Contact forms with arbitrarily large systolic ratio: a construction without plugs
aa r X i v : . [ m a t h . S G ] M a r CONTACT FORMS WITH ARBITRARILY LARGE SYSTOLICRATIO: A CONSTRUCTION WITHOUT PLUGS
MURAT SA ˘GLAM
Abstract.
If a contact form on a (2 n + 1)-dimensional closed contact man-ifold admits closed Reeb orbits, then its systolic ration is defined to be thequotient of ( n + 1)-th power of the shortest period of Reeb orbits by the con-tact volume. We prove that every co-oriented contact structure on any closedcontact manifold admits a contact form with arbitrarily large systolic ratio.This statement generalizes the result of Abbondandolo et al. in dimensionthree to higher dimensions. The proof is inductive and uses the three dimen-sional result as its basis step and relies on the Giroux correspondence for theinductive step. The proof does not require any plug construction that is usedby Abbondandolo et al. and by the author in the previous version of the proof. Contents
1. Introduction 12. Generalities on Giroux’s correspondence in higher dimensions 33. The result 9References 171.
Introduction
A classical question in Riemannian geometry is the existence of an upper boundon the length of the shortest non-constant closed geodesic in terms of the Riemann-ian area on a given closed surface. More specifically on a given closed surface S ,one studies the functional(1) ρ ( S, g ) = l min ( S, g ) area( S, g ) , on the space of all Riemannian metrics, which is invariant under scaling. Here, l min ( S, g ) denotes the length of the shortest non-constant closed geodesic and area(
S, g )denotes the area of S with respect to the metric g .In 1949, Loewner showed that if in (1), l min ( S, g ) is replaced by sys ( S, g ), namelythe length of a shortest non-contractible geodesic, the corresponding ratio ρ nc ( T , · )admits an optimal bound. In 1952, Pu proved the existence of an optimal boundon ρ nc ( RP , · ). In fact, in both statements the metrics that maximize ρ nc do notadmit any contractible geodesic and hence they also maximize (1). In early 80’s,Gromov proved that ρ nc ( S, · ) ≤ S but this bound is in general non-optimal [Gro83]. In fact in [Gro83], Gromov studied the so called systolic ratio in Date : March 2020. any dimension and showed that for any essential n-dimensional closed manifold M , ρ nc ( M, g ) = sys ( M, g ) n vol( M, g )admits an upper bound, which depends only on the dimension. On the other hand,in late 80’s Croke gave the first upper bound on ρ ( S · ) [Cro88], which was laterimproved by several authors.A natural direction for the generalization of the problem is weakening the Rie-mannian assumption on the metric. In fact, the ratios ρ and ρ nc generalize to theFinsler setting by replacing the Riemannian area with the Holmes-Thompson areaand the bounds on ρ generalize to the Finsler case [APBT16]. For the detailedaccount of results about the systolic ratio in Riemannian and Finsler geometry, werefer to [ABHS18a] and [ABHS18b].The systolic ratio ρ naturally generalizes to contact geometry. The contact sys-tolic ratio on a closed contact manifold ( V, ξ ) is defined to be the scaling invariantfunctional ρ ( V, α ) := T min ( V, α ) n +1 vol( V, α )on the space of all contact forms on (
V, ξ ). Here, T min ( V, α ) denotes the minimumamong the periods of all orbits of the Reeb vector field R α andvol( V, α ) := Z V α ∧ ( dα ) n is the contact volume of V associated to the contact form α .We note that the contact systolic ratio is not merely a generalization of thenotion to a dynamical system but it is strongly related to the classical question. Infact, given a smooth Finsler manifold ( M, F ), the canonical Liouville 1-form pdq on the cotangent bundle T ∗ S , restricts to a contact form α F on the unit cotangentbundle S ∗ F M . In this case, the Reeb flow is nothing but the geodesic flow restrictedto S ∗ F M and up to a universal constant, the contact volume vol( S ∗ F M, α F ) is theHolmes-Thompson volume of ( M, F ). Hence the contact systolic ratio of ( S ∗ F M, α F )recovers the classical systolic ratio of ( M, F ).But it turns out that it is not possible to bound the contact systolic ratio globally.In the case of the tight 3-sphere ( S , ξ st ), it was shown in [ABHS18a] that thesystolic ratio can be made arbitrarily large. Yet it was also shown that the Zollcontact forms , namely the contact forms for which all Reeb orbits are closed andshare the same minimal period, are maximizers of the functional ρ ( S , · ) if thefunctional is restricted to a C -neighbourhood of all Zoll contact forms. For anycontact 3-manifold ( M, ξ ), the non-existence of a global bound on ρ ( M, · ) is laterproved by the same authors in [ABHS18b] whereas in [BK18], the local bound on ρ ( S , · ) was generalized to all contact 3-manifolds that admit Zoll contact forms.The aim of this paper is to prove that the contact systolic ratio is unbounded inany dimension. Here we need to point that ρ ( V, α ) makes sense only if the Reebvector field R α admits a closed orbit. If dim V = 3, by a result of Taubes [Tau07],we know that any contact form on V admits a closed Reeb orbit but in higherdimensions, this might not be the case. Since we aim for the non-existence of abound on ρ , it is legitimate for us to ignore this issue.The main result of this paper is as follows. Theorem 1.1.
Let ( V, ξ ) be a closed connected co-oriented contact manifold andlet C > be given. Then there exists a contact form α on V such that ker α = ξ and ρ ( V, α ) ≥ C. ONTACT FORMS WITH ARBITRARILY LARGE SYSTOLIC RATIO 3
In [ABHS18b], the above statement is proven in dimension three. The strategyof the proof is as follows. On a given closed co-oriented contact three manifold, oneconstructs a contact form, for which the Reeb flow is Zoll on an invariant domainthat occupies arbitrarily large portion of the total contact volume and away fromthis domain the periods of closed Reeb orbits are bounded away from zero. Thenone modifies the contact form in this large portion with suitable plugs so that themost of the contact volume is eaten up but the minimal period is still boundedaway from zero. The construction of the initial contact form is carried out on asupported open book decomposition. The author of this paper provided a proof ofTheorem 1.1 in [Sag18], which is a direct generalization of the proof of [ABHS18b].Here we present a much simpler proof, which is an inductive proof and relies onthe three dimensional result and again the results of Giroux on higher dimensionalopen books [Gir03, Gir17]. We construct the desired contact form directly on asupported open book without any plug construction. Instead we use the inductionhypothesis, which says that the binding of the open book admits a contact formwith large systolic ratio. The three dimensional result of [ABHS18b] serves as thebasis step of the induction. We note that the construction given here does notapply to dimension three since in this case the binding of the open book is onedimensional and the systolic ratio is always one the binding. In that sense, theplug construction seems to be essential for dimension three.
