Lagrangian Rabinowitz Floer Homology

• Lagrangian Rabinowitz Floer Homology is a Floer-type invariant defined on a symplectic manifold paired with a Lagrangian submanifold, capturing key features like Hamiltonian chords and leaf-wise intersections.
• Its framework involves constructing an action functional with a Lagrange multiplier and grading by the transverse Maslov index to build a robust Floer chain complex.
• LRFH yields rigidity results that obstruct Lagrangian displaceability and informs computations relevant to dynamics, including applications in space mission design.

Lagrangian Rabinowitz Floer homology (LRFH) is a Floer-type invariant associated with a pair consisting of a (possibly non-exact) symplectic manifold and a Lagrangian submanifold intersecting a hypersurface of restricted or virtual contact type. LRFH extends the machinery of classical Floer homology to address fundamental questions in symplectic topology, such as the existence of Hamiltonian or Reeb chords, leaf-wise intersection points, and rigidity phenomena involving Lagrangian submanifolds and energy hypersurfaces. Its definition and computational schemes have played a central role in symplectic rigidity, the study of non-displaceability, and geometric problems in dynamics such as the two-boost problem in space mission design (Cieliebak et al., 11 Dec 2024, Merry, 2010).

1. Geometric and Analytical Foundations

The typical framework involves a smooth manifold $Q$ (commonly $Q = \mathbb{R}^n$), its cotangent bundle $T^*Q$ with canonical 1-form $\lambda$ and symplectic form $\omega = d\lambda$. Consider an autonomous Hamiltonian $H : T^*Q \to \mathbb{R}$ with regular energy level $\Sigma = H^{-1}(c)$ and a Lagrangian $L$ (e.g., a Lagrangian fiber $T^*_{q_0} Q$ over $q_0 \in Q$). The path space relevant for LRFH consists of Sobolev paths $v \in W^{1,2}([0,1], T^*Q)$ with endpoints on given Lagrangian fibers, e.g., $v(0) \in T^*_{q_0} Q$, $v(1) \in T^*_{q_1} Q$.

The Lagrangian Rabinowitz action functional is defined as

$$A^{H-c}_{q_0, q_1}(v, \eta) = \int_0^1 \lambda(\dot v(t))\,dt - \eta \int_0^1 (H(v(t)) - c)\,dt,$$

where the real parameter $\eta$ acts as a Lagrange multiplier imposing the energy constraint $H(v(t)) = c$. Critical points $(v, \eta)$ correspond to solutions of

$\dot v(t) = \eta\, X_H(v(t)), \qquad H(v(t)) = c,$

so that $v$ parameterizes a Hamiltonian chord of period $\eta$ on $\Sigma$ connecting the two specified Lagrangian submanifolds (fibers) (Cieliebak et al., 11 Dec 2024, Merry, 2010).

2. Floer Chain Complex Construction and Grading

The Floer chain complex for LRFH is generated by (nondegenerate) critical points of $A^{H-c}_{q_0, q_1}$. The $(v, \eta)$ serve as generators, graded by a version of the Maslov index—specifically, the transverse Maslov index $\mu(v, \eta) \in \frac{1}{2}\mathbb{Z}$ (Robbin–Salamon index). For manifolds with Morse–Bott nondegeneracy, Morse functions on the critical manifold further refine the grading. The boundary operator

$$\partial x = \sum_{\substack{y \in \mathrm{Crit}\,A\ \mu(y) = \mu(x) - 1}} \#(\overline{\mathcal{M}}(x,y))\, y$$

counts, modulo 2, isolated negative gradient trajectories (Floer trajectories) connecting generators of index difference 1. The boundary squared satisfies $\partial^2 = 0$ by standard Floer-theoretic arguments, provided suitable compactness is achieved (Cieliebak et al., 11 Dec 2024, Merry, 2010).

The following table summarizes these components:

Object	Definition	Role in LRFH
Action functional	$A^{H-c}_{q_0, q_1}(v, \eta)$	Generates chain complex
Generator	Critical point $(v, \eta)$	Floer chain complex element
Grading	Transverse Maslov index $\mu(v, \eta) \in \frac{1}{2}\mathbb{Z}$	Assigns degree to chains
Boundary operator	Counts Floer trajectories between critical points (mod 2)	Satisfies $\partial^2 = 0$

3. Compactness, Invariance, and Well-Posedness

Noncompactness of the energy hypersurface $\Sigma$ poses the central analytical challenge in the LRFH framework. Several tools ensure compactness of moduli spaces:

• Hamiltonian cutoff: Modifications by compactly supported $C^\infty$-functions $\varphi$ outside large sets, ensuring $H$ agrees with the original inside a compact that contains all chords of interest.
• $L^\infty$-bounds: Application of the Aleksandrov maximum principle to plurisubharmonic functions such as $F = \frac{1}{2}|q|^2$ or $F = \frac{1}{2}|p|^2$ to control Floer solutions.
• A priori estimates: Derived from the action–energy relations for the Lagrange multiplier $\eta$ and the Sobolev norms of solutions.

