On a Theorem by Schlenk

In this paper we prove a generalisation of Schlenk's theorem about the existence of contractible periodic Reeb orbits on stable, displaceable hypersurfaces in symplectically aspherical, geometrically bounded, symplectic manifolds, to a forcing result for contractible twisted periodic Reeb orbits. We make use of holomorphic curve techniques for a suitable generalisation of the Rabinowitz action functional in the stable case in order to prove the forcing result. As in Schlenk's theorem, we derive a lower bound for the displacement energy of the displaceable hypersurface in terms of the action value of such periodic orbits. The main application is a forcing result for noncontractible periodic Reeb orbits on quotients of certain symmetric star-shaped hypersurfaces. Either there exist two geometrically distinct noncontractible periodic Reeb orbits or the period of the noncontractible periodic Reeb orbit is small. This theorem can be applied to many physical systems including the H\'enon-Heiles Hamiltonian and Stark-Zeeman systems. Further applications include a new proof of the well-known fact that the displacement energy is a relative symplectic capacity on $\mathbb{R}^{2n}$ and that the Hofer metric is indeed a metric.


Introduction
In [Bäh23], a generalisation of Rabinowitz-Floer homology was constructed.Rabinowitz-Floer homology is the Morse-Bott homology in the sense of Floer associated with the Rabinowitz action functional introduced by Kai Cieliebak and Urs Frauenfelder in 2009.The main application of this generalisation was to prove an existence result for noncontractible periodic Reeb orbits on quotients of certain symmetric star-shaped hypersurfaces in C n , n ≥ 2.More precisely, let Σ ⊆ C n be a compact and connected star-shaped hypersurface invariant under the rotation φ : C n → C n , φ(z 1 , . . ., z n ) := e 2πik 1 /m z 1 , . . ., e 2πikn/m z n for some even m ≥ 2 and k 1 , . . ., k n ∈ Z coprime to m.Then Σ/Z m admits a noncontractible periodic Reeb orbit generating the fundamental group π 1 (S 2n−1 /Z m ) ∼ = Z m .For a proof see [Bäh23, Theorem 1.2] and [ALM22, Theorem 1.1] for the more general result, removing the restriction of m being even.The existence of noncontractible periodic Reeb orbits on lens spaces is extremely relevant and attracts much attention in celestial mechanics as mentioned in [LZ22,Introduction] or [HS16].We quickly recall the setup for the proof of this result.Let (W, λ) be a connected Liouville domain with connected boundary ∂W and consider a Liouville automorphism φ ∈ Aut(W, λ), that is, φ ∈ Diff(W ) is of finite order and there exists a unique function f φ ∈ C ∞ (Int W ) such that φ * λ − λ = df φ .The main step was to construct a homology theory for the twisted Rabinowitz action functional on the completion (M, λ) of (W, λ), where L φ M := {γ ∈ C ∞ (R, M ) : γ(t + 1) = φ(γ(t)) ∀t ∈ R} denotes the twisted loop space of M and φ.Twisted loops play a significant role in physical systems with symmetries, see for example [CFV23, Section 6.2].Consider the chain complex RFC φ (∂W, M ) generated by a suitable Morse function on the critical manifold Crit(A H φ ), where with R ∈ X(∂W ) denoting the Reeb vector field.We then define twisted Rabinowitz-Floer homology as the Morse-Bott homology with coefficients in Z 2 by where the boundary map ∂ counts twisted negative gradient flow lines modulo two with respect to a suitable dλ-compatible almost complex structure on M .This homology theory has the following crucial properties.
2. Twisted Rabinowitz-Floer homology is indeed a generalisation of the standard Rabinowitz-Floer homology RFH(∂W, M ) defined in [CF09], as contradicting property 4.However, if Fix(φ| ∂W ) = ∅, then one cannot directly conclude the existence of a twisted periodic Reeb orbit on ∂W .This is for example the case for the rotation φ : C n → C n from the beginning.So the best one can hope for is some kind of forcing result to hold.More precisely, if we know that there exists a sufficiently well-behaved twisted periodic Reeb orbit, then this forces the existence of another one.The above observation is already a forcing result, as Fix(φ| ∂W ) is precisely the set of all constant twisted periodic Reeb orbits on the contact manifold ∂W .

