As above, let (V, λ) be the completion of a Liouville domain ˜V with contact boundary M = ∂V˜, let Σ ⊂ V be an exact contact hypersurface bounding a compact Liouville domainW and letH be a defining Hamiltonian for Σ.
Definition 12. The spectrum spec(Σ, α) of a contact manifold Σ with contact form α is the set of real numbers η∈R such that the ordinary differential equation v˙ =ηRα has a 1-periodic solution. We denote with P(α)⊂C∞(S1,Σ)×Rthe set of pairs (v, η), such that η ∈spec(Σ, α) and v˙ =ηRα.
Remark. We identify a pair (v, η)∈ P(α) with theη-periodic Reeb orbit ˜v(t) :=v(t/η).
Note that we do not exclude the cases η = 0 and η < 0. For η < 0, we have that v is again a Reeb orbit with period |η|, but run through in the opposite direction. For η = 0, the loop v is just constant. The pairs (v,0) ∈ P(α) are hence in one-to-one correspondence to the points of Σ.
LetL =C∞(S1, V) denote the free (smooth) loop space ofV. TheRabinowitz action functional on V associated to H is given by
AH :L ×R→R, AH(v, η) :=
1 0
λ( ˙v(t))−ηH(v(t)) dt.
One can think ofAH as the Lagrange multiplier action functional of classical mechanics.
In Section 2.1, we will see that the critical points (v, η) of AH are characterized by 0 = ˙v−ηXH and 0 =
1 0
H(v(t))dt.
The first equation implies that dtdH(v) = dH( ˙v) = dH(ηXH) = 0, so that H(v) is constant and hence 0 by the second equation. As H is a defining Hamiltonian with H−1(0) = Σ andXH|Σ=Rα, we find that these equations are therefore equivalent to
v(t)∈Σ ∀t and v˙ =ηRα. (2)
This shows that the set of critical points crit AH
of AH consists of the closed Reeb trajectories v on Σ with periodη, or equivalently that crit
AH
=P(α).
Definition 13. A family of ω-compatible almost complex structures J :S1 ×R→End(T V), (t, n)→→Jt(·, n) is called admissible (of class Cℓ/smooth) if:
• as a map with domain S1 ×V ×R we have that J is of class Cℓ/smooth.
• J ist- andn-independent cylindrical at the unbounded end in V. This means that on M×[R,∞) for R >0 sufficiently large J is of the form
J|ξM =J0 and J ∂
∂r =Rλ,
whereJ0 is any compatible almost complex structure on the contact structure ξM of M andRλ is the vector field whose restriction to the contact hypersurface M× {r}
equals the Reeb vector field. Equivalently, we can require that d(er)◦J =−λ on M×[R,∞).
• the family isCℓ-bounded, meaning that supn||Jt(·, n)||Cℓ <∞with respect to some background norm || · || on M.
Remark. The reason for the dependency ofJ on the additional parametern, not found in the literature until recently in [8] and [1], is based in the transversality problem for Rabinowitz-Floer homology and will become clear in the proof of the local Transversality Theorem 38.
In Section 2.1, formula (6), we will see that such a family of almost complex structures J can be used to define a metric g onL ×R, which yields a gradient ∇AH forAH. The explicit formula for ∇AH is given in (7).
Definition 14. An AH-gradient trajectory is a solution (v, η) ∈ C∞(R×S1, V)× C∞(R,R) of the Rabinowitz-Floer equation, which is the following partial differential equation:
∂s(v, η) = ∇AH(v, η) ⇔
∂sv+Jt(v, η)
∂tv−ηXH(v)
= 0
∂sη +
1 0
H
v(s, t)
dt= 0. (3) The next two lemmas characterize the AH-gradient trajectories which connect critical points (v±, η±)∈crit
AH
=P(α).
Lemma 15. If (v, η)∈ P(α), then AH(v, η) =η. Thus AH(P(α)) =spec(Σ, α).
