The contact Banach-Mazur distance and large scale geometry of overtwisted contact forms
TTHE CONTACT BANACH-MAZUR DISTANCEAND LARGE SCALE GEOMETRY OF OVERTWISTED CONTACTFORMS
THOMAS MELISTAS
Abstract.
In the symplectic realm, a distance between open starshaped domainsin Liouville manifolds was recently defined. This is the symplectic Banach-Mazurdistance. It was proposed by Ostrover and Polterovich and developed by Ostro-ver, Polterovich, Usher, Gutt, Zhang and Stojisavljević. The natural question is,can an analogous distance in the contact realm be defined? One idea is to definethe distance on contact hypersurfaces of Liouville manifolds and another one oncontact forms supporting isomorphic contact structures. Rosen and Zhang recentlydefined such a distance working with manifolds that are prequantizations of Liou-ville manifolds. They also considered a distance on contact forms supporting thesame contact structure on a contact manifold Y . This allowed them to view thespace of contact forms supporting isomorphic contact structures on a manifold Y as a pseudometric space, study its properties, and derive interesting results. In thiswork, we do something similar, yet the distance we define is less restrictive. More-over, viewing contact homology algebra as a persistence module, focusing purelyon the overtwisted case and exploiting the fact that the contact homology of over-twisted contact structures vanishes, allows us to bi-Lipschitz embed part of the2-dimensional Euclidean space into the space of overtwisted contact forms support-ing a given contact structure on a smooth closed manifold Y . Contents
List of Figures 21. Introduction 21.1. Organization 41.2. Definitions 42. Statement of the Results 82.1. Main Results 82.2. The Rosen-Zhang definition of d CBM a r X i v : . [ m a t h . S G ] S e p THE CONTACT BANACH-MAZUR DISTANCE
List of Figures A (cid:48) ( r )
294 The source H (cid:15)
355 The path defining the particular Lutz twist 376 The support of the compensating function ν l
387 The functions used for smoothing 448 The function g used in the definition of a ( − -fold Dehn twist 489 The functions h ( r ) and h ( r ) Introduction
Let ( Y, ξ ) be a closed, co-oriented contact manifold of dimension n − . A con-sequence of Gray’s stability theorem is that the space of contact structures up todiffeomorphism Ξ( Y ) /Dif f ( Y ) on an odd dimensional closed manifold Y is discrete.In other words, there are no non-trivial deformations of a contact structure. Theelements of this space are contactomorphism classes of contact structures definedon the smooth manifold Y . Although from a topological point of view there is nodifference for the contact structure up to isotopy, the dynamics depend highly onthe particular 1-form co-orienting the manifold Y , hence they can be vastly different.Looking at everything from the dynamics perspective, allows to ask the questions likehow “large” is a class in Ξ( Y ) /Dif f ( Y ) , or in other words how far apart are two rep-resentatives of the same contactomorphism class. The idea on how to measure theirdistance comes from an analogy with the case of symplectic manifolds. In particular,for open Liouville domains the idea, which is inspired by convex geometry and theBanach-Mazur distance, is to look at the optimal way to interleave them, see [Ush18].Interleaving is usually achieved by rescaling, so the way to rescale is a key issuethat needs to be addressed when attempting to define a Banach-Mazur distance.Some very interesting attempts have already been made. One can work as in section1.2.1 of [RZ20] where Rosen and Zhang work with the case of fiberwise star-shapeddomains U in the contactization of a Liouville manifold W , namely W × S and withsubdomains of contact manifolds that are boundaries of Liouville domains. Other-wise, one can look at their setup in section 1.2.2, in which they define a distancebetween contact forms supporting isomorphic contact structures. This distance canin turn be used to define a distance between closed Liouville fillable contact mani-folds. In subsection 2.2, we briefly recall part of their work, mainly focusing on thedistance between contact forms as it is most relevant here and compare results to ours. HE CONTACT BANACH-MAZUR DISTANCE 3
The most natural space in which two contactomorphic contact manifolds shouldbe interleaved appears to be their common symplectization. This idea stems fromthe fact that Liouville manifolds decompose as the union of the symplectization ofa hypersurface of restricted contact type Y (which can be viewed as the boundaryof the corresponding Liouville domain bounded by Y ) and their core or skeleton.To be more specific, the symplectization of the boundary Y of a Liouville domain W sits naturally in the completion (cid:99) W of the Liouville domain. The key advantageis that the symplectization also works in the case of absence of a core, namely theovertwisted or more generally in the non-fillable case. The main difference betweena fillable and a non-fillable Y appears to be the existence of a core, so this indicatesthat the distance may be defined when restricting to the same contactomorphismclass in both the fillable and the non-fillable case. In this work, we approach theproblem similarly to the second setup of Rosen and Zhang [RZ20], i.e. the case ofcontact forms supporting isomorphic contact structures, yet we allow more flexibilityusing appropriate embeddings in the symplectization, which we call cs-embeddings(because a Contact manifold is embedded into its Symplectization), resulting in adistance which does not only depend on conformal factors. The distance that we usehere is the contact Banach-Mazur distance which is defined in subsection 1.2. Wedenote it by d CBM .The main result of this work is a mix of quantitative, dynamical and topologicalnature. Let C Y,ξot denote the space of contact forms on the closed manifold Y , support-ing the co-oriented overtwisted contact structure ξ , which are positive with respectto the co-orientation. Let also H be the lower half-space H in R and d ∞ denote themetric induced from the norm || · || ∞ in R . Theorem 1.1 (Main Theorem) . There exists a bi-Lipschitz embedding F : ( H , d ∞ ) → ( C Y,ξot , d