Preliminaries on Twisted Stable Hypersurfaces
Definition 2.1 (Stable Hypersurface, [CFP10, p. 1774]).Let (M, ω) be a connected symplectic manifold.A stable hypersurface in (M, ω) is a compact and connected hypersurface Σ ⊆ M such that the following conditions hold: 1. Σ is separating, that is, M \ Σ consists of two connected components M ± , where M − is bounded and M + is unbounded.
2. There exists a vector field X in a neighbourhood of Σ such that X is outward-pointing to Σ ∪ M − and ker ω| Σ ⊆ ker L X ω| Σ .
Example 2.1 (Star-Shaped Hypersurfaces).Consider the Liouville automorphism for m ≥ 2 an integer and k 1 , . . ., k n ∈ Z coprime to m.Let f ∈ C ∞ (S 2n−1 ) be a positive function such that f • φ = f .Then the star-shaped hypersurface is a contact manifold with φ-invariant contact form λ| Σ f , where (y j dx j − x j dy j ) = i 4 n j=1 (z j dz j − z j dz j ) with complex coordinates z j = x j + iy j .Indeed, by [FK18, Lemma 12.2.2],we have that for the defining Hamiltonian function Hence (Σ f , λ| Σ f ) is a contact manifold as the Liouville vector field satisfies i X dλ = λ and is outward-pointing as Finally, we conclude that is the Reeb vector field.
Example 2.2 (Magnetic Torus, [CFP10, Section 6.1]).Let T n be the standard flat torus for n ≥ 2 and let J : R n → R n be an antisymmetric nonzero linear map.Define ρ ∈ Ω 2 (T n ) by setting ρ(•, •) := ⟨•, J•⟩ and denote by ω ρ = dp ∧ dq + π * ρ the magnetic symplectic form on Define A := (J| im J ) −1 and α ∈ Ω 1 (im J) by Then Σ k is a displaceable stable Hamiltonian manifold for every k > 0 by [CFP10, Proposition 6.3].The stabilising form λ on Σ k is given by where pr ⊥ : R n → ker J, pr ∥ : R n → im J, pr : denote the projections with respect to the orthogonal splitting Let φ ∈ Diff(T n ) be an isometry of finite order satisfying and consider the cotangent lift Then clearly φ(Σ k ) = Σ k as φ is an isometry and Dφ † is of finite order as φ is.Moreover, we have that Dφ † ∈ Symp(T * T n , ω ρ ), as Dφ † ∈ Symp(T * T n , dp ∧ dq) and Dφ † preserves ρ by assumption (2).Lastly, we have that φ * λ = λ by considering formula (1) together with assumption (2), and thus also φ where φ F := ϕ X F 1 denotes the time-1-map of the smooth flow of the time-dependent Hamiltonian vector field X Ft .
Definition 2.5 (Symplectic Asphericity, [MS12, p. 302]).A connected symplectic manifold (M, ω) is said to be symplectically aspherical, if Equivalently, (M, ω) is symplectically aspherical if and only if for the de-Rham-homology class Example 2.4 (Magnetic Torus).The magnetic torus (T * T n , ω ρ ) from Example 2.2 is symplectically aspherical as ω ρ = dλ θ is exact with for all q ∈ T n by [CFP10, Lemma 6.2], where π : T * T n → T n denotes the projection.Alternatively, the magnetic cotangent bundle (T * T n , ω ρ ) is symplectically aspherical as we have and a filling v ∈ C ∞ (D, M ) on the unit disc such that v(e 2πit ) = v(t) for all t ∈ T. We denote the space of all contractible twisted periodic loops of M and φ by Λ φ M .
