Floer homological methods in symplectic geometry and applications

我的研究项目的目的是双重的。第一个是关于整体截面曲面，它是理解低维动力系统如平面限制性三体问题的主要工具。动力学系统通常承认对称性，但整体截面没有对称性。因此，我们构造了一个在对称性下不变的圆盘形整体截面曲面，并通过拉伸辛同调梯度流线的颈部构造了一个新的圆盘形整体截面曲面。这种新方法有两个优点。一种是如果由梯度流线生成圆盘状整体截面曲面，则其生成轨道（圆盘状整体截面曲面的边界）具有小于或等于某一辛容度的周期，这部分地回答了Hofer-Wysocki-Zehnder提出的一个结构上的公开问题.另一个优点是，这种拉伸颈部的方法适用于更一般的情况。例如，在由于拓扑或几何原因不能存在盘状全局截面的情况下，我们仍然能够找到一个具有很好的链接性质的spanning-like周期轨道。我的项目的另一个目标是Rabinowitz Floer同调，它非常适合于研究自治Hamilton系统。这是一个与Peter阿尔伯斯（明斯特大学）的联合项目。我们将Rabinowitz Floer同调的构造推广到弱单调辛流形上，并在弱单调辛流形上找到辛同调与Rabinowitz Floer同调之间的一个广义联系.特别是通过计算闭辛流形上弱单调负线丛的Rabinowitz Floer同调，证明了辛同调的消失等价于辛非球面辛流形上Rabinowitz Floer同调的消失，并研究了Rabinowitz Floer同调的动力学应用.我们想证明Gromoll-Meyer型条件意味着存在无穷多个叶向交叉点和无穷多个制动轨道。此外，在一个机械哈密顿函数的势壁是断开的情况下，我们的目标是找到制动轨道，制动多次的不同组件的势壁。

The aim of my research project is two-fold. The first one is concerned with global surfaces of section which are major tools to understand low dimensional dynamical systems such as the planar restricted 3-body problem. Dynamical systems often admit symmetries but a global surface of section does not see symmetry features. Therefore we will construct a disk-like global surface of section which is invariant under the symmetry.We also develop a new construction of disk-like global surfaces of section by stretching the neck of gradient flow lines of symplectic homology. There are two advantages of this new approach. One is that if a disk-like global surface of section is produced from a gradient flow line, its spanning orbit (the boundary of a disk-like global surface of section) has period less than or equal to a certain symplectic capacity and this partially answers a structural open question raised by Hofer-Wysocki-Zehnder. Another advantage is that this stretching the neck method is applicable to more general situations. For instance, in situations in which for topological or geometrical reasons a disk-like global surface of section cannot exist, we are still able to find a spanning-like periodic orbit which has nice a linking property.Another goal of my project is about Rabinowitz Floer homology which is well suited to studying autonomous Hamiltonian systems. This is a joint project with Peter Albers (Universität Münster). We will extend the construction of Rabinowitz Floer homology to weakly monotone symplectic manifolds and find a generalized connection between symplectic homology and Rabinowitz Floer homology in the weakly monotone case. In particular by computing Rabinowitz Floer homology for weakly monotone negative line bundles over closed symplectic manifolds, we will disprove that vanishing of symplectic homology is equivalent to vanishing of Rabinowitz Floer homology which is true for symplectically aspherical symplectic manifolds.We will also study dynamical applications of Rabinowitz Floer homology. We want to show that Gromoll-Meyer type conditions imply the existence of infinitely many leafwise intersections and infinitely many brake orbits. Moreover in case that a potential wall of a mechanical Hamiltonian function is disconnected, our goal is to find brake orbits which brake multiple times on different components of the potential wall.