Acknowledgements.
I thank Marcelo Alves, who pointed out that such a proofshould work and motivated this paper. I thank Alberto Abbondandolo for hiscomments on this manuscript. This work is part of a project in the SFB/TRR 191‘Symplectic Structures in Geometry, Algebra and Dynamics’, funded by the DFG.2.
Generalities on Giroux’s correspondence in higher dimensions
We first summarize the necessary definitions and results concerning the Giroux’scorrespondence between the contact structures and supported open books in higherdimensions. For the details, we refer to [Gir03] and [Gir17].Let F be a 2 n -dimensional domain with boundary K and let F o denote theinterior of F . A symplectic form ω ∈ Ω ( F o ) is called an ideal Liouville structure ,abbreviated by ILS, on F if it admits a primitive λ ∈ Ω ( F o ) such that for some/anysmooth function(2) u : F → [0 , + ∞ ) , where K = u − (0) is a regular level set,the 1-form uλ on F o extends to a smooth 1-form β on F , which is a contact formalong K .If such a 2-form ω exists, then the pair ( F, ω ) is called an ideal Liouville domain ,abbriviated as ILD, and any primitive λ of above property is called an ideal Liou-ville form , abbriviated as ILF. It turns out that given an ILD ( F, ω ), the contactstructure ξ := ker( β | T K )depends on the 2-form ω but not on λ or u , see Proposition 2 in [Gir17]. Moreover,once λ is chosen, one can recover all possible (positive) contact forms on ( K, ξ )by restricting the extension of uλ to K as u moves among the functions with theproperty (2). Hence the pair ( K, ξ ) is called the ideal contact bounday of (
F, ω ).We note that the orientation of K that is determined by the co-oriented contactstructure ξ coincides with the orientation of K as the boundary of ( F, ω ).A very useful feature of an ILD is that the vicinity of its bounday admits anexplicit parametrization by means of which any ILF has a very nice form.
Lemma 2.1.
Let ( F, ω ) be an ILD and λ be an ILF. Let u be a function satisfy-ing (2) and let β be the extension of uλ . Then for any contact form α on ( K, ξ ) , MURAT SA ˘GLAM there exists an embedding ı : [0 , + ∞ ) × K → F such that ı ∗ λ = 1 r α and ı (0 , q ) = q for all q ∈ K, where r ∈ [0 , + ∞ ) .Proof. The above statement is a reformulation of Proposition 3 in [Gir17]. We givea similar but more explicit proof.Let dim F = 2 n . Using ω = dλ = d ( β/u ) on F ◦ we compute(3) ω n = ( d ( β/u )) n = u − n − ( u dβ + nβ ∧ du ) ∧ ( dβ ) n − = u n − µ where we put µ := ( u dβ + nβ ∧ du ) ∧ ( dβ ) n − . Note that by (3), µ is a smooth positive volume form on F . Define the smoothvector field X on F by(4) ı X µ = − nβ ∧ ( dβ ) n − . Since β is by assumption a positive contact form on K , β ∧ ( dβ ) n − is a positivevolume form on K . Recall that the Liouville vector field Y of λ is the vector fieldon F ◦ defined by ı Y dλ = λ . Using β = uλ on F ◦ we compute − nβ ∧ ( dβ ) n − = − nu n λ ∧ ( dλ ) n − = − u n ı Y ω n = − u − ı Y µ. Comparing with (4) we find Y = − uX . Applying ı X to (4) we see that β ( X ) ≡ F ◦ . Hence on F ◦ ,(5) L X β = ı X dβ = − u ı Y ( du ∧ λ + udλ ) = − u ( du ( Y ) λ + uλ ) = 1 u ( du ( X ) − β. This shows that du ( X ) = 1 along K and that the function u ( du ( X ) −
1) is smoothon F .Since F is compact and X points inwards on K , the flow φ t of X is well-defined atevery point of K and for every non-negative time. We define the smooth embeddingΦ : [0 , + ∞ ) × K → F, ( t, q ) φ t ( q ) . By construction we have Φ ∗ X = ∂ t . Put ˆ β := Φ ∗ β , ˆ u := Φ ∗ u , and ˆ λ := Φ ∗ λ . Theidentities β ( X ) = 0 and (5) say that on [0 , + ∞ ) × K ,(6) ˆ β ( ∂ t ) = 0 , ˆ β t = ˆ u t − u ˆ β. The solution of the problem (6) with initial condition β ( q ) = β (0 , q ) isˆ β ( t, q ) = exp (cid:18)Z t ˆ u t ( τ, q ) − u ( τ, q ) dτ (cid:19) β ( q )and therefore ˆ λ ( t, q ) = 1ˆ u ( t, q ) exp (cid:18)Z t ˆ u t ( τ, q ) − u ( τ, q ) dτ (cid:19) β ( q ) . Now let α be a positive contact form on K . Then there is a positive function κ on K such that β = κα . On (0 , + ∞ ) × K define the functionΛ( t, q ) = κ ( q )ˆ u ( t, q ) exp (cid:18)Z t ˆ u t ( τ, q ) − u ( τ, q ) dτ (cid:19) . ONTACT FORMS WITH ARBITRARILY LARGE SYSTOLIC RATIO 5
Then ˆ λ = Λ α . It is clear that Λ >
0, and lim t → Λ( t, q ) = + ∞ for all q ∈ K . Wenote that ∂ Λ ∂t = − Λˆ u < t, q ) = Λ(1 , q ) exp (cid:18) − Z t u ( τ, q ) dτ (cid:19) . On [0 , + ∞ ) × K , ˆ u is bounded from above since F is compact. Therefore lim t → + ∞ Λ( t, q ) =0 for all q ∈ K . It follows that Λ( · , q ) is a diffeomorphism from (0 , + ∞ ) onto (0 , + ∞ )for all q . Hence there exists a positive smooth function f on (0 , + ∞ ) × K such that(8) Λ( f ( r, q ) , q ) = 1 r ∀ ( r, q ) ∈ (0 , + ∞ ) × K. Define the embeddingΨ : (0 , + ∞ ) × K → [0 , + ∞ ) × K, ( r, q ) ( f ( r, q ) , q ) . By construction Ψ ∗ ˆ λ = r α . We claim that Ψ extends to a smooth embedding of[0 , + ∞ ) × K with Ψ(0 , q ) = (0 , q ). Postponing the proof of the claim, we note that ı = Φ ◦ Ψ is the desired embedding. The rest of the statement of the lemma followsimmediately from the identity ı ∗ λ = r α .We want to show that the function Ψ : [0 , + ∞ ) × K → [0 , + ∞ ) × K withΨ(0 , q ) = 0 for all q ∈ K is a smooth embedding. We first combine (7) and (8) andget(9) Λ(1 , q ) − exp Z f ( r,q )1 u ( τ, q ) dτ ! = r. We consider the function g ( t, q ) := Λ(1 , q ) − exp (cid:18)Z t u ( τ, q ) dτ (cid:19) on (0 , + ∞ ) × K and we define˜Ψ : (0 , + ∞ ) × K → (0 , + ∞ ) × K, ˜Ψ( t, q ) = ( g ( t, q ) , q ) . Then we have ˜Ψ ◦ Ψ( r, q ) = ˜Ψ( f ( r, q ) , q ) = ( g ( f ( r, q ) , q ) , q ) = ( r, q )on (0 , + ∞ ) × K . We claim that ˜Ψ extends smoothly on [0 , + ∞ ) × K with ˜Ψ(0 , q ) =(0 , q ) for all q ∈ K . In order to see this, we define v ( t, q ) := Z t ˆ u t ( τ, q ) − u ( τ, q ) dτ. We note that by the integrand above is the restriction of the function ( du ( X ) − /u ,which is smooth and bounded on F by (5), to the subset [0 , + ∞ ) × K . Hence theintegrand is smooth and bounded on [0 , + ∞ ) × K and so is the fuction v . For any t ∈ (0 , + ∞ ) and q ∈ K , e v = exp (cid:18)Z t ˆ u t − u dτ (cid:19) = exp (cid:18) log ˆ u ( t, q ) − log ˆ u (1 , q ) − Z t u dτ (cid:19) ⇒ exp (cid:18)Z t ˆ u t − u dτ (cid:19) = ˆ u ( t, q )ˆ u (1 , q ) exp (cid:18) − Z t u dτ (cid:19) ⇒ g ( t, q ) = Λ(1 , q ) − e − v ( t,q ) ˆ u ( t, q )ˆ u (1 , q ) . MURAT SA ˘GLAM
