The Constrained Symplectic Area Functional and its Floer Homology

TL;DR

This paper develops a new Floer homology for Reeb orbits on Liouville domain boundaries, based on a symplectic area functional with a vanishing mean value constraint, avoiding Lagrange multipliers and enabling intrinsic product structures.

Contribution

It introduces Constrained Floer Homology (CFH), linking it to Rabinowitz Floer homology, and simplifies the Fredholm theory while establishing compactness under geometric conditions.

Findings

CFH shares chain groups with RFH

Fredholm theory for CFH reduces to RFH

Established a priori bounds for moduli space compactness

Abstract

This paper introduces a new Floer homology for periodic Reeb orbits on the boundaries of Liouville domains. The construction of this Constrained Floer Homology (CFH) is based on the symplectic area functional, restricted to loops satisfying a vanishing Hamiltonian mean value condition. While CFH shares its chain groups with Rabinowitz Floer homology (RFH), it avoids the use of a Lagrange multiplier, enabling a more intrinsic product structure. Our first main result shows that the Fredholm theory for CFH reduces to that of RFH: in particular, the standard Morse-Bott condition is sufficient. We then establish the required a priori bounds to ensure compactness of the moduli spaces. A key technical challenge is the non-local term that arises when differentiating along the constraint. To control it, we impose the additional geometric assumption that the Liouville vector field is of gradient type - i.e., that the ambient manifold satisfies a strengthened Weinstein condition.