Proof: Since H|Σ = 0 and ˙v =ηRand im(v)⊂Σ if (v, η)∈ P(α), we may calculate AH(v, η) =
1 0
λ( ˙v) +ηH(v) dt =
1 0
λ(ηR)dt =
1 0
η dt=η.
Lemma 16. If (v, η) is a non-stationary AH-gradient trajectory between critical points (v±, η±)∈ P(α), i.e. where lim
s→±∞(v, η) = (v±, η±), then η+ > η−.
Proof: With|| · || as the norm of the metricg and Lemma 15, we calculate η+−η−=AH(v+, η+)− AH(v−, η−) =
∞
−∞
d
dsAH(v, η)ds
=
∞
−∞
g
∇AH(v, η), ∂s(v, η) ds
(3)=
∞
−∞
||∇AH(v, η)||2ds
>0.
Definition 17. The quantity E(v, η) :=∞
−∞||∇AH(v, η)||2ds ≥0 is called the energy of the AH-gradient trajectory (v, η). If lim
s→±∞(v, η) = (v±, η±) ∈ P(α), then Lemma 16 tells us that E(v, η) =η+−η−.
Since any pair (v, η)∈ P(α), η̸= 0, yields a whole S1-family of points in P(α) by time shift, P(α) never consists of isolated points. In order to build a Floer-type homology with basisP(α), we therefore have to use Morse-Bott techniques. This is why we impose the following non-degeneracy assumption on the Reeb flow ϕt on Σ:
The set Nη ⊂ Σ formed by the η-periodic Reeb orbits is a closed submanifold for each η ∈R and TpNη = ker (Dpϕη−id) holds for all p∈ Nη.
(MB) Note that we do not assume that the rankdλ|Nη is locally constant, as was done in [14].
As far as we see this is not needed. The assumption (MB) is generically satisfied, as shown in [14], appendix B. Moreover, (MB) implies thatP(α) is a proper submanifold of L ×R (see Theorem 23). Note that the components of P(α) with fixed η∈spec(Σ, α) correspond to the submanifolds Nη of Σ via the map (v, η) →→ v(0). By abuse of notation, we will write Nη also for the components of P(α), i.e. we consider Nη either as a submanifold of L ×R or as a submanifold of Σ, depending on the context.
There are several approaches to deal with Morse-Bott situations. The one that we will use here, flows with cascades, was developed by Urs Frauenfelder, [25], and Fr´ed´eric Bourgeois, [5], based on an idea from Piunikhin, Salamon, and Schwarz. It uses flows on the critical manifolds without further perturbation, which makes the computations we have in mind easier. For that, we choose an additional Morse function h and a suitable metric gh on P(α) such that the restrictions hη := h|Nη and gη := gh|Nη form a Morse-Smale pair on Nη for every η ∈ spec(Σ, α). This is equivalent to hη being a Morse function onNη for which the stable and unstable manifolds Ws(p) resp. Wu(q) of the ∇ghh-gradient flow intersect transversally for each pair p, q ∈crit(hη) .
Definition 18. An h-Morse flow line y ∈ C∞
R, crit AH
is a solution of
˙
y=∇h(y), where ∇h is the gradient of h with respect to gh.
Definition 19. For c−, c+ ∈crit(h) and m ∈N, we call a trajectory from c− to c+ with m cascades a tupel
(x, t) =
(xk)1≤k≤m,(tk)1≤k≤m−1
,
consisting of AH-gradient trajectories xk = (vk, ηk) and real numbers tk ≥ 0 such that there exist (possibly constant) h-Morse flow lines yk, 0≤k≤m, with
i. lim
s→−∞y0(s) = c−, lim
s→−∞x1(s) =y0(0),
ii. lim
s→∞xk(s) =yk(0), lim
s→−∞xk+1(s) =yk(tk),
iii. lim
s→∞xm(s) =ym(0), lim
s→∞ym(s) =c+.
A trajectory with 0 cascades from c− to c+ is an h-Morse flow line from c− to c+. See Figure 1 for an illustration.
x1
x2
y0
Nη1
y2
y1
c−
c+
Nη−
Nη+
Fig. 1: A flow line with 2 cascades passing through the critical manifoldsNη−,Nη1,Nη+.
