切触流形上Reeb流的闭轨道及其几何性质

本课题涉及哈密顿动力系统、辛拓扑与切触拓扑、微分几何等多个学科。闭轨道的存在性、多重性、稳定性及其相关几何性质是哈密顿系统与辛动力系统的核心内容之一。本项目基于近期我们建立了2n维欧氏空间中紧星型超曲面上闭特征的共振恒等式及Ekeland-Hofer理论，并在闭特征的多重性和稳定性研究上取得了新的进展，同时国外同行使用辛几何中的Floer同调和切触同调理论在这些问题上得到了相关的成果。我们主要研究以下几个课题: （1） 研究2n维欧氏空间中紧星型超曲面上闭特征数目的下界。 （2） 将紧凸超曲面上闭特征问题的方法与Floer同调、切触同调理论方法相结合研究更一般切触流形上闭轨道的多重性和稳定性问题。 （3） 研究与切触流形上闭轨道相关的拟全纯曲线和有限能量面的几何性质。 （4） 研究与闭特征问题密切相关的Finsler非单连通流形上的闭测地线多重性问题。

This project involves several subjects such as Hamiltonian Dynamics, Symplectic and Contact Topology, Differential Geometry and so on. The problems of existence, multiplicity, stability and geometrical property of closed orbits are central in Hamiltonian systems and Symplectic dynamics. This project is based on the resonance identities and Ekeland-Hofer theory for closed characteristics on compact star-shaped hypersurfaces in the 2n-dimensional Euclidean space which we established recently and the progresses we made on the multiplicity and stability of closed characteristics, at the same time, our foreign colleagues use Floer homology and contact homology of Symplectic Geometry to obtain related results on these problems. Mostly we will study the following topics: (1)The lower bound of the number of closed characteristics on compact star-shaped hypersurfaces in the 2n-dimensional Euclidean space. (2) Combining the methods of Floer homology and contact homology with the methods used in the problem of closed characteristics on compact convex hypersurfaces, we study the problem of multiplicity and stability of closed orbits on general contact manifolds. (3) The geometrical property of pseudo holomorphic curves and finite energy surfaces related to the closed orbits on contact manifolds. (4) The problem of multiplicity of closed geodesics on non-simply connected Finsler manifolds which is closely related to the problem of closed characteristics.