CBM ) . The core of the proof of the main theorem is to be able to control the action levelfor which the identity becomes an exact element in the filtered contact homologyalgebra. In 3 dimensions, our goal will be to modify a construction by Wendl in[Wen05], so as to be able to know precisely what is the action level for which theunit in the contact homology algebra CH ( Y, λ ot ) of an overtwisted contact manifoldbecomes exact. This is the subject of section 4.2. In higher dimensions, we followBourgeois and Van Koert’s approach from [BvK10] which uses the characterization ofovertwisted contact manifolds as negatively stabilized open books. This is discussedin 5.2. As will be explained, in all dimensions, the action level for which the unitin the contact homology algebra becomes exact corresponds to the right endpoint ofthe largest finite bar in the barcode.The importance of this control is also justified by the following well known obser-vation. We know that the vanishing level of the class of the unit controls all othervanishing levels just by using Leibniz rule. This can be seen as follows. If y repre-sents a class in the contact homology algebra, then we need to find an element thatmaps to y under ∂ , i.e. show that any element is exact. If x is the orbit boundingthe pseudoholomorphic plane, then ∂x = 1 . Thus, using Leibniz rule we see that ∂ ( xy ) = ( ∂x ) y ± x∂y = y (note that y is closed as it represents a class). If y hasaction A ( y ) , then x · y has action A ( x ) + A ( y ) , hence the vanishing level of the class [ y ] is at most A ( x ) + A ( y ) , which shows that the length of its corresponding finite bar THE CONTACT BANACH-MAZUR DISTANCE is A ( x ) + A ( y ) − A ( y ) = A ( x ) = l , hence in the case of contact homology algebra,the bar of the unit is the longest and most essential one. This is essentially anotherapplication of the argument used to show the vanishing of contact homology of over-twisted structures, if we have proven the existence of a unique pseudoholomorphicplane bounded by a Reeb orbit.1.1. Organization.
In subsection 1.2 we provide the main definitions needed tostudy this work. In subsection 2.1 we describe our main results more thoroughly. Inshort, these amount to defining the contact Banach-Mazur pseudodistance betweencontact forms and using it to show that the lower half-space in R bi-Lipschitz em-beds in the space of overtwisted contact forms supporting a given overtwisted contactstructure. In subsection 2.2 we recall the relevant definitions from [RZ20] where adifferent and more restrictive flavor of the distance is defined and compare their re-sults to ours. In particular, using symplectic folding we exhibit the main differencesbetween our definitions. Section 3 is devoted to recalling the construction of (filtered)contact homology which viewed as a persistence module gave us the idea to producethe bi-Lipschitz embedding. Section 4 provides the proof of the main bi-Lipschitzembedding theorem, theorem 2.5. Section 5 addresses the question of extending theresult of the previous section to higher dimensions. Finally, section 6 describes thesituation if one would like to attempt to bi-Lipschitz embed R n into the space ofovertwisted contact forms supporting isomorphic contact structures for n > . Thereis no definite answer provided there. Acknowledgements.
I am very grateful to my advisor, Michael Usher, for hisguidance, patience and support and for teaching me so many interesting things duringthis work. I would also like to thank Leonid Polterovich and Jun Zhang for helpfuldiscussions and insightful comments. This work was partially supported by the NSFthrough the grant DMS-1509213.1.2.
Definitions.
Throughout this paper, unless otherwise stated, Y will be a (2 n − -dimensionalclosed manifold with a co-oriented contact structure ξ . The contact Banach-Mazurdistance is a distance between 2 co-orientation compatible contact forms on Y havingthe same kernel, the contact hyperplane field ξ . Fixing a hypersurface of restrictedcontact type Y (see definition 1.6) in a Liouville manifold W , the distance can alsobe defined between contact hypersurfaces of restricted contact type that are in theimage of the Liouville flow starting at Y and flowing for either positive or negativetime, not necessarily uniformly.In what follows, we are going to need the notion of the symplectization of a contactmanifold ( Y, ξ ) . There are two versions of the definition. One of them does not requirethe choice of a contact form in order to be defined and it is a special line subbundleof the cotangent bundle of Y . The other one involves the choice of a co-orientingcontact form for Y . This choice of global contact form for Y yields a splitting of thesymplectization SY as a trivial principal R + -bundle, R + × Y . The advantage of thefirst one is obvious, while the advantage of the second one is of course the convenienceof being able to perform hands on computations. We start with the latter one. HE CONTACT BANACH-MAZUR DISTANCE 5
Definition 1.2.
We define the symplectization of ( Y, ξ = ker ( λ )) to be the manifold ( S λ Y = M = R + × Y, ω = d ( rλ )) , where r is the real positive coordinate.One easily checks that ( S λ Y, ω ) is a symplectic manifold. The fact that ω is closed isimmediate since it is exact. The non-degeneracy is equivalent to the contact conditionfor λ . Note that implicitly in this definition we chose a global form λ for Y . Thechoice-free definition of the symplectization is as follows. We fix a co-orientation for ξ . Definition 1.3.
