Symplectic Topology, Symplectic Submanifolds and Floer Theory

摘要：作者：Ely kerman本论文由三个研究辛流形的各种不变量与其所支持的哈密顿流的周期轨道之间关系的课题组成。Kerman最近的工作表明，其中一个不变量hofer - zehnder容量对于某些辛子流形的管状邻域是有限的。利用biran的分解定理，给出了紧卡勒流形的几种新的辛交现象。第一个项目的目标是研究这些新的交点结果，这些结果表明紧卡勒流形的许多基本辛性质是由它所承认的Biran分解决定的。第二个项目是与V.L. ginzburg和B. Gurel共同努力的。它涉及到哈密顿花同调的一个广义版本的构造，在这个版本中，周期轨道无关同伦类可以通过计算带穿孔的微扰全纯曲线的广义花微分来相互作用。这一建构是由Eliashberg、Givental和Hofer的辛场论推动的。由此产生的理论也应该具有丰富的代数结构，以及各种各样的应用，包括对弱精确辛流形的Hofer-Zehnder容量的新计算。第三个课题是证明在hamilton的非退化辛临界子流形附近的所有水平集中都存在周期轨道的猜想。这是阿诺德关于带电粒子在磁场中运动的周期轨道的一些类似猜想的推广。第一步是为底层变分原理构造一个er类型不变量。一旦它被严格定义，这将很快导致许多新的存在结果。我们也希望这个不变量可以用来扩充辛场论，允许人们沿着某些非接触型的超曲面分裂辛流形。哈密顿流被用来模拟许多重要的物理系统，其中能量是守恒的。这样的系统包括在相互引力作用下运动的行星和卫星，在电磁场中运动的带电粒子，以及不可压缩理想流体的流动。这些情绪通常是相当复杂的，开始理解它们的整体行为的一种方法是寻找重复的模式，即周期轨道。虽然大多数哈密顿流有许多周期轨道，但通常很难在固定的能级上建立它们的存在性。这个问题是研究哈密顿流的一个中心主题，在现代，已经被证明与定义流的空间形状密切相关。本提案中的项目研究了这种关系的各个方面。在前两个项目中，我们使用哈密顿流来定义和计算辛不变量。最后一个项目涉及一个新的辛不变量的构造，它将导致描述带电粒子在磁场中的运动的哈密顿流的周期轨道的新的存在性结果。

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.