In other words, a trajectory with cascades is an alternating sequence of segments of h-Morse flow lines and whole AH-gradient trajectories. The space of all such trajectories with m cascades from c− toc+ is denoted by M(c −, c+, m). The moduli space
M(c−, c+, m) :=M(c −, c+, m) Rm
is obtained by dividing out the free Rm-action on the m cascades given by time shifts.
If m = 0, we also divide by R (not by R0 ∼= 1), as there is still an R-action by time shift now on the h-Morse flow line. In Theorem 38 we show that M(c−, c+, m) is a manifold for generic choices of Jt(·, n). We denote the moduli space of all trajectories with cascades fromc− toc+ by
M(c−, c+) :=
m∈N
M(c−, c+, m).
Due to Theorem 23, there are only finitely many non-empty critical manifolds Nη with η ∈ [η−, η+] for each η± ∈ R, so that the above union is in fact finite. Indeed, each cascade reduces the period η (cf. Lemma 16) and connects critical manifolds Nη1 and Nη2, where η− ≤η1 < η2 ≤η+.
Theorem 46 states thatM(c−, c+) still carries the structure of a manifold and Theorem 48 asserts that it is compact. Its zero-dimensional component M0(c−, c+) is hence a finite set.
Now we define the Rabinowitz-Floer homology. The chain complex RF C(H, h) is given as theZ2-vector space consisting of formal sums
ξ =
c∈crit(h)
ξc·c,
where the coefficientsξc∈Z2 satisfy the following finiteness condition:
#{c∈crit(h)|ξc̸= 0 ∧ AH(c)≥κ}<∞ for all κ∈R. (4) SoRF C(H, h) is a Novikov-completion of the Z2-vector space generated by the critical points of h. Let #2M0(c−, c+) ∈ Z2 denote the cardinality of M0(c−, c+) modulo 2.
The boundary operator∂F is then defined to be the (infinite) linear extension of
∂Fc+ =
c−∈crit(h)
#2M0(c−, c+)·c−, c+∈crit(h). (5) To see that∂F is well-defined, we have to show that the right hand side still satisfies the finiteness condition (4):
Indeed, it follows from Lemma 16 that ∂F reduces the action, i.e. #2M0(c−, c+) ̸= 0 only ifAH(c−)≤ AH(c+). Moreover, we show in Theorem 23, that for any c+and every a∈R, there are only finitely manyc−∈crit(h) with a≤ AH(c−)≤ AH(c+). Therefore
∂Fc+ satisfies (4) for any c+. In the general case, where ∂F ξc·c
:=
ξc·∂Fc, condition (4) follows as both the sum
ξc·cand ∂Fc satisfy the finiteness condition.
It follows from standard Floer-techniques by considering the 1-dimensional component ofM(c −, c+), that∂F ◦∂F = 0. Hence, ∂F is a boundary operator on RF C(H, h). The Rabinowitz-Floer homology of (V,Σ) with respect to the Hamiltonian H and the Morse-function h is the homology of the chain complex
RF C(H, h), ∂F , i.e.
RF H(H, h) := ker∂F im∂F .
We prove in Corollary 56 thatRF H(H, h) only depends on the contact manifold (Σ, ξ) and the filling Liouville domainW, thus allowing us to writeRF H(W,(Σ, ξ)), where we omit ξ whenever it is clear from the context.
In Section 3.4, we show that RF H(W,Σ) can be given a Z-grading under the following assumptions:
(A) The map i∗ :π1(Σ)→π1(W) induced by the inclusion is injective.
(B) The integralIc1 :π2(W)→Z of the first Chern classc1(T W)vanishes on spheres.
Forc= (v, η)∈crit(h)⊂ P(α), the degree µ(c) is a half integer given by the formula µ(c) =µCZ(v) +indh(c)− 1
2dimcNη+ 1 2,
where indh(c) is the Morse index of c with respect to h, µCZ(v) is the (transversal) Conley-Zehnder index ofv and dimcNη is the local dimension ofNη atc.