Note that Λ(1 , q ) = 0. The above expression says that g smooth on [0 , + ∞ ) × K and g (0 , q ) = 0 for any q ∈ K . We compute g t ( t, q ) = Λ(1 , q ) − e − v ( t,q ) (cid:20) − v t ( t, q ) ˆ u ( t, q )ˆ u (1 , q ) + ˆ u t ( t, q )ˆ u (1 , q ) (cid:21) = Λ(1 , q ) − e − v ( t,q ) (cid:20) − (ˆ u t ( t, q ) − u ( t, q ) ˆ u ( t, q )ˆ u (1 , q ) + ˆ u t ( t, q )ˆ u (1 , q ) (cid:21) = Λ(1 , q ) − e − v ( t,q ) u (1 , q ) > . Now it is clear that D ˜Ψ( t, q ) is invertible. By the inverse function theorem, theextension of Ψ over [0 , + ∞ ) × K is continuously differentiable and in fact smoothsince ˜Ψ is smooth. (cid:3) Ideal Liouville domains are particularly useful for clarifying the existence anduniqueness of the contact structures supported by open books in higher dimensions.We first recollect some facts on open books.An open book in a closed manifold V is a pair ( K, Θ) where(ob1) K ⊂ V is a closed co-dimension two submanifold with trivial normal bundle;(ob2) Θ : V \ K → S = R / π Z is a locally trivial fibration such that K has aneighbourhood U , which admits a parametrization ( re ix , q ) ∈ D × K ∼ = U so that Θ reads as Θ( re ix , q ) = x on U .The submanifold K is called the binding of the open book and the closures of thefibres of Θ are called the pages . All the pages are compact manifolds, for whichthe binding is the common boundary. We note that the canonical orientation of S induces co-orientations on the pages and the binding. Hence if V is oriented thenso are the pages and the binding. Another way of defining an open book is thefollowing. Let h : V → C be a smooth function such that(1) h vanishes transversely;(2) Θ := h/ | h | : V \ K → S has no critical points, where K := h − (0).Then the pair ( K, Θ) is an open book in V . Moreover, any open book in V may berecovered via a defining function h as above and such a defining function is uniqueup to multiplication by a positive function on V .Given an open book ( K, Θ) in a closed manifold V , one finds a vector filed X ,refered as a spinning vector field , on V such that(m1) X lifts to a smooth vector field on the manifold with boundary obtainedfrom V by a real oriented blow-up along K ;(m2) X = 0 on K and (Θ ∗ dx )( X ) = 2 π on V \ K .Then the time-one-map of the flow of X is a diffeomorphism φ : F → F of the 0th-page F := Θ − (0) ∪ K , which fixes K pointwise. The isotopy class [ φ ] of φ among the diffeomorphisms of F , which fixes K pointwise, is called the monodromy of the open book and it turns out that the open book is characterized by the pair( F, [ φ ]). Namely, given the pair ( F, φ ), one defines the mapping torus
M T ( F, φ ) := ([0 , π ] × F ) (cid:14) ∼ ; (2 π, q ) ∼ (0 , φ ( q )) , which is a manifold with boundary. One has the natural fibrationˆΘ : M T ( F, φ ) → S , where all fibres are diffeomorphic to F and there is a natural parametrization of thefibre ˆΘ − (0) via the restriction of the above quotient map to { } × F . It turns out ONTACT FORMS WITH ARBITRARILY LARGE SYSTOLIC RATIO 7 that if φ ′ ∈ [ φ ], then there is a diffeomorphism between M T ( F, φ ) and
M T ( F, φ ′ )that respects the fibrations over S and the natural parametrizations of the 0-thpages. Now given M T ( F, φ ), one collapses its boundary, which is diffeomorphic to S × K , to K and obtains so called the abstract open book OB ( F, φ ). In fact, theclosed manifold OB ( F, φ ) admits an open book given by the pair ( K, Θ) where Θis induced from ˆΘ. Moreover, for φ ′ ∈ [ φ ], the diffeomorphism between M T ( F, φ )and
M T ( F, φ ′ ) descends to a diffeomorphism between corresponding abstract openbooks. In particular, V and OB ( F, φ ) may be identified together with their openbook structures. We note that one may choose a vector field X that is actuallysmooth on V (compare with (m1)) and even 1-periodic near K . But it is not possibleto obtain any given representative of the monodromy class via such a vector field,see Remark 12 in [Gir17]. In fact, in order to obtain all representatives of themonodromy class, one needs to sweep out the whole affine space of spinning vectorfields.Open books meet with the contact topology via the following definition. Let V be a closed manifold and ξ be a co-oriented contact structure on V . We say ξ is supported by an open book ( K, Θ) on V if there is a contact form α on ( V, ξ ), thatis ξ = ker α , such that • α restricts to a (positive) contact form on K ; • dα restricts to a (positive) symplectic form on each fibre of Θ.It turns out that given a closed contact manifold V , the isotopy classes of co-orientedcontact structures are in one-to-one correspondence with (equivalence classes of)supporting open books. This statement is a very rough summary of what is calledthe Gioux correspondence. We will recall certain pieces of this celebrated statementin detail. Theorem 2.1. (Theorem 10 in [Gir03] ) Any contact structure on a closed manifoldis supported by an open book with Weinstein pages.