We define the symplectization of ( Y, ξ ) to be M = SY = (cid:91) y ∈ Y S y Y ,where S y Y := (cid:40) β ∈ T ∗ y Y − { } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ker ( β ) = ξ y and β > on vectors positively transverse to ξ y (cid:41) The symplectization of Y is a submanifold of its cotangent bundle T ∗ Y . As it isknown, T ∗ Y comes naturally equipped with the canonical, or tautological, or Liou-ville 1-form θ and it turns out that dθ is a symplectic form when restricted to SY .Thus, ( SY, dθ ) is a symplectic manifold. There is an identification between the twoversions of the symplectization which sends d ( rλ ) → dθ . We will primarily work withthe latter formulation of the definition as it requires no reference to a contact form λ , yet when concrete calculations are needed we will work using the former one.We denote by L = L θ the Liouville vector field of the symplectization SY , i.e. theunique vector field on ( SY, ω ) satisfying i L ω = i L dθ = θ . One can see that ( Y, α ) sits in a standard way as a contact hypersurface Y α inside SY . This is understoodas follows. Y gets identified with the graph of its contact form α inside SY which isviewed as a subbundle of T ∗ Y . It is important to remember that we will denote itsidentification by Y α . Different choices of a contact form with kernel ξ yield differentembeddings of ( Y, ξ ) into SY and different splittings of SY .When we need to focus on the contact dynamics instead of just the contact structureitself, we use the notion of a strict contact manifold. A strict contact manifold is aclosed manifold Y equipped with a co-orienting contact form α . The term is notstandard in the literature, yet it is very useful here as we work with contact formsand not only contact structures. Definition 1.4.
By a cs-embedding of a strict contact manifold ( X, α ) to ( SY, dθ ) we mean an embedding φ : ( X, α ) → ( SY, dθ ) with φ ∗ ( θ + η ) = α , where η is anexact, compactly supported 1-form on SY .Figure 1 illustrates this definition.This definition is equivalent to the statement that X embeds as a hypersurfacetransverse to the Liouville vector field defined by i L ω = θ + η . Note that if η = 0 ,then the Liouville vector field is simply L = p∂ p , where p are the cotangent fibercoordinates. So, the relaxed condition allows our embeddings to be transverse tosome Liouville vector field dictated by θ + η and not just the standard one.We denote the Liouville flow for time t by L t . Note that under this flow, the con-tact form λ used to decompose SY gets multiplied by e ln( r ) = r . This is because theflow of the Liouville vector field L conformally expands volume since by definition THE CONTACT BANACH-MAZUR DISTANCE SY φ ( Y ) L θ + η supp ( η ) Figure 1.
A cs-embedding L L ω = ω , where L denotes the Lie derivative operation. The relationship between t and r is t = ln( r ) .A way to produce such cs-embeddings is to postcompose the standard embeddinginduced by α , by a compactly supported symplectic isotopy Φ t . Remark 1.5.
The symplectic isotopy is automatically Hamiltonian. The flux deter-mines whether the symplectic isotopy is a Hamiltonian. Looking at [MS17], section10.2, the flux homomorphism corresponds to a homomorphism π ( SY ) → R , definedby γ (cid:55)→ (cid:90) (cid:90) ω ( X t ( γ ( s )) , ˙ γ ( s )) dsdt, γ : R / Z → SY where X t the vector field generating the isotopy. Lemma 10.2.1 in [MS17] states thatthe right hand side above depends only on the homotopy class of γ and the homotopyclass of Φ t with fixed endpoints. The value of Flux( { Φ t } ) on the loop γ is the areaswept by the loop under the symplectic isotopy Φ t . Any loop in the compact supportof Φ t is homotopic to one outside of the support of the symplectic isotopy Φ t . Thus,the flux of any loop is equal to zero and hence Φ t is Hamiltonian.As stated in the first paragraph of this section, the following definition will also beuseful. Definition 1.6.
Let ( W, θ ) be an exact symplectic manifold. A codimension-onesmooth submanifold Y ⊂ W is said to be a restricted contact type hypersurface of ( W, θ ) if the Liouville vector field L is transverse to Y , i.e. ∀ y ∈ Y we have L y / ∈ T y Y .One application of this definition will be, in the case that Y is fillable, to relate d CBM with d c /d SBM as we can view Y as the contact type boundary of a starshapeddomain.We will define a partial order to the set of co-orientation compatible contact formshaving kernel ξ . First, we provide some preliminary definitions. Definition 1.7.
Let Y β be the standard embedding of ( Y, β ) in SY as the imageof the form β in T ∗ Y . Then we define W ( β ) = { p ∈ SY | < p ( v ) ≤ β ( v ) , ∀ v ∈ T Y such that β ( v ) > } . HE CONTACT BANACH-MAZUR DISTANCE 7
If we choose a contact form, namely a splitting for SY , the above definition turnsinto the following one which is more suitable for calculations. Definition 1.8.
Let Y β be the standard embedding of ( Y, β ) in ( S β Y = R + × Y, d ( rβ )) . Then we define W ( β ) = { ( s, y ) ∈ SY | s ≤ } .The partial order is defined as follows. Definition 1.9. α ≺ β iff there is a cs-embedding in the sense of definition 1.4, φ : ( Y, α ) → SY such that φ ( Y ) ⊂ W ( β ) . Remark 1.10.