Definition 2.7 (Twisted Rabinowitz Action Functional).Let (Σ, ω| Σ , λ) be a twisted stable hypersurface in a symplectically aspherical symplectic manifold (M, ω).For a defining Hamiltonian function H for Σ with H • φ = H, we define the twisted Rabinowitz action functional Let X ∈ X(γ) be a twisted variation, that is, X is a vector field along γ and satisfies the condition Then a routine computation shows that where R ∈ X(Σ) is the stable Reeb vector field.If J is a φ-invariant almost complex structure compatible with ω, then the gradient grad e sJ pds + q = φ(q), e τ J p = Dφ −1 (q) T p, and ∥p∥ 2 = 2k.
A computation similar to [CFP10, p. 1843] shows 2. T x C = ker Hess A (x) for all x ∈ C for the Hessian Hess A of A .
1.There are constants C 0 , C 1 > 0 with 2. The sectional curvature of the metric is bounded above, and its injectivity radius is bounded away from zero.

A Forcing Theorem for Twisted Periodic Reeb Orbits
Let (W, λ) be a connected Liouville domain with connected boundary Σ := ∂W .Let (M, λ) be the completion of (W, λ) and φ ∈ Aut(W, λ) a Liouville automorphism, that is, φ ∈ Diff(W ) is a diffeomorphism of finite order such that φ * λ = λ.In this setup, the kernel of the twisted Rabinowitz action functional A H φ admits the canonical description Theorem 2.1.Let Σ ⊆ C n , n ≥ 2, be a compact and connected star-shaped hypersurface invariant under the rotation for some m ≥ 2 and k 1 , . . ., k n ∈ Z coprime to m. Assume that there exists a nondegenerate noncontractible simple periodic Reeb orbit (γ 0 , τ 0 ) on Σ/Z m .Then there does exist a noncontractible periodic Reeb orbit (γ, τ ) on Σ/Z m such that 0 < τ − τ 0 ≤ e(Σ).
Consequently, we have two cases.
2. If γ is a p-fold iterate of γ 0 , then the period τ 0 is small in the sense of Remark 2.2.We cannot conclude the existence of two geometrically distinct noncontractible periodic Reeb orbits as in [ALM22, Theorem 1.2] from Theorem 2.1 even under the additional assumption that Σ is dynamically convex.Indeed, the sphere Σ = S 2n−1 is dynamically convex by the Hofer-Wysocki-Zehnder Theorem [FK18, Theorem 12.2.1],and both cases do occur there.
Example 2.8 (The Hénon-Heiles Hamiltonian, [Sch20, Section 2]).Consider the mechanical Hamiltonian function This Hamiltonian function is known as the Hénon-Heiles Hamiltonian.On R 4 ∼ = C 2 consider the coordinates z := q 1 + iq 2 and w := We have that φ * λ = λ for For every 0 < k < 1 6 , the regular energy surface H −1 (k) contains a strictly convex sphere-like component Σ k ∼ = S 3 .The resulting quotient Σ k /Z 3 is diffeomorphic to the lens space L(3, 1), but not contactomorphic to it with the standard contact distribution.Here we write L(m, k 2 ) for the lens space S 3 /Z m from Example 2.1 with k 1 = 1.Instead, the quotient Σ k /Z 3 is contactomorphic to L(3, 2) with its standard contact distribution.This is mainly due to the use of different coordinates.By a shooting argument, one can show that there exist at least two Z 3 -symmetric periodic orbits on Σ k .In fact, by [Sch20, Corollary 2.5], there exist infinitely many periodic orbits on Σ k .
Then K is invariant under the rotation φ and thus Σ/Z 4 is diffeomorphic to the lens space L(4, 1), but again due to the choice of nonstandard coordinates not contactomorphic to it.It is a delicate question in Contact Topology to decide the correct value of k 2 ̸ = 1, such that the obtained lens space L(4, 1) in Hill's lunar problem is contactomorphic to L(4, k 2 ).The tight contact structures on the lens spaces L(m, k 2 ) are classified up to isotopy by [Hon00, Therorem 2.1], so in principle it should be possible to obtain the correct value of k 2 .