The above statement is the core part of the correspondence between supportedopen books and contact structures. In fact the existence statement for the oppositedirection is relatively easy to achieve, especially in dimension three. Namely, givenan open book in a 3-dimensional closed manifold, it is not hard to construct acontact form on the corresponding abstract open book, whose kernel is supported.It turns out that in higher dimensions, one needs to a have an exact symplectic pageand a symplectic monodromy in order to construct a contact form on an abstractopen book, whose kernel is supported, see Proposition 9 in [Gir03] and Proposition17 in [Gir17]. We will carry out such a construction in the next section. Concerningthe uniqueness features of the Giroux correspondence, we are mainly interested inone side, namely the ”uniqueness” of supported contact structures. It turns outthat such a statement is again more involved in higher dimensions. Philosophically,given an open book, the symplectic geometry of the pages determines the supportedcontact structures and in dimension three, any two symplectic structure on a pageare isotopic since they are simply two area forms on a given surface. But in higherdimensions, this is not the case.In [Gir17], Giroux introduced the notion of a Liouville open book, which clearsout the technicalities that pointed above.A
Liouville open book , abbreviated as LOB, in a closed manifold V is a tripple( K, Θ , ( ω x ) x ∈ S ) where(lob1) ( K, Θ) is an open book on V with pages F x = Θ − ( x ) ∪ K , x ∈ S ;(lob2) ( F x , ω x ) is an ILD for all x ∈ S and the following holds: there is a definingfunction h : V → C for ( K, Θ) and a 1-form β on V such that the restriction MURAT SA ˘GLAM of d ( β/ | h | ) to each page is an ILF. More precisely, ω x = d ( β/ | h | ) | T F ox for all x ∈ S .The 1-form β in (lob2) is called a binding 1-form associated to h . Note that if h ′ isanother defining function for ( K, Θ), then h ′ = κh for some positive function κ on V and β ′ := κβ is a binding 1-form associated to h ′ . We also note that for a fixeddefining function, the set of associated binding 1-forms is an affine space.Similar to classical open books, LOB’s are characterized by the monodromy,which now has to be symplectic. Namely, one considers a symplectically spinningvector field , that is a vector filed X satisfying (m1)-(m2) and generating the kernelof a closed 2-form on V \ K , which restricts to ω x for all x ∈ S . Given such avector field, the time-one-map of its flow, say φ , is a diffeomorphism of F := F ,which fixes K and preserves ω := ω . The isotopy class [ φ ], among the symplecticdiffeomorphisms that fixes K , is called the symplectic monodromy and characterizesthe given LOB. For the construction of a LOB in the abstract open book OB ( F, φ ),where φ ∗ ω = ω , we refer to Propostion 17 in [Gir17] and our construction in thenext section.Similar to the classical open books, symplectically spinning vector fields form anaffine space and all representatives of the symplectic monodromy may be obtainedby sweeping out this affine space. It turns out that the obvious choice of a symplec-tically spinning vector field is actually smooth and by modifying a given binding1-form along Θ, it is possible to get a symplectically spinning vector filed, whoseflow is 1-periodic near the binding. Lemma 2.2. (Lemma 15 in [Gir17] ) Let ( K, Θ , ( ω x ) x ∈ S ) be a LOB on a closedmanifold V and h : V → C be a defining function for ( K, Θ) . Then for everybinding 1-form β , the vector field X on V \ K spanning the kernel of d ( β/ | h | ) andsatisfying (Θ ∗ dx )( X ) = 2 π extends to a smooth vector field on V which is zeroalong K. Furthermore, β can be chosen so that X is 1-periodic near K. Natural sources of LOB’s are contact manifolds, namely we have the followingstatement.
Proposition 2.1. (Proposition 18 in [Gir17] ) Let ( V, ξ ) be a closed contact mani-fold, and ( K, Θ) be a supporting open book with defining function h : V → C . Thenthe contact forms α on ( V, ξ ) such that d ( α/ | h | ) induces an ideal Liouville structureon each page form a non-empty convex cone. Let ( K, Θ , ( ω x ) x ∈ S ) be a LOB on a closed manifold V with a defining function h . A co-oriented contact structure ξ on V is said to be symplectically supported by( K, Θ , ( ω x ) x ∈ S ) if there exists a contact form α on ( V, ξ ) such that α is a binding1-form of the LOB associated to h .By our remark following the definition of the binding 1-form, the definition ofbeing symplectically supported is independent of the given defining function. Butthe crucial fact is that once a defining function is fixed, a contact binding 1-formis unique whenever it exists, see Remark 20 in [Gir17]. Hence, once a definingfunction h is fixed, there is a one-to-one correspondence between contact structuressupported by ( K, Θ , ( ω x ) x ∈ S ) and contact binding 1-forms associated to h . Nowgiven two contact structures ξ and ξ supported by ( K, Θ , ( ω x ) x ∈ S ), there existunique contact binding 1-forms α and α respectively. Since the set of binding1-forms associated to h is affine, there is a path ( β t ) t ∈ [0 , of binding 1-forms suchthat β = α and β = α . Then by modifying β t ’s along the 1-form Θ ∗ dx , onegets a path of contact forms ( β ct ) t ∈ [0 , and a homotopy (cid:0) ( β st ) t ∈ [0 , (cid:1) s ∈ [0 ,c ] betweenthe paths ( β t ) t ∈ [0 , and ( β ct ) t ∈ [0 , such that ONTACT FORMS WITH ARBITRARILY LARGE SYSTOLIC RATIO 9 • for all s ∈ [0 , c ] and t ∈ [0 , β st is a binding 1-form for ( K, Θ , ( ω x ) x ∈ S )associated to h (since β t ’s stay the same along the pages through the mod-ification); • for all s ∈ [0 , c ], β s and β s are contact forms (since if β t is already a contactform then it keeps being a contact form through the modification).In particular, whenever β st is a contact form, ker β st is symplectically supportedby ( K, Θ , ( ω x ) x ∈ S ) and β st is the unique contact binding 1-form associated to h .This tells us that the concatenation of the paths (ker β s ) s ∈ [0 ,c ] , (ker β ct ) t ∈ [0 , and(ker β c − s ) s ∈ [0 ,c ] gives an isotopy between ξ and ξ along the contact structuresthat are symplectically supported by ( K, Θ , ( ω x ) x ∈ S ). In fact the following moregeneral statement holds. Proposition 2.2. (Proposition 21 in [Gir17] ) On a closed manifold, contact struc-tures supported by a given Liouville open book form a non-empty and weakly con-tractible subset in the space of all contact structures. The result
We prove the following version of Theorem 1.1.