Note that later we will use the notation (cid:22) for another partial order,so a warning should be given here. (cid:22) will be referring to Rosen-Zhang partial order.Recall that an example of a cs-embedding is produced by postcomposing the stan-dard embedding by symplectic isotopies. The case when this isotopy has emptysupport corresponds to the partial order (cid:22) as the following example shows.
Example 1.11.
If already Y α ⊂ W ( β ) , then we can take the support of the isotopyto be ∅ . One such example is when there are contactomorphisms such that φ ∗ ( α ) = h ( y ) λ , ψ ∗ ( β ) = h ( y ) λ and h ( y ) ≤ h ( y ) , i.e. using notation that will be mademore precise in section 2.2, α (cid:22) β . The obvious obstruction in that setting is thevolume of Y α being larger than the volume of Y β . Definition 1.12.
Let ( Y, α ) , ( Y, β ) be two contact manifolds in the same contacto-morphism class and ( SY, dθ ) their common symplectization. We define the contactBanach-Mazur distance between α and β to be d CBM ( α, β ) := inf { ln C ∈ [0 , ∞ ) | α ≺ C · β, β ≺ C · α } In view of definition 1.3, it is obvious that if ( Y, α ) is contactomorphic to ( Y, β ) then they have the same symplectization. So, the reference to the symplectizationin the definition above is not ambiguous. The fact that this is a pseudodistance isproved in the following section.We will be measuring the volume of the image of a cs-embedding of a contactmanifold Y into the relevant symplectization as follows. Definition 1.13.
Let θ be the canonical form of the symplectization and α := θ | φ ( Y ) .Then V ol ( φ ( Y )) := (cid:82) φ ( Y ) α ∧ ( dα ) n − Dealing with contact forms and not just contact structures provides the advan-tage of being able to obtain dynamical (and not just topological) information aboutcontact manifolds, thus being able to obtain obstructions (for instance by using thebarcodes of corresponding persistence modules of contact homologies) to the existenceof symplectic cobordisms between them or symplectic embeddings of their respectivefillings, otherwise not detected considering the contact structure itself. Of course, inthe overtwisted case, fillings are excluded by a theorem of Gromov-Eliashberg whichstates that if a contact manifold is fillable, then it is tight.As a last introductory note, the distance defined above is easily seen to be nontrivial since contact forms yielding different volume for Y are at positive distanceapart. As we will see later on, the definition of this distance is not semi-vacuousby depending only on volume, as it is also possible for contact forms with the samevolume to be positive distance apart. THE CONTACT BANACH-MAZUR DISTANCE Statement of the Results
Main Results.
In this section we provide the main results and we only give some of the moststraightforward proofs. The rest of the proofs are given in following sections as wefirst need to recall some tools and ideas from the literature for each one respectively.
Theorem 2.1. d CBM is a pseudodistance on the space of contact forms supportingthe contact structure ξ on the contact manifold ( Y, ξ ) .Proof. We have to show non-negativity, symmetry and the triangle inequality. Thedistance is by definition non-negative. Symmetry is also immediate from the defini-tion.We show in claim 2.3 that ≺ is transitive. Using this, the triangle inequality canbe shown as follows. We have d CBM ( α, β ) = inf { ln( l ) | α ≺ lβ, β ≺ lα } = ln( L ) and d CBM ( β, γ ) = inf { ln( m ) | γ ≺ mβ, β ≺ mγ } = ln( M ) By transitivity, if α ≺ lβ ≺ lmγ and γ ≺ mβ ≺ lmα we obtain α ≺ lmγ and γ ≺ lmαd CBM ( α, γ ) = inf { ln( C ) | α ≺ Cγ, γ ≺ Cα } ≤ ln( LM )= ln( L ) + ln( M ) = d CBM ( α, β ) + d CBM ( β, γ ) (cid:3) Remark 2.2.
The distance captures dynamical information and degenerates as oneexpects, since two strictly contactomorphic manifolds have distance 0. Indeed, it isclear that if two contact manifolds ( Y, α ) and ( Y, β ) are strictly contactomorphic,i.e. there exists a diffeomorphism f : Y → Y such that f ∗ ( β ) = α , the embeddings φ : Y → SY and φ ◦ f : Y → SY yield d CBM ( Y α , Y β ) = 0 Claim 2.3. ≺ is transitive. For the proof we will need the following lemma. We give the following definition fornotational convenience. We know that a diffeomorphism φ of a manifold Y inducesa map F φ on T ∗ Y given by F φ ( x, p ) = ( φ ( x ) , ( φ − ) ∗ p ) . Then we define W ( φ ∗ β ) := F φ ( W ( β )) = W (( φ − ) ∗ β ) . Lemma 2.4.