Example 2.10 (Stark-Zeeman Systems).Planar Stark-Zeeman systems as in [CFK17] and [CFZ23] generalise many important physical systems including the diamagnetic Kepler problem and the restricted three body problem [Mor22].By [CFK17, Corollary 1], for energy values below the first critical value, the Moser regularised energy hypersurfaces are diffeomorphic to the unit cotangent bundles S * S n .In particular, for n = 2 we obtain S * S 2 ∼ = RP 3 , a real projective space.
Theorem 2.1 immediately follows from a more general result.
Theorem 2.2 (Forcing).Let Σ be a twisted stable displaceable hypersurface in a symplectically aspherical, geometrically bounded, symplectic manifold (M, ω) for a symplectomorphism φ ∈ Symp(M, ω) of finite order ord(φ) and suppose that v 0 is a contractible twisted periodic Reeb orbit on Σ belonging to a Morse-Bott component C. Then there exists a contractible twisted periodic Reeb orbit v / ∈ C such that Remark 2.3.The case (M, Σ) = (C n , S 2n−1 ) with the rotation shows that the estimate in Theorem 2.2 is sharp.
Applying Theorem 2.2 to the Morse-Bott component Fix(φ| Σ ) from Example 2.6 yields the following corollary.
Corollary 2.1.Let Σ be a twisted stable displaceable hypersurface in a symplectically aspherical, geometrically bounded, symplectic manifold (M, ω) for φ ∈ Symp(M, ω) with Fix(φ| Σ ) ̸ = ∅.Then there does exist a nonconstant contractible twisted periodic Reeb orbit v such that In particular, if we take φ = id M in Corollary 2.1, we recover Schlenk's theorem as stated in [CFP10, Theorem 4.9] about the existence of contractible closed characteristics on stable, displaceable hypersurfaces with energy less or equal to the displacement energy of the hypersurface.Schlenk proved this result in [Sch06, Theorem 1.1] using quite different methods.
Example 2.11 (Magnetic Torus).We can apply Theorem 2.2 and its Corollary 2.1 to the magnetic torus in Example 2.2.Indeed, (T * T n , ω ρ ) is geometrically bounded by Example 2.7 and symplectically aspherical by Example 2.4.Moreover, Σ k is stable and displaceable for every energy value k > 0. Thus for every contractible twisted periodic Reeb orbit v 0 belonging to a Morse-Bott component, there does exist a contractible twisted periodic Reeb orbit v with Further applications of Theorem 2.2 and its Corollary 2.1 are the content of the next section.The proof of Theorem 2.2 is given in Section 4. It is also the aim of future research to numerically investigate the Examples 2.8, 2.9 and 2.10, that is, finding upper bounds of the displacement energy and minimal periods.

The Hofer Distance and Relative Symplectic Capacities
Computing the displacement energy 2.4 is usually quite difficult.Sometimes it is possible to give upper bounds on the displacement energy as in [Iri14, Theorem 1] or lower bounds as for any nonempty open subset A ⊆ M of a symplectic manifold (M, ω) we have e(A) > 0 as in [BHS18, Theorem 1.1].Corollary 2.1 has two immediate consequences.First, the existence of a nonconstant contractible twisted periodic Reeb orbit on any twisted stable displaceable hypersurface.Second, the existence of a lower bound for the displacement energy via the action value of this critical point.If the hypersurface is of contact type, this action value is precisely the period of the parametrised periodic Reeb orbit.We illustrate the usefulness of the second implication and give dynamical proofs of standard results.Recall, that a relative symplectic capacity on R 2n is a map c which assigns to each subset A ⊆ R The periodic Reeb flow on ∂B 2n (r) is given by Hence the parametrised periodic Reeb orbits are (ϕ Rr (z), τ ) with τ ∈ πr 2 Z.But Corollary 2.1 implies the existence of a nonconstant closed periodic Reeb orbit (v, τ ) on the contact hypersurface ∂B 2n (r) such that where This is only possible for τ = πr 2 and the statement follows.