Theorem 3.1.
Let ( V, ξ ) be a closed co-oriented contact manifold such that dim V ≥ . Then for any ε > , there exists a contact form α on ( V, ξ ) satisfying T min ( α ) ≥ / and vol( V, α ) ≤ ε . The rest of this section is devoted to the proof of Theorem 3.1. We prove thestatement by induction on dim V = 2 n + 1. For n = 1 the statement follows fromthe main result of [ABHS18b]. Now assume that the statement is true for n − V, ξ ) be given such that dim V = 2 n + 1. By Theorem 2.1, there is an openbook ( K, Θ) in V supported by ξ . Let F x := Θ − ( x ), x ∈ S = R / π Z denote thepages of the open book and let h : V → C be a defining function for ( K, Θ). Wewant to construct a contact form on the abstract open book defined via the 0thpage, namely F := Θ − (0) ∪ K. (10)By Proposition 2.1, there is a contact form α on ( V, ξ ) such that ( K, Θ , d ( α/ | h | ) T F ox )is a LOB, which supports ξ symplectically. By Lemma 2.2, we modify the contactbinding form α only along Θ and obtain a binding 1-form ˆ α , not necessarily contact,such that the associated symplectically spinning vector field X is 1-periodic near K . Hence the time-one-map of the flow of X gives us a diffeomorphism ψ : F → F such that(11) ψ ∗ ( dλ ) = dλ where λ ∈ Ω ( F o ) is the ILF given by λ := (ˆ α/ | h | ) | T F o = ( α/ | h | ) | T F o (12)and ψ = id on some neighbourhood of K in F . Now our aim is to recover V asthe abstract open book induced by the pair ( F, ψ ) and to define a contact form onthe abstract open book with the desired properties. We first consider the mappingtorus
M T ( F, ψ ) := ([0 , π ] × F ) (cid:14) ((2 π, p ) ∼ (0 , ψ ( p ))) . Since ψ = id on some neighbourhood of K , ∂M T ( F, ψ ) has an open neighbourhoodgiven as a product of K with an annulus, in which we collapse the boundary and getthe abstract open book OB ( F, ψ ). We postpone the precise collapsing procedurefor the moment since it would involve precise choices of coordinates but we note that the abstract open book is independent of these choices. We note the followingidentifications
M T ( F o , ψ ) = M T ( F, ψ ) \ ∂M T ( F, ψ ) = OB ( F, ψ ) \ K. A family of contact form away from the binding.
On [0 , × F o , we definea family of 1-forms α s = dx + s ( λ + β ( x ) λ ψ )(13)where λ ψ := ψ ∗ λ − λ , s is a positive real parameter and β : [0 , π ] → [0 ,
1] isa smooth function such that β (0) = 0, β (2 π ) = 1 and supp( β ′ ) ⊂ (0 , π ). Bythe choice of β , α s descends to a family of 1-forms on M T ( F , ψ ). We have thefollowing observations. Lemma 3.1.
There exists s > , depending on ψ, λ, β such that α s is a contactform on M T ( F , ψ ) for all s ∈ (0 , s ] .Proof. Since dλ ψ = 0, we get dα s = s ( β ′ dx ∧ λ ψ + dλ ) ⇒ ( dα s ) n = s n (cid:0) ( n − β ′ dx ∧ λ ψ ∧ ( dλ ) n − + ( dλ ) n (cid:1) and α s ∧ ( dα s ) n = [ dx + s ( λ + βλ ψ )] ∧ s n (cid:2) ( n − β ′ dx ∧ λ ψ ∧ ( dλ ) n − + ( dλ ) n (cid:3) ⇒ α s ∧ ( dα s ) n s n = dx ∧ ( dλ ) n + sβ ′ λ ∧ dx ∧ λ ψ ∧ ( dλ ) n − . Note that dx ∧ ( dλ ) n is a volume form and the top degree form λ ∧ dx ∧ λ ψ ∧ ( dλ ) n − is compactly supported in M T ( F , ψ ). Hence there exists s > s ∈ (0 , s ]. (cid:3) We study the Reeb vector field R α s of α s on M T ( F o , ψ ). We define the vectorfield Y on M T ( F o , ψ ) so that it is tangent to { x } × F for each x and satisfies ı Y dλ = − β ′ λ ψ along { x } × F for each x . Since ψ is compactly supported in F , Y is compactlysupported in M T ( F o , ψ ). We compute ı ( ∂ x + Y ) dα = s (cid:0) ı ( ∂ x + Y ) β ′ dx ∧ λ ψ + ı ( ∂ x + Y ) dλ (cid:1) = s (cid:0) β ′ dx ( ∂ x + Y ) λ ψ − λ ψ ( ∂ x + Y ) β ′ dx + ı Y dλ (cid:1) = s ( β ′ λ ψ + dλ ( Y, Y ) β ′ dx − β ′ λ ψ )= 0 . Hence on
M T ( F , ψ ), the Reeb vector field of α reads as R α s = ∂ x + Yα s ( ∂ x + Y ) . (14)We note that R α s = ∂ x near K . Since the ∂ x component of R α s never vanishesand Y is tangent to the pages, R α s is transverse to F o × { x } for all x . Hence F isa global hypersurface of sections for R α s on M T ( F , ψ ). We have the first-return-time map τ s : F o → R , τ s ( p ) = inf { t > | φ tR αs (0 , p ) ∈ { } × F o } (15)and the first-return mapΥ : F o → F o ; (0 , Υ( p )) = φ τ s ( p ) R αs (0 , p ) , ∀ p ∈ F o . (16) ONTACT FORMS WITH ARBITRARILY LARGE SYSTOLIC RATIO 11
Remark 3.1.