Consider W ( β ) and a cs-embedding ψ : ( Y, β ) → ( SY, d ( θ + η )) , foran exact one form η , compactly supported in a neighborhood of ψ ( Y ) , such that theLiouville vector field L θ + η associated to θ + η is transverse to ψ ( Y ) . Then, we havean embedding F : W ( β ) → W (( ψ ∗ ) − β ) ⊆ ( SY, d ( θ + η )) with F ∗ ( θ + η ) = rβ .Proof. Let ψ : ( Y, β ) → ( SY, d ( θ + η )) be the assumed cs-embedding, i.e. ψ ∗ ( θ + η ) = β . As before, we denote the flow for time t of the Liouville field L α associated to theprimitive α by L tα . Define F : ( SY, rβ ) → ( SY, d ( θ + η )) by F ( L ln( t ) rβ s β ( y )) = L ln( t ) θ + η ( ψ ( y )) , ∀ y ∈ Y, t ∈ R where s β the standard embedding of Y as the graph of the form β in its sym-plectization and L ln( t ) α the Liouville flow in the symplectization ( SY, dα ) . We show HE CONTACT BANACH-MAZUR DISTANCE 9 F ∗ ( θ + η ) = rβ . The tangent spaces in the source and target symplectizations splitas (cid:104) L rβ (cid:105) ⊕ (cid:104) R rβ (cid:105) ⊕ ξ and (cid:104) L θ + η (cid:105) ⊕ (cid:104) R θ + η (cid:105) ⊕ ξ . L α denotes the Liouville vector fieldof ( SY, dα ) .It will be enough to show that F ∗ ( L rβ ) = ( L θ + η ) and ∀ t and Φ s := L ln( t ) θ + η ◦ ψ ◦ s − β ◦ L − ln( t ) rβ : L ln( t ) rβ ( s β ( y )) → L ln( t ) θ + η ( ψ ( y )) we have Φ ∗ t ( θ + η ) = rβ , i.e. for fixed t , the hyper-surfaces (cid:8) L ln( t ) rβ ( s β ( y )) (cid:12)(cid:12) y ∈ Y (cid:9) and (cid:8) L ln( t ) θ + η ( ψ ( y )) (cid:12)(cid:12) y ∈ Y (cid:9) are strictly contactomorphic.For the first statement we have, F ∗ (( L rβ ) L ln( τ ) rβ s β ( y ) ) = ddt (cid:12)(cid:12) t =0 F ( L ln( t ) rβ ( L ln( τ ) rβ s β ( y )) = ddt (cid:12)(cid:12) t =0 F ( L ln( t ) ln( τ ) rβ s β ( y )) = ddt (cid:12)(cid:12) t =0 F ( L ln( t + τ ) rβ s β ( y )) = ddt (cid:12)(cid:12) t =0 F ( L ln( t + τ ) θ + η ψ ( y )) = ( L θ + η ) F ( L ln( τ ) rβ s β ( y )) For the second statement we have, Φ ∗ t ( θ + η ) = ( L − ln( t ) rβ ) ∗ ◦ ( s − β ) ∗ ◦ ψ ∗ ( t ( θ + η )) = ( L − ln( t ) rβ ) ∗ ( trβ ) = rβ Restricting F to Y × { t ≤ } yields the required embedding. (cid:3) We prove claim 2.3, namely that ≺ is transitive. Proof.
Let α ≺ β and β ≺ γ . We show α ≺ γ . The assumption means that thereexist embeddings φ : ( Y, α ) → ( SY, d ( rβ ) and ψ : ( Y, β ) → ( SY, d ( rγ ) such that φ ( Y ) ⊂ W ( β ) , ψ ( Y ) ⊂ W ( γ ) , φ ∗ ( rβ + η ) = α and ψ ∗ ( rγ + η ) = β for two compactlysupported exact one forms η , η . Let F be the map defined in the previous lemma.Under F , φ ( Y ) maps into W (( ψ ∗ ) − β ) and in particular into W ( γ ) . Moreover, setting Φ := F ◦ φ : ( Y, α ) → ( SY, d ( rγ )) , we have a cs-embedding of ( Y, α ) in ( SY, d ( rγ )) .Indeed, Φ( Y ) ⊆ W (( ψ ∗ ) − β ) ⊆ W ( γ ) and Φ ∗ ( rγ + η + η ) = ( F ◦ φ ) ∗ ( rγ + η + η ) = φ ∗ ( rβ + η ) = α , for any compactly supported exact one form η such that F ∗ ( η ) = η .Thus, we have shown that α ≺ γ . (cid:3) The main result, already described in the introduction, is the following. Fix aco-oriented overtwisted contact structure ξ on Y . T Y /ξ is a trivial line bundle so let X ∈ ( T Y /ξ ) ⊥ be a global section. Consider C Y,ξot = { α ∈ T ∗ Y | α ( X ) > , α ∧ ( dα ) n (cid:54) = 0 , ker ( α ) = ξ } Moreover, the norm || · || ∞ on R induces a metric on R and in particular the half-space H = { ( x, y ) ∈ R | y < } . We denote it by d ∞ . Theorem 2.5.
Let ( Y, ξ = ker ( α )) be an overtwisted closed contact manifold. Thereexists a bi-Lipschitz embedding ( H , d ∞ ) → ( C Y,ξot , d
CBM ) Note that there is no assumption on the dimension of the contact manifold Y . Themain tool used in the proof of this theorem in 3 dimensions is the Lutz twist whichis recalled before the proof of this theorem in subsection 4.1. Overtwisted contactstructures in 3 dimensions are classified by the homotopy type of the plane field ξ and full Lutz twists do not alter the homotopy type of the plane field we start with.So, we can modify the overtwisted contact form representing the contact structure ξ and consequently the dynamics on Y without changing the contact structure. Indimensions higher than 3, the situation is similar for reasons that will be explained insection 5. The classification in higher dimensions is again homotopy theoretic as wasrecently shown in [BEM15], yet there is some subtlety involved when generalizing the Lutz twist. This generalization is proposed in [EP11],[EP16] by Etnyre andPancholi. There are drawbacks though with the main one being that half Lutz twistsdo not behave as expected (e.g. they alter the diffeomorphism type of the originalmanifold). Moreover, the pseudoholomorphic curve analysis is quite involved. Insteadof using the generalization of the Lutz twist in higher dimensions, we will mostly followthe negatively stabilized open book decomposition approach as in [BvK10]. This isbecause the pseudoholomorphic curve analysis is already carried out. We explain thisin subsection 4.1. Remark 2.6.
