Symplectic Topology, Symplectic Submanifolds and Floer Theory

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AbstractAward: DMS-0405994Principal Investigator: Ely KermanThis proposal is comprised of three projects which concern therelation between various invariants of a symplectic manifold andthe periodic orbits of the Hamiltonian flows which it supports.Recent work by Kerman shows that one of these invariants, theHofer-Zehnder capacity, is finite for tubular neighborhoods ofcertain symplectic submanifolds. Using a decomposition theorem ofBiran, this implies several new kinds of symplectic intersectionphenomena for compact Kahler manifolds. The goal of the firstproject is to study these new intersection results which suggestthat many basic symplectic properties of a compact Kahlermanifold are determined by the Biran decompositions itadmits. The second project is a joint effort with V.L. Ginzburgand B. Gurel. It involves the construction of a generalizedversion of Hamiltonian Floer homology in which periodic orbits indifferent homotopy classes are allowed to interact via ageneralized Floer differential that counts perturbed holomorphiccurves with punctures. The construction is motivated by theSymplectic Field Theory of Eliashberg, Givental and Hofer. Theresulting theory should also have a rich algebraic structure, aswell as a variety of applications including new calculations ofthe Hofer-Zehnder capacity for weakly-exact symplecticmanifolds. The third project is a program to prove a conjecturewhich asserts the existence of periodic orbits on all level setsnear a nondegenerate symplectic critical submanifold of aHamiltonian. This is a generalization of some similar conjecturesof Arnold which concern periodic orbits of a charged particlemoving in a magnetic field. The first step is to construct aFloer-type invariant for the underlying variationalprinciple. Once it is rigorously defined, this should quicklylead to many new existence results. It is also hoped that thisinvariant can be used to augment Symplectic Field Theory byallowing one to split a symplectic manifold along certainhypersurfaces which are not of contact type.Hamiltonian flows are used to model many important physicalsystems in which energy is conserved. Such systems includeplanets and satellites moving under their mutual gravitationalattraction, a charged particle moving in an electro-magneticfield, and the flow of an incompressible ideal fluid. Thesemotions are often quite complex and one way to begin tounderstand their global behavior is to look for repeatingpatterns, i.e., periodic orbits. While most Hamiltonian flowshave many periodic orbits, it is usually a difficult problem toestablish their existence at a fixed energy level. This problemis a central theme in the study of Hamiltonian flows and, inmodern times, has been shown to be deeply related to the shape ofthe space on which the flow is defined. The projects in thisproposal study various aspects of this relation. In the first twoprojects we use Hamiltonian flows to define and computesymplectic invariants. The last project involves the constructionof a new symplectic invariant which should lead to new existenceresults for periodic orbits of Hamiltonian flows which describethe motion of a charged particle in a magnetic field.