Proof.If A ⊆ M is not displaceable, we have that e(A) = +∞ and thus there is nothing to show.Moreover, if A is not compact, we define e(A) := sup So we can assume that A is displaceable by a compactly supported Hamiltonian symplectomorphism φ F ∈ Ham c (R 2n , dy ∧ dx).As A is displaceable and has nonempty interior, there exists a closed ball B(r) of radius r such that Since the displacement energy is a normalised relative symplectic capacity by Proposition 3.1, we conclude that e(A) ≥ e(B(r)) = πr 2 > 0.
Proof.Let φ ∈ Ham c (R 2n , dy ∧ dx) be not equal to the identity.Thus there exists a set A with nonempty interior such that φ(A) ∩ A = ∅.Thus Lemma 3.2 implies This proves the statement.

Physical Systems and the Mañé Critical Value
Proposition 3.3.Let (T * M, dp ∧ dq, H) be a Hamiltonian system for a compact configuration space M and define e 0 (H) : where π T * M : T * M → M denotes the projection.Suppose that Σ k := H −1 (k) with k < e 0 (H) is a φ-twisted stable regular energy surface admitting a contractible twisted periodic Reeb orbit (q 0 , p 0 ) belonging to a Morse-Bott component C. Then there exists a contractible twisted periodic Reeb orbit (q, p) / ∈ C such that Proof.We claim that e(Σ k ) < +∞ for all k < e 0 (H).In particular, every energy hypersurface Σ k is displaceable in the geometrically bounded and symplectically aspherical symplectic manifold (T * M, dp ∧ dq) since T * M is an exact symplectic form with canonical Liouville form pdq.
As k < e 0 (H), we can displace Σ k into the missing fibres.The explicit compactly supported Hamiltonian symplectomorphism achieving that is constructed in [Con06, Proposition 8.2].Hence if Σ k is twisted stable and k < e 0 (H), we conclude the existence of such a contractible periodic Reeb orbit from Theorem 2.2.
Example 3.1 (Magnetic Torus).Let M be a compact manifold and θ ∈ Ω 1 (M ).Then the map φ θ : (T * M, dp ∧ dq) → (T * M, ω dθ ), φ θ (q, p) := (q, p − θ q ) is an exact symplectomorphism.Indeed, for every (q, p) ∈ T * M and v ∈ T T * (q,p) M we compute , where λ ∈ Ω 1 (T * M ) denotes the canonical Liouville form and φ −θ • φ θ = id T * M .A mechanical Hamiltonian function for some potential function V ∈ C ∞ (M ) is transformed under φ θ to a magnetic Hamiltonian function H θ = φ * θ H given by In the case of a magnetic torus as in Example 2.11, we have that Thus if k > 0, the intersection of Σ k = H −1 θ (k) with T * q T n is a sphere centred at θ q for every q ∈ T n .For more details see [AS19, Example 5.2].Consequently, we have that e 0 = 0 and Proposition 3.3 cannot be applied.Note that the Mañé critical value c is infinite in this case because a nonzero ρ has no bounded primitives in R n .Remark 3.2.In the setting of Proposition 3.3, if H is a Tonelli Hamiltonian function, that is, H is strictly fibrewise convex and superlinear, then any stable energy level of H does contain a periodic Reeb orbit by [MP10].See also [Abb13,Theorem (iv)].H(q, θ q ), where the infimum is taken over all 1-forms θ on the universal covering manifold M with dθ = ρ, and H ∈ C ∞ (T * M ) denotes the lift of H.We always have that c ≥ e 0 (H).
If k > c, then the Rabninowitz-Floer homology RFH * (Σ k , T * M ) of [CFP10] is well-defined and does not vanish.In particular, Σ k is not displaceable.Thus we cannot apply Theorem 2.2 in that case.

Proof of Theorem 2.2
The proof of Theorem 2.2 uses a method called a "homotopy of homotopies argument".Fix ε > 0 and choose a Hamiltonian function F ∈ C ∞ c (M × [0, 1]) satisfying This is possible by definition of the displacement energy 2.4.Next we need to carefully choose a twisted defining Hamiltonian function H for the stable hypersurface Σ.We postpone the construction of this Hamiltonian function and explain the main idea of the proof.Choose a smooth family ( for all r ∈ [0, +∞).Define a family of twisted Rabinowitz action functionals For a suitable φ-invariant ω-compatible almost complex structure we consider the moduli space Note that always (v 0 , τ 0 , 0) ∈ M and that such a φ-invariant ω-compatible almost complex structure always exists by [MS17, Lemma 5.5.6].The gradient grad A r of A r is taken with respect to the metric then M is compact.