Since any two norms on a finite dimensional vector space are equiv-alent, we could have chosen to use any norm. For convenience we choose || · || ∞ . Itturns out that the proof shows something slightly stronger than the statement of thetheorem.The proof of this theorem in 3 dimensions will be provided in section 4, as we willneed first to recall some basic tools, in order to modify contact forms and prove theexistence of a certain unique pseudoholomorphic disk bounded by a Reeb orbit. Theextension to higher dimensions will be given in section 5.It is clear that any two contactomorphic manifolds are finite distance apart. Thus,the question that arises is, can we provide upper bounds on d CBM ? A more elab-orate answer is provided in terms of the bi-Lipschitz embedding theorem 2.5, yet astraightforward answer is given with respect to the positive function f which relatesthe two forms. Proposition 2.7.
Let ( Y, α ) , ( Y, β ) two closed contactomorphic contact manifolds(i.e. β = f α for some smooth f : Y → R + ). Then, d CBM ( α, β ) ≤ max { ln(max( f )) , − ln(min( f )) } Proof.
We embed ( Y, α ) into its symplectization SY in the standard way, i.e. ψ (( Y, α )) = { }× Y . Using ψ , ( Y, β ) embeds as ψ (( Y, β )) = ( f, Y ) . A value for k that will provide α ≺ kβ is max { min ( f ) , } .Now, if we embed ( Y, β ) in the standard way, i.e. φ (( Y, β )) = { } × Y , then ( Y, α ) embeds as φ (( Y, α )) = ( f , Y ) . A value for k that will provide β ≺ kα is max { f ) , } = max { max( f ) , } .Note that a value for k that works for both directions is max (cid:8) max( f ) , f ) (cid:9) sincethis is always ≥ . Taking logarithms as in the definition of d CBM we have max (cid:110) ln(max( f )) , ln( 1min( f ) ) (cid:111) = max (cid:8) ln(max( f )) , − ln(min( f )) (cid:9) (cid:3) We remark that cleverer embeddings may provide sharper upper bounds for thedistance. We provide such an example using symplectic folding. This is example 2.27.The proof of the above also works in the case of the so called conformal factor dis-tance (to be defined shortly), yielding upper bounds. The conformal factor distancecan be defined both on the space of contact forms supporting the same contact struc-ture ξ on Y and on the contactomorphism group Cont + ( Y, ξ ) of contactomorphisms HE CONTACT BANACH-MAZUR DISTANCE 11 preserving the co-orientation of ξ . We begin with the former.If ( Y, α ) and ( Y, β ) are contactomorphic, then there exist φ : Y → Y and a smoothfunction f : Y → R + such that φ ∗ α = f β . On the other hand, if α , β support thesame contact structure, then Y α and Y β with the contact structures induced by theLiouville vector field in the symplectization, are of course contactomorphic. This canbe seen from [CE12], Lemma 11.4, which states Lemma 2.8.
Let Σ , (cid:101) Σ be hypersurfaces in a Liouville manifold ( V, ω, X ) such thatfollowing the trajectories of X defines a diffeomorphism Γ : Σ → (cid:101) Σ . Then Γ is acontactomorphism for the contact structures induced by i X ω . Definition 2.9. If α , β support the same contact structure on Y, then define d CF ( α, β ) := inf φ { max Y | ln( f ) | (cid:12)(cid:12)(cid:12) φ ∗ ( α ) = f β } The conformal factor norm on
Cont + ( Y, ξ ) is defined as follows. Definition 2.10.
Let ξ = ker ( α ) and φ ∈ Cont + ( Y, ξ ) such that φ ∗ α = f α . Then, | φ | CF := max Y | ln( f ) | . Definition 2.11.
The conformal factor distance is d CF ( φ, ψ ) := | φ − ψ | CF The Rosen-Zhang definition of d CBM . In this subsection, we briefly recall definitions and results from [RZ20]. The proofscan be found in the relevant reference. We then compare their results to ours. Forclarity, we denote Rosen and Zhang’s version of the distance by ∆ CBM . It appearsthat, if we restrict the set of contactomorphisms we are working with to the identitycomponent of the contactomorphism group
Cont ( Y, ξ ) , this definition is similar tothe definition of the conformal factor distance between two contact forms that wasdefined in the previous section. This is the content of proposition 2.16. Let ( Y, ξ ) a co-oriented contact manifold. Their strategy is first to define a distance on the space offorms supporting the same contact structure and then make geometric sense out of it.Making a choice of a contact form α supporting ξ , they view the space of all suchcontact forms supporting ξ as an orbit space O ξ ( α ) := C ∞ ( Y, R ) · { α } where the action of an element f ∈ C ∞ ( Y, R ) is given by multiplication of the form α by e f , i.e. based on a previous remark regarding the effect of the Liouville flow onthe contact form in the symplectization, this multiplication is equivalent to flowingfor time t = ln( e f ) = f . Note that since f is not constant, this time is not uniform.This will be helpful in the definition of the distance on hypersurfaces of restrictedcontact type.For any form β supporting an isomorphic structure to ξ , there is a contactomor-phism φ β such that β = φ ∗ β α = e f β α . A partial order is defined on O ξ ( α ) asfollows. Definition 2.12. α (cid:22) β iff f α ≤ f β pointwise. Somehow notationally absurd, but chosen so as to remember that “ (cid:22) ” comes purelyby an inequality between conformal factors “ ≤ ”, we have the following. Proposition 2.13. If α (cid:22) β , then α ≺ β Proof.
