基于切触同调的 Reeb 向量场低能量面上周期轨道存在性研究

The theory of periodic orbits of Hamiltonian systems plays an important role in the study of Dynamical systems, and has received extensive concerns from many scholars. The Reeb vector field on odd dimension contact manifold has many comparable properties with the Hamiltonian system on even dimension symplectic manifold. In this project we will mainly study the existence and multiplicity of periodic orbits of Reeb vector field on contact manifold, and the influence of Hamiltonian system on symplectic manifold, which is filling of the contact manifold, on the existence of periodic orbits of corresponding Reeb orbits of contact form, with contact structures supported by planar open books. For the manifold admitting Stein filling, with ashperical structure, we study the multiplicity of periodic orbits for every low energy level of the Hamiltonian function. For the manifold admitting weak filling, we study the existence of periodic orbits for every low energy level of the Hamiltonian function. Because of degenerate, we first analysis the mean index, and give the estimate of iterated index. We also adopt the Gysin-type exact sequence between the symplectic and contact manifolds. The non-vanishing propertity of contact homology ensures the existence the periodic orbits with certain characteristics. We hope the results obtained in this project will give more clear explain of Hamiltonian systems and will have a good theoretical guiding for practical problems such as the motion of a charge in a magnetic field.

辛流形上哈密顿系统的周期轨道在动力系统研究当中扮演着重要的角色，并且与偶数维辛流形上的哈密顿向量场相关的奇数维切触流形上的Reeb向量场有很多对应的性质。 本项目计划研究切触流形上 Reeb 向量场生成的动力系统周期轨道的存在性与多重性，以及研究辛流形上周期轨道生成的辛同调与相应切触流形上切触同调之间的联系。拟把指标迭代理论，同调理论的最新研究方法和研究成果推广到可 Stein 填充以及弱填充两类切触流形上； 建立辛同调与切触同调之间的 Gysin-型正合序列关系， 对于可 Stein 填充以及弱填充两类切触流形，证明其上的 Reeb向量场的周期轨道生成的切触同调非平凡性；对于电荷在磁场中的运动模型,证明每个低能量面上存在一周期轨道。本项目的实施和成果将深化对哈密顿系统性质的理解,并为电荷在磁场中周期运动等方面实际问题的研究提供新思路。

研究切触流形上 Reeb 向量场生成的动力系统周期轨道的存在性与多重性, 以及研究辛流形上周期轨道生成的辛同调与相应切触流形上切触同调之间的联系. 把指标迭代理论, 同调理论的最新研究方法和研究成果推广到可 Stein 填充以及弱填充两类切触流形上; 建立辛同调与切触同调之间的 Gysin-型正合序列关系, 对于可 Stein 填充以及弱填充两类切触流形, 证明其上的 Reeb向量场的周期轨道生成的切触同调非平凡性;..对于具有扭转性质的 $\bar{\partial}$ Neumann 算子. 研究在有界多次调和穷竭定义函数条件下, 解算子的紧性及全局正则性.