Under suitable hypotheses (e.g., $H^{-1}(c)$ noncompact of restricted contact type, all chords and moduli spaces contained in a fixed compact for all allowed Hamiltonian perturbations), the LRFH is well-defined and independent of the perturbations. This is formalized in the invariance theorems such as Theorem 2.7 in (Cieliebak et al., 11 Dec 2024) and analogous results in (Merry, 2010).

4. Computation and Exact Sequences

LRFH can be computed explicitly in certain geometric settings. For instance, in cotangent bundle situations with conormal Lagrangians, there exists an exact sequence relating LRFH to Morse homology on path spaces. The Abbondandolo–Schwarz short exact sequence is

$$0 \longrightarrow H_*(P(M,S)) \longrightarrow \mathrm{RFH}_*(\Sigma, N^*S) \longrightarrow H^{-* + 2d - n + 1}(P(M,S)) \longrightarrow 0,$$

where $P(M,S) = \{q : [0,1] \to M \mid q(0), q(1) \in S\}$ is the space of paths with boundary on $S \subset M$ and $d, n$ are the dimensions of $S$ and $M$, respectively. For twisted cotangent bundles subject to Mañé supercriticality, this yields explicit homological computations in terms of singular homology of $P(M, S)$. In many untwisted cases, infinite Betti numbers result in infinite-dimensional LRFH (Merry, 2010).

Computationally, for the “Copernican” Hamiltonian

$$H_0(q_1,q_2,p_1,p_2) = \tfrac{1}{2}(p_1^2 + p_2^2) + p_1q_2 - p_2q_1,$$

the positive-action part of $\mathrm{RFH}_*^+$ vanishes in all gradings except $\tfrac{1}{2}$, where it is isomorphic to $\mathbb{Z}_2$, and there is a unique chord of positive action, confirming the sharpness of rigidity results (Cieliebak et al., 11 Dec 2024).

5. Rigidity, Chord Existence, and Applications

The principal geometric consequence of non-vanishing LRFH is the existence of Hamiltonian or Reeb chords between specified Lagrangian submanifolds and energy hypersurfaces. In particular, the Rabinowitz–Floer principle asserts that nontrivial LRFH guarantees at least one such orbit. In space mission dynamics, this underpins solutions to the two-boost problem: for any energy $c > 0$ and distinct points $q_0 \neq q_1$, there exists at least one Hamiltonian chord on $H_0^{-1}(c)$ connecting $T^*_{q_0} Q$ to $T^*_{q_1} Q$. Results extend to Hamiltonians perturbed by compactly supported potentials and to cases with potentials decaying at infinity (Cieliebak et al., 11 Dec 2024).

In symplectic topology, LRFH provides obstructions to Lagrangian displaceability and lower bounds for symplectic capacities. Non-vanishing of LRFH detects Reeb chords and implies existence of infinitely many relative leaf-wise intersection points for generic diffeomorphisms, as realized in the context of twisted cotangent bundles. In such scenarios, the infinite-dimensionality of LRFH implies rigidity properties and the absence of displaceability for pairs of hypersurfaces and Lagrangians (Merry, 2010).

6. Extensions, Virtual Contact, and Further Directions

LRFH has been extended to the virtually contact setting, where the underlying symplectic form is not exact. The theory is constructed on the universal cover, with conditions ensuring the existence of a bounded primitive of the lifted symplectic form. When these virtual exactness and contact conditions are met for both the hypersurface and the Lagrangian, standard compactness and Floer-theoretic properties are preserved.

Frauenfelder’s cascade approach addresses Morse–Bott degeneracies from constant chords. The theory accommodates significant generalizations:

• Twisted cotangent bundles with non-exact forms.
• Conormal Lagrangians for submanifolds.
• Detection of relative leaf-wise intersections under compactly supported Hamiltonian isotopies.

A plausible implication is the utility of LRFH as an invariant capturing deep symplectic rigidity phenomena across both exact and non-exact settings, tying together the dynamics of Reeb flows, existence of chords, and algebraic invariants bridging Morse-theoretic and Floer-theoretic landscapes (Merry, 2010, Cieliebak et al., 11 Dec 2024).