We can identify the embeddings of ( Y, α ) and ( Y, β ) into SY with the graphsboth α and β in SY . Since α (cid:22) β , i.e. f α ≤ f β , we get that the image of ( Y, α ) underthe standard cs-embedding as the graph of the form α , which we denote s α , satisfies s α ( Y ) ⊂ W ( β ) . Moreover, s ∗ α ( θ ) = α . Thus, both requirements in the definition of acs-embedding are satisfied for the standard embedding. (cid:3) Now we recall definition 1.12 from [RZ20] which is their definition of the Banach-Mazur distance on the space of forms supporting ξ . Denote by Cont ( Y, ξ ) the identitycomponent of the contactomorphism group of ( Y, ξ ) . Definition 2.14.
For any α, β ∈ O ξ ( α ) , we define ∆ CBM ( α, β ) := inf (cid:110) ln C ≥ | ∃ φ ∈ Cont ( Y, ξ ) s.t. C α (cid:22) φ ∗ β (cid:22) Cα (cid:111) The condition in the definition is explicitly f α − ln C ≤ f β ◦ φ + g φ,α ≤ f α + ln C where φ ∗ β = e g φ,α α .All expected properties hold according to the following proposition which is propo-sition 2.8 in [RZ20]. Proposition 2.15.
For any α , α , α ∈ O ξ ( α ) we have • ∆ CBM ( α , α ) ≥ and ∆ CBM ( α , α ) = 0 • ∆ CBM ( α , α ) = ∆ CBM ( α , α ) • ∆ CBM ( α , α ) ≤ ∆ CBM ( α , α ) + ∆ CBM ( α , α ) • ∆ CBM ( φ ∗ α , ψ ∗ α ) = ∆ CBM ( α , α ) for any φ, ψ ∈ Cont ( Y, ξ ) Since all information can be read off of the conformal factors one expects thefollowing.
Proposition 2.16. ∆ CBM = d CF when restricted to Cont ( Y, ξ ) .Proof. We have d CF ( α, β ) = inf φ ∈ Cont ( Y,ξ ) { max Y | ln( f ) | (cid:12)(cid:12) φ ∗ ( α ) = f β } and also ∆ CBM ( α, β ) = inf { ln( C ) (cid:12)(cid:12) ∃ φ ∈ Cont ( Y, ξ ) s.t. C β (cid:22) φ ∗ ( α ) (cid:22) Cβ } We rewrite the second distance in order to make it look more like the first one. ∆ CBM ( α, β ) = ∆ CBM ( β, α ) = inf φ ∈ Cont ( Y,ξ ) { ln( C ) (cid:12)(cid:12) C β (cid:22) φ ∗ ( α ) = f β (cid:22) Cβ } = inf φ ∈ Cont ( Y,ξ ) { ln(max Y ( f )) (cid:12)(cid:12) φ ∗ ( α ) = f β } = inf φ ∈ Cont ( Y,ξ ) { max Y | ln( f ) | (cid:12)(cid:12) φ ∗ ( α ) = f β } = d CF ( α, β ) (cid:3) HE CONTACT BANACH-MAZUR DISTANCE 13
The way to make geometric sense out of this definition of the distance betweenforms is as follows. Let ( Y, α ) be a closed Liouville fillable contact manifold sothat there exists a domain ( W, ω, L ) with ∂W = Y, ( ι L ω ) | Y = α and complete flowfor t < . The completion of W is denoted by (cid:99) W and is SY (cid:116) Core ( W ) with SY being the symplectization. The coordinates on SY are ( u, x ) . In these coordinates, Y = { u = 1 } , W = { u ≤ } , Core ( Y ) = { u = 0 } . Pick α = e f α ∈ O ξ ( α ) , for some f : Y → R . Define the corresponding Liouville domain W α = { ( u, x ) ∈ (cid:99) W | u < e f ( x ) } Remark 2.17.
Note that W ( α ) is not the same as W α as the former refers to asubset of the symplectization and the latter to the Liouville domain bounded by thecontact hypersurface equipped with the form e f α . If α = α , so e f = 1 , we havethat Core ( (cid:99) W ) (cid:116) W ( α ) = W α . The difference becomes apparent in the absence ofcore.The following was not explicitly defined in [RZ20], yet it was implied by theirstability result, Theorem 1.14. Definition 2.18.
The contact Banach-Mazur distance between domains W α i is de-fined to be ∆ CBM ( W α , W α ) := ∆ CBM ( α , α ) .Using d CBM as defined definition 1.9, we can also provide the following definitionbetween such Liouville manifolds.
Definition 2.19.
The contact Banach-Mazur distance between domains W α i is de-fined to be d CBM ( W α , W α ) := d CBM ( α , α ) .We now recall the definition of the coarse and symplectic Banach-Mazur distances. Definition 2.20.
