Floer homological methods in symplectic geometry and applications

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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.