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Floer homological methods in symplectic geometry and applications

辛几何中的Floer同调方法及其应用

基本信息

批准号：
252380623
负责人：
Dr. Jungsoo Kang
金额：
--
依托单位：
Mathematisches Institut
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2014
资助国家：
德国
起止时间：
2013-12-31 至 2016-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/252380623?language=en
关键词：
Floer homological methods symplectic geometry

项目摘要

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.

我的研究项目有两个目的。第一种是关于整体截面曲面的，它是理解低维动力系统的主要工具，例如平面受限三体问题。动力系统通常承认对称性，但整体截面表面看不到对称性。因此，我们将构造一个在对称下不变的圆盘状整体截面曲面，并通过拉伸辛同调的梯度流线的颈来构造一种新的圆盘状整体截面曲面。这种新方法有两个优点。一种是由梯度流线产生的圆盘状整体截面面，其生成轨道(圆盘状整体截面面的边界)的周期小于或等于一定的辛容量，这部分回答了Hofer-Wysocki-Zehnder提出的一个结构公开问题。另一个优点是，这种伸展脖子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圆盘状全局截面曲面的情况下，我们仍然能够找到具有良好链接性的生成类周期轨道。我的另一个目标是关于Rabinowitz Floer同调，它非常适合于研究自治的哈密顿系统。这是与彼得·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，并在弱单调情形下找到了辛同调与Rabinowitz-Floer同调之间的广义联系。特别地，通过计算闭辛流形上弱单调负线丛的Rabinowitz Floer同调，我们将证明辛同调的消失等价于辛非球面上的Rabinowitz Floer同调的消失。我们还将研究Rabinowitz Floer同调的动力学应用。我们想要证明Gromoll-Meyer类型条件蕴含着无限多个叶交叉和无限多个制动轨道的存在。此外，在机械哈密顿函数的势壁不连通的情况下，我们的目标是找到在势壁的不同分量上制动多次的制动轨道。