As a corollary of Lemma 4.1 we get Theorem 2.2.Indeed, the moduli space M is the zero level set of a Fredholm section of a bundle over a Banach manifold.As v 0 belongs to a Morse-Bott component, the Fredholm section is regular at the point v 0 , that is, the linearisation of the gradient flow equation is surjective there.By compactness, we can therefore perturb the Fredholm section to make it transverse.Hence M is a compact smooth manifold with boundary

8.
Step 2: There exists r 0 ∈ R such that r ≤ r 0 for all (u, τ, r) ∈ M .Crucial is the existence of a constant δ > 0 such that This is proven along the lines of [CF09, Lemma 3.9].With this inequality and Step 1 we estimate and thus we can set Step 3: There exists a constant C > 0 such that ∥τ ∥ ∞ ≤ C for all (u, τ, r) ∈ M .This is a delicate estimate based on the explicit construction of the defining Hamiltonian H for Σ as well as an extension of the stabilising form λ. The bound on the Lagrange multiplier is derived by comparing the twisted Rabinowitz action functional to a different action functional.This modified version of the twisted Rabinowitz action functional is obtained using a suitable extension of the φ-invariant stabilising form λ ∈ Ω 1 (Σ) to a compactly supported form β λ ∈ Ω 1 (M ).The precise constructions can be found in [CFP10, Section 4. The main point in the choice of the φ-invariant H ∈ C ∞ (M ), β λ ∈ Ω 1 (M ) and the ω-compatible φ-invariant almost complex structure J is to make sure, that the properties 1. d A 0 (v, τ )(X, η) = m grad A 0 (v, τ ), (X, η) , 2. (m − m) (X, η), (X, η) ≤ 0, are true for all (v, τ ) ∈ Λ φ M ×R and (X, η) ∈ T (v,τ ) Λ φ M ×R.These two conditions ensure that the difference A 0 − A 0 is a Liapunov function for the negative gradient flow lines of the twisted Rabinowitz action functional A 0 .The uniform bound on the Lagrange multiplier τ now follows from Steps 1 and 2. For details see [CFP10, p. 1808].The only subtle difference in our case is, that everything needs to be φ-invariant.However, this is no problem as we explain now.The construction of H, β λ and J is based on the existence of a stable tubular neighbourhood of Σ, that is, a pair (ρ 0 , ψ) with ρ 0 > 0 and ψ : (−ρ 0 , ρ 0 ) × Σ → M an embedding such that .Indeed, the uniform L ∞ -bound on the sequence (u k ) follows from the assumption that (M, ω) is geometrically bounded and the uniform L ∞ -bound on the derivatives (Du k ) follows from the absence of bubbling as (M, ω) is symplectically aspherical.In particular, there cannot exist a nonconstant J-holomorphic sphere when the sequence of derivatives is unbounded [MS12, Section 4.2].Denote the limit of this subsequence by (u, τ, r).This limit clearly satisfies the first equation in (3), thus one only needs to check the asymptotic conditions in (3).Again by compactness, (u, τ ) converges to critical points (w ± , τ ± ) of A 0 at its asymptotic ends.We claim that A r (u(s), τ (s), s) ∈ −∥F ∥ + A 0 (v 0 , τ 0 ), ∥F ∥ + A 0 (v 0 , τ 0 ) ∀s ∈ R. (6)

Remark 3. 3 .
The proof of Proposition 3.3 does not work for higher energy values in general.This is due to a theorem of Will Merry in [Mer11, Theorem 1.1] and [Mer11, Remark 1.7].Let H ∈ C ∞ (T * M ) be a Tonelli Hamiltonian function.Define the Mañé critical value c := inf θ sup q∈ M