Motivation: Defining a "product structure on Rabinowitz-Floer homology".
Main problem: In [1], Kai Cieliebak and Urs Frauenfelder defined Rabinowitz-Floer homology (RFH) using the Rabinowitz action functional
RFH has been successfully applied and adapted to a multitude of problems in symplectic and contact geometry, but when it comes to defining an instrinic product structure, the Lagrange multiplier 𝜏 is ill-behaved (see e.g. [2], Appendix A).
Idea: Instead of (1), we use the (mean-value)-constrained symplectic action functional
which is the restriction of the symplectic action functional to
The two functionals have the same critical points, but different gradient flow equations.
Research Questions:
What can be said about the moduli space of gradient flow cylinders of (2)?
This problem is adressed in [3], where Fredholm-properties and compactness are shown and a new homology called Constrained Floer Homology (CFH) is defined.
After defining the pair-of-pants product for CFH, what algebraic properties does it satisfy?
Currently, I am studying the compactness properties of the "pair-of-pants"-moduli spaces.
Is there a (canonical) isomorphism from CFH to RFH?
Links:
[1] Kai Cieliebak and Urs Frauenfelder. ‘A Floer homology for exact contact embeddings’. In: Pacific Journal of Mathematics 239.2 (2009), pp. 251 –316.
DOI: 10.2140/pjm.2009.239.251.
[2] Urs Frauenfelder. ‘The Gradient flow equation of Rabinowitz action functional in a symplectization’. (2022).
DOI: 10.48550/ARXIV.2202.00281.
[3] Emilia Konrad. 'The Floer homology of the (mean value)-constrained symplectic action functional'. (2025)
Draft