We note that since R α s is multiple of the vector field ∂ x + Y andlatter is independent of s . Hence the return map Υ is independent of s , whichjustifies the absence of the subscript in (16). We note that for all s ∈ (0 , s ] τ s ≡ , Υ = id on F \ supp ( ψ ) . (17) Lemma 3.2.
There exists s < s such that for all s ∈ (0 , s ] , ≤ T min ( α s )(18) on F o .Proof. We have dx ( R α s ) = 1 α s ( ∂ x + Y ) = 11 + s ( λ ( Y ) + βλ ψ ( Y )) , which converges to 1 uniformly as s →
0. This follows from the fact that Y is compactly supported. Then τ s converges uniformly to 1 and there exists some s < s such that for all s ∈ (0 , s ], 1 / ≤ τ s on F o . The statement then follows. (cid:3) A family of contact forms near the binding.
By inductive hypothesis anda suitable re-scaling, we know that for any ε > σ ε on( K, ξ | K ) such that vol( K, σ ε ) ≤ ε and T min ( σ ε ) ≥ . (19)Given σ ε , by Lemma 2.1 there is an embedding ı ε : [0 , + ∞ ) × K ֒ → F s . t . ı ∗ ε λ = 1 r σ ε . (20)Then there exists r ε > depending only on ψ and σ ε such that(21) ı ε ([0 , r ε ] × K ) ∩ supp( ψ ) = ∅ . We define F ε := F \ ([0 , r ε ) × K )(22)and note that near the boundary of M T ( F ε , ψ ), (13) reads as α s = dx + sr σ ε . (23) Lemma 3.3.
For every ε > and s ∈ (0 , r ε / there exist smooth functions f, g : [0 , r ε ] → R with the following properties. (f1) f ( r ) = s/r near r = r ε and f ( r ) = 1 near r = 0 . (f2) f ( r ε /
2) = 1 / − /r ε ≤ f ′ ≤ , r ε ] and f ′ < r ε / , r ε ].(g1) g = 1 on [ r ε / , r ε ] and g ( r ) = r / near r = 0 . (g2) 0 ≤ g ′ ≤ /r ε on [0 , r ε ] and < g ′ on (0 , r ε / . The easy proof is left to the reader. For later use we define h := f g ′ − f ′ g and notethat(24) 0 < h ≤ r ε + 2 r ε = 6 r ε on (0 , r ε ].In fact, h = − f ′ g > r ε / , r ε ] and h ≥ f g ′ ≥ g ′ / > on (0 , r ε ].Given ε > s ∈ (0 , r ε / α s,ε ( x, r, q ) = g ( r ) dx + f ( r ) σ ε ( q )(25)on [0 , r ε ] × S × K . We note that by (f1) and (g1), α s,ε = r dx + σ ε near r = 0 and therefore α s,ε is smooth on r ε D × K . Lemma 3.4.
For ε > and s ∈ (0 , r ε / , α s,ε is a contact form on r ε D × K .Proof. We compute α s,ε ∧ ( dα s,ε ) n = ( g dx + f σ ε ) ∧ ( g ′ dr ∧ dx + f ′ dr ∧ σ ε + f dσ ε ) n = n h f n − (cid:0) dr ∧ dx ∧ σ ε ∧ ( dσ ε ) n − (cid:1) . (26)By (24), f n − h > α s,ε is a contact form away from K . Near K wehave h ( r ) = r , so that there α s,ε ∧ ( dα s,ε ) n reads n (cid:0) rdr ∧ dx ∧ σ ε ∧ ( dσ ε ) n − (cid:1) , which is a positive volume form at any point on K . (cid:3) An easy computation shows that away from K, the Reeb vector field reads as(27) R α s,ε ( x, r, q ) = − f ′ h ∂ x + g ′ h R σ ε ( q )and has the flow(28) φ tα s,ε ( x, r, q ) = (cid:18) x − f ′ ( r ) h ( r ) t, r, φ g ′ ( r ) th ( r ) σ ε ( q ) (cid:19) , where φ tσ ε is the flow of R σ ε . For (0 , q ) ∈ D × K , we have(29) R α s,ε (0 , q ) = R σ ε ( q ) , φ tR αs,ε (0 , q ) = (cid:0) , φ tσ ε ( q ) (cid:1) . We consider possible closed orbits of R σ ε . Assume φ Tα s,ε ( x, r, q ) = ( x, r, q ) for some T >
0. We have the following cases: • If r ∈ [ r ε / , r ε ], by (g1) φ tα s,ε ( x, r, q ) = (cid:0) x − t, r, φ σ ε ( q ) (cid:1) = ( x − t, r, q ) . Son in order this orbit to close up, one needs T ≥ π . • If r ∈ (0 , r ε / g ′ ( r ) h ( r ) > g ′ ( r ) Th ( r ) ≥ T min ( σ ε ) ≥ . This is a necessary condition for the projection of the orbit to K to closeup. We note that by (f2) h/g ′ = f − f ′ g/g ′ ≥ f ≥ / ⇒ T ≥ h/g ′ ≥ / • If r = 0, then by (29), T ≥ T min ( σ ε ) ≥ T min ( α s,ε ) ≥ r ε D × K . ONTACT FORMS WITH ARBITRARILY LARGE SYSTOLIC RATIO 13
A family of contact forms on OB ( F, ψ ) . For any ε > s ∈ (0 , r ε / α s,ε = α s on M T ( F ε , ψ ) g ( r ) dx + f ( r ) σ ε on r ε D × K (31)on the abstract open book OB ( F, ψ ) =
M T ( F ε , ψ ) ∪ ( r ε D × K )where α s is defined by (13) and f and g are given by Lemma 3.3. By (23) and theproperties (f1) and (g1), α s,ε is a well-defined contact form on OB ( F, ψ ). We firstestimate the volume. Z OB ( F,ψ ) α s,ε ∧ ( dα s,ε ) n = Z MT ( F ε ,ψ ) α s ∧ ( dα s ) n + Z r ε D × K α s,ε ∧ ( dα s,ε ) n . For s ∈ (0 , s ], we have Z MT ( F ε ,ψ ) α s ∧ ( dα s ) n = Z F ε τ s ( dα s | { }× F ε ) n = Z F ε τ s s n ( dλ ) n ≤ s n Z F ε ( dλ ) n , where we use the bound on τ s given in the proof of Lemma 18 for the last inequality.For the second term we have Z r ε D × K α s,ε ∧ ( dα s,ε ) n = Z r ε D × K nhf n − (cid:0) dr ∧ dx ∧ σ ε ∧ ( dσ ε ) n − (cid:1) = 2 πn vol( K, σ ε ) Z r ε hf n − dr = 2 πnε Z r ε hf n − dr ≤ πnε Z r ε hdr ≤ πnε Z r ε r ε dr = 12 πnε. Hence for s ∈ (0 , s ] we get(32) vol( α s,ε ) ≤ s n Z F ε ( dλ ) n + 12 πnε. Now given any ε >
0, we choose ε > πnε ≤ ε /
2. Once ε is chosen, r ε and R F ε ( dλ ) n are fixed. Then we choose s > s < min s , r ε , ε R F ε ( dλ ) n ! n . Since s < r ε / α s,ε is well-defined on OB ( F, ψ ). Since s < s and s < (cid:16) ε R Fε ( dλ ) n (cid:17) n ,we get Z MT ( F ε ,ψ ) α s ∧ ( dα s ) n ≤ ε / α s,ε ) ≤ ε . Finally since s ≤ s and s < r ε /
2, we deduce from (18) and (30) that T min ( α s,ε ) ≥ / . Lemma 3.5.
