Floer homological methods in symplectic geometry and applications
Final Report Abstract
Most of the proposed projects have been carried out. In what follows, I will briefly describe the achievements, unexpected phenomena, and the related future projects. Concerning the first goal, Urs Frauenfelder and I gave a construction of invariant disklike global surfaces of section in the presence of symmetry. In order to do this, we studied properties of pseudo-holomorphic curves which are invariant under symmetry, and this provides an additional tool set in symplectic geometry. We also explained in the paper that how this research is related and can be applied to the planar circular restricted three-body problem. In a follow-up project, I investigated the Poincaré return map of an invariant disk-like global surface of section. In this case, the symmetry reduces to the reflection along the “middle line” of an invariant global surface of section, and the return map is reversible with respect to the reflection. Then one can define a notion of symmetric periodic points in such a way that they correspond to symmetric periodic orbits of the original dynamics. I am currently working on the following open question: does a periodic Reeb orbit with minimal period always bound a disk-like global surface of section in a strictly convex hypersurface in R4? One of the involved methods is a new construction of a pseudo-holomorphic plane using Floer homology, which was a part of the proposed projects. Speaking of the second goal, Peter Albers and I studied the Rabinowitz Floer homology on negative line bundles. Negative line bundles over closed integral symplectic manifolds are open symplectic manifolds which are not always symplectically aspherical. However, they have bounded geometry and admit relatively easy Reeb dynamics, and therefore they are nice examples to see how symplectic invariants behave in open symplectic manifolds. We extended a construction of Rabinowitz Floer homology to weakly monotone negative line bundles. Then we proved that the Rabinowitz Floer homology vanishes for a sphere sub-bundle in the negative line bundle when the radius is smaller than a certain quantity. This quantity is sharp in examples as explained below. This result should be compared with the fact that the symplectic homology in this setting does not necessarily vanish. After our paper is published, Sara Venkatesh contacted us and sent a preprint where she computed the Rabinowitz Floer homology for the O(−1)-bundle over CP1. The computation says that in that case the Rabinowitz Floer homology vanishes when the radius of the sphere sub-bundle is smaller than 1/√π and does not vanish (in fact being isomorphic to the Novikov ring) when the radius is larger than that. She also conjectured about computations of the Rabinowitz Floer homology for more general negative line bundles. Interestingly these computations are predicted by means of mirror symmetry. Peter Albers and I are now working on defining a refined version of Rabinowitz Floer homology for negative line bundles using winding numbers. This new Rabinowitz Floer homology seems to have at least two advantages. First, it does not necessarily vanish. Therefore it has potential applications to the orderability problems of prequantization bundles which arise as sphere sub-bundles of negative line bundles. Second, it provides a way to understand the structures of the Rabinowitz Floer chain complex by comparing the ordinary Rabinowitz Floer homology and the refined version of it. A preliminary investigation tells us that it is possible to understand the Rabinowitz-Floer boundary operators completely for the O(−1)-bundle over CP1, and consequently to recover Vekantesh’s computation.
Publications
- On the minimal number of periodic orbits on some hypersurfaces in R2n , Ann. Inst. Fourier (Grenoble) 66 (2016), no. 6, 2485–2505
Jean Gutt and Jungsoo Kang
- Real holomorphic curves and invariant global surfaces of section, Proc. Lond. Math. Soc. (3) 112 (2016), no. 3, 477–511
Urs Frauenfelder and Jungsoo Kang
(See online at https://doi.org/10.1112/plms/pdw003) - Vanishing of Rabinowitz Floer homology on negative line bundles, Math. Z. 285 (2017), no. 1-2, 493–517
Peter Albers and Jungsoo Kang
(See online at https://doi.org/10.1007/s00209-016-1718-6) - On reversible maps and symmetric periodic points, Ergodic Theory Dynam. Systems 38 (2018), no. 4, 1479–1498
Jungsoo Kang
(See online at https://doi.org/10.1017/etds.2016.71)
