits, it is usually a difficult problem to establish their existence at a fixed energy level. This problem is a central theme in the study of Hamiltonian flows and, in modern times, has been shown to be deeply related to the shape of the space on which the flow is defined. The projects in this proposal study various aspects of this relation. In the first two projects we use Hamiltonian flows to define and compute symplectic invariants. The last project involves the construction of a new symplectic invariant which should lead to new existence results for periodic orbits of Hamiltonian flows which describe the motion of a charged particle in a magnetic field.