After applying a diffeomorphism, ker α s,ε is isotopic to ker α . Before proving the above lemma, we note that by Gray’s stability theorem, thereis a diffeomorphism ρ : OB ( F, ψ ) → OB ( F, ψ ) such that ker ρ ∗ α s,ε = ker α . Since T min and the volume are invariant under diffeomorphisms, we have T min ( ρ ∗ α s,ε ) ≥ / ρ ∗ α s,ε ) ≤ ε . Hence the proof of Theorem 3.1 is complete. Proof. (Lemma 3.5) We first want to show that the obvious open book structureon OB ( F, ψ ) is a Liouville open book with the contact binding form α s,ε . Let˜Θ : OB ( F, ψ ) \ K → S be the fibration induced by the projection M T ( F, ψ ) → S . We pick a suitabledefining function ˜ h as follows. We define a smooth function˜ u : F → [0 , ∞ )such that for some suitably chosen d > δ > u ( r, q ) = r for ( r, q ) ∈ [0 , r ε ] × K ,(df2) ˜ u ≡ d on ([0 , r ε + δ ) × K ) c and supp ( ψ ) ⊂ ([0 , r ε + δ ] × K ) c .(df3) ˜ u depends only on r and ∂∂r ˜ u ≥ , r ε + δ ] × K .Note that on supp ( ψ ), ˜ u is constant. Hence the S -invariant extension of ˜ u isa well-defined smooth function on M T ( F, φ ), which constitutes the function | ˜ h | .Pairing | ˜ h | with ˜Θ leads to a well-defined defining function ˜ h for the open book( K, ˜Θ) on OB ( F, ψ ). Note that on r ε D × K , ˜ h is simply the projection to the disc,which is smooth. First we need to check the following. Claim 1: d ( α s,ε / | ˜ h | ) induces an ideal Liouville structure on each fibre of ˜Θ .Proof. We put ˜ λ x := ( α s,ε / | ˜ h | ) | T ( { x }× F ) (33)where { x } × F o = ˜Θ − ( x ). • On { x } × (0 , r ε ] × K : by (df1) we have˜ λ x = f ( r ) r σ ε . (34) Hence up to positive constants, we get d ˜ λ x = f ′ r − fr dr ∧ σ ε + fr dσ ε ⇒ ( d ˜ λ x ) n = f n − f ′ r − fr n +1 dr ∧ σ ε ∧ ( dσ ε ) n − . We note that due to the parametrization (20), dr ∧ σ ε ∧ ( dσ ε ) n − is anegative volume form. By (f1) and (f3), f ′ r − f < d ˜ λ x is apositive symplectic form. • On { x } × [ r ε , r ε + δ ] × K : Note that by (df2), ψ = id on this set. Hencewe have ˜ λ x = sr ˜ u σ ε . (35) Then up to positive constants d ˜ λ x = − ˜ u + r ˜ u r r ˜ u dr ∧ σ ε + 1 r ˜ u dσ ε ⇒ ( d ˜ λ x ) n = − ˜ u + r ˜ u r r n +1 ˜ u n +1 dr ∧ σ ε ∧ ( dσ ε ) n − . By (df3), ˜ u + r ˜ u r > ONTACT FORMS WITH ARBITRARILY LARGE SYSTOLIC RATIO 15 • On { x } × ([0 , r ε + δ ) × K ) c : By (df3), ˆ u ≡ d and˜ λ x = sd ( λ + β ( x ) λ ψ ) ⇒ d ˜ λ x = sd dλ (36) which is clearly symplectic. (cid:3) Now we are in the following situation. On OB ( F, ψ ), we have the Liouville openbook (cid:16) K, ˜Θ , d ( α/ | h | ) | T ( { x }× F ) (cid:17) , (37)which is symplectically supported by the contact structure ξ = ker α . Here α , ξ and h stand for the objects induced by the correspondence between V and OB ( F, ψ )due to the symplectically spinning vector field X on V . Now we have a secondLiouville open book (cid:16) K, ˜Θ , d ( α s,ε / | ˜ h | ) | T ( { x }× F ) (cid:17) , (38)which is symplectically supported by the contact structure ker α s,ε . Note that bythe equations (34), (35) and (36), the ideal Liouville structures ( d ( α s,ε / | ˜ h | ) | T ( { x }× F ) ) x ∈ S are invariant under ∂ x . Claim 2.
There exists a diffeomorphism
Φ : OB ( F, ψ ) → OB ( F, ψ )(39) such that Φ ◦ ˜Θ = ˜Θ ◦ Φ and the restriction of Φ to each fibre is symplectic, that is,for all x ∈ S , Φ ∗ d ( α s,ε / | ˜ h | ) | T ( { x }× F ) = d ( α/ | h | ) T ( { x }× F ) . Proof.
We have the following ideal Liouville structures on the 0-th page:(40) ˜ ω := d ( α s,ε / | ˜ h | ) | T ( { }× F ) , (41) ω := d ( α/ | h | ) | T ( { }× F ) = dλ. We first show that ω t := (1 − t ) ω + t ˜ ω is symplectic on F o for all t ∈ [0 , λ t = (1 − t ) λ + t ˜ λ, t ∈ [0 , F o for all t , where λ is the primitive of ω given by (12) and˜ λ is the primitive of ˜ ω given by (33). Again we compute dλ t on separate pieces of F o . • On { x } × (0 , r ε ] × K : By (34) we have λ t = (1 − t ) 1 r σ ε + t fr σ ε = κ ( r ) r σ ε where κ = (1 − t ) + tf . We have κ > κ ′ < κ ′ r − κ < • On { x } × [ r ε , r ε + δ ] × K : By (35) we have λ t = (1 − t ) 1 r σ ε + t sr ˜ u σ ε = κ ( r ) r σ ε where κ = (1 − t ) + ts/ ˜ u . We have κ > κ ′ ≤ κ ′ r − κ < • On { x } × ([0 , r ε + δ ) × K ) c : By (36) we have dλ t = (1 − t ) dλ + t sd dλ = ((1 − t ) + ts/d ) dλ. Hence ω t = dλ t is symplectic on F o for all t ∈ [0 , t as the angle coordinate x . We thenhave a smooth path of ideal Liouville structures ( ω x ) x ∈ [0 , such that ω = ω and ω = e ω . Moreover by (20) and (f1), ω = ˜ ω = d (cid:18) r σ ε (cid:19) near K . Applying the standard Moser argument to the path ( ω x ) x ∈ [0 , π ] , we get asmooth isotopy ( ψ x ) x ∈ [0 , π ] of F such that(Ψ1) ψ = id;(Ψ2) ψ x = id near K for all x ∈ [0 , π ];(Ψ3) ψ ∗ x ω x = ω = ω for all x ∈ [0 , π ].Note that the ILS’s we consider coincide on a neighbourhood of K so one simplyapplies the Moser trick to the objects with compact support.Now we define Φ : [0 , π ] × F → [0 , π ] × F byΦ( x, p ) := (cid:0) x, ψ π ◦ ψ − x ◦ ψ − ◦ ψ x ( p ) (cid:1) (43)where ψ is the the monodromy that we fixed at the outset of the proof. We notethat Φ(2 π, p ) = (cid:0) π, ψ − ◦ ψ π ( p ) (cid:1) , and by (Ψ1), Φ(0 , ψ ( p )) = (0 , ψ π ( p )) = (cid:0) , ψ ( ψ − ◦ ψ π ( p )) (cid:1) . Hence Φ descends to a diffeomorphism on
M T ( F, ψ ). Since ψ = id near K and ψ x = id near K for each x by (Ψ2), we have that Φ = id on a neighbourhood of ∂M T ( F, ψ ). Hence Φ descends to a diffeomorphism on OB ( F, ψ ). By definition, Φcommutes with e Θ.Now recall that ∂ x is a symplectically spinning vector field for both LOBs (37)and (38). In view of (40) and (41) and identifying { x } × F ◦ with { } × F ◦ via theflow of ∂ x , we can therefore identify d (cid:0) α/ | h | (cid:1) | T ( { x }× F ◦ ) with ω | T ( { x }× F ◦ ) := ω,d (cid:0) α s,ε / | ˜ h | (cid:1) | T ( { x }× F ◦ ) with e ω | T ( { x }× F ◦ ) := e ω. Also recall that ψ ∗ ω = ω . Since ∂ x generates the monodromy ψ and ∂ x preserves e ω ,we also have ψ ∗ e ω = e ω . Therefore, ψ ∗ ω x = ω x for all x ∈ [0 , π ]. Inserting (43) and ONTACT FORMS WITH ARBITRARILY LARGE SYSTOLIC RATIO 17 using (Ψ3) we obtain, with the abbreviation F ◦ x = T ( { x } × F ◦ ),Φ ∗ d (cid:0) α s,ε / | ˜ h | (cid:1) | F ◦ x = Φ ∗ e ω | F ◦ x = (cid:0) ψ π ◦ ψ − x ◦ ψ − ◦ ψ x (cid:1) ∗ e ω | F ◦ x = ψ ∗ x ( ψ − ) ∗ ( ψ − x ) ∗ ψ ∗ π ω π | F ◦ x = ψ ∗ x ( ψ − ) ∗ ( ψ − x ) ∗ ω | F ◦ x = ψ ∗ x ( ψ − ) ∗ ω x | F ◦ x = ψ ∗ x ω x | F ◦ x = ω | F ◦ x = ω | F ◦ x = d ( α/ | h | ) | F ◦ x . The proof of the claim is complete. (cid:3)
Now ker Φ ∗ α s,ε and ker α are two contact structures on OB ( F, ψ ), which sym-plectically support the Liouville open book (37). Hence they are isotopic by Propo-sition 2.2. (cid:3)
References [ABHS18a] A. Abbondandolo, B. Bramham, U. L. Hryniewicz and P. A. S. Salom˜ao,
Sharp sys-tolic inequalities for Reeb flows on the three-sphere , Invent. Math. 211 (2018), 687-778.[ABHS18b] A. Abbondandolo, B. Bramham, U. L. Hryniewicz and P. A. S. Salom˜ao,
Contactforms with large systolic ratio , Annali della Scuola Normale di Pisa - Classe di Scienze, (toappear).[APBT16] J. C. Alvarez Paiva, F. Balacheff, and K. Tzanev,
Isosystolic inequalities for opticalhypersurfaces , Adv. Math. 301, 2016.[BK18] G. Benedetti and J. Kang,
A local systolic-diastolic inequality in contact and symplecticgeometry , Preprint, arXiv:1801.00539.[Cro88] C. B. Croke,
Area and length of the shortest closed geodesic , J. Differential Geom. 18,1988.[Gir03] E. Giroux,
G´eom´etrie de contact: de la dimension trois vers les dimensions sup´erieures ,Preprint, arXiv:math/0305129.[Gir17] E. Giroux,
Ideal liouville domains - a cool gadget , Preprint, arXiv:1708.08855.[Gro83] M. Gromov,
Filling Riemannian manifolds , J. Differential Geom. 18, 1983.[Sag18] M. Sa˘glam,
Contact forms with large systolic ratio in arbitrary dimensions , Preprint,arXiv:1806.01967.[Tau07] C. H. Taubes,
The Seiberg-Witten equations and the Weinstein conjecture , Geom. Topol.11, 2007.
Murat Sa˘glam, Fakult¨at f¨ur Mathematik, Ruhr-Universit¨at Bochum
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