For two open star-shaped domains
U, V of a Liouville manifold ( W, ω, L ) we define their coarse Banach Mazur distance by d c ( U, V ) := inf { ln( C ) > |∃ ( φ, ψ ) s.t. U φ (cid:44) −→ CV and V ψ (cid:44) −→ CU } where φ (cid:44) −→ means that between the two starshaped domains U, V there exists a Hamil-tonian isotopy { φ t } t ∈ [0 , defined on W such that φ = Id and φ ( U ) ⊂ V .The stronger version d SBM of this distance is defined by additionally requiringthat the composition ψ ◦ φ is isotopic inside CU to the identity map on W throughHamiltonian isotopies. This is what is called the unknottedness condition.Let P ( a, b ) := B ( a ) × B ( b ) ⊂ C . As far as we know, the first time a similarquestion was raised was in [FHW94], where the authors show that if a ≤ b < c and a + b > c , then the embeddings φ , φ : → P ( c, c ) ◦ given by φ ( w, z ) = ( w, z ) and φ ( w, z ) = ( z, w ) are not isotopic through compactly supported symplectomorphismsof P ( c, c ) ◦ . Recently, a stronger notion of knottedness appeared. If A ⊂ U ⊂ C n , asymplectic embedding φ : A → U is called knotted if there is no symplectomorphism ψ : U → U with ψ ( A ) = φ ( A ) . Some examples of knotted embeddings in this strongersense were produced by Usher and Gutt in [GU19], see theorem 1.10. The authorsexhibit examples of toric domains in R which using filtered positive S -equivariantsymplectic homology are shown to be knotted.The following stability result, theorem 1.14 in [RZ20], holds. Theorem 2.21.
For any α , α ∈ O ξ ( α ) , we have d c ( W α , W α ) ≤ d SBM ( W α , W α ) ≤ ∆ CBM ( W α , W α ) It is obvious from these definitions that the rescaling takes place in SY using theLiouville vector field corresponding to α . This is the major restriction when workingwith ∆ CBM as one has to make a choice, that is to select the reference form α . Thisis equivalent to picking the Liouville vector field by which we rescale our contactmanifolds. This appears to be very restrictive and all relevant information can beread off of the conformal factors of the forms in question, namely the function f suchthat φ ∗ β = e f α . By allowing cs-embeddings as we do in the definition in this article,the Liouville vector field is allowed to be modified within the compact support of theexact form η . So, we can realize that the distance defined in here is finer than the onein [RZ20] or in other words d CBM ≤ ∆ CBM . We provide a proof for this statementbelow.
Theorem 2.22.
Let ( Y, ξ ) a closed contact manifold and α , α two contact formswith ξ = ker ( α ) = ker ( α ) . Then we have d CBM ( α , α ) ≤ ∆ CBM ( α , α ) . For the proof of this theorem we will need the following lemma.
Lemma 2.23.
Let ( Y, ξ ) a closed contact manifold, φ ∈ Cont ( Y, ξ ) and α , α twocontact forms supporting ξ . If C α ≤ φ ∗ ( α ) ≤ Cα , then C (cid:22) φ ∗ ( α ) (cid:22) Cα .Proof. The assumption C α ≤ φ ∗ ( α ) ≤ Cα implies that there are embeddings of ( Y, C α ) , ( Y, φ ∗ ( α )) and ( Y, Cα ) in SY as the graphs of C α , φ ∗ ( α ) and Cα in T ∗ Y . Moreover, the setting C (cid:22) φ ∗ ( α ) (cid:22) Cα requires the existence of embeddings Φ : ( Y, C α ) → SY and Ψ : (
Y, α ) → SY with Φ ∗ ( θ + η ) = C α , Ψ ∗ ( θ + η ) = α ,for some compactly supported exact one forms η , η . For any one form β on Y , wedefine the map s β : Y → SY given by y (cid:55)→ ( y, β y ) .For φ ∈ Cont ( Y, ξ ) we also define F φ ∈ Symp ( SY ) by p (cid:55)→ ( φ − ) ∗ p . We set Ψ := s φ ∗ α ◦ φ − : ( Y, α ) → SY and Φ := F φ ◦ s C α : ( Y, C α ) → SY and we checkthat they have the desired properties.First, Ψ ∗ ( θ ) = ( s φ ∗ α ◦ ( φ − )) ∗ θ = ( φ − ) ∗ ◦ ( s φ ∗ α ) ∗ θ = ( φ − ) ∗ ( φ ∗ α ) = α andalso Φ ∗ ( θ ) = ( F φ ◦ s C α ) ∗ = s ∗ C α ◦ F ∗ φ θ = s ∗ C α θ = C α . It is also immediate that Ψ( Y ) ⊆ W ( Cα ) and Φ( Y ) ⊆ W ( α ) . Thus, we showed C (cid:22) φ ∗ ( α ) (cid:22) Cα asrequired. (cid:3) We now prove theorem 2.22.
Proof.
From the previous lemma we have that the setting C α ≤ φ ∗ ( α ) ≤ Cα induces two embeddings Ψ , Φ required in the definition of d CBM . This proves theresult. (cid:3)
The setting we have talked about so far can be a bit more general by using d CBM / ∆ CBM in defining a distance between hypersurfaces of restricted contact typeinside Liouville manifolds (cid:99) W .Pick a hypersurface of restricted contact type Y in ( W, θ ) and define α := θ | Y .For every smooth g : Y → R we have a hypersurface of restricted contact type Y g := { L g ( y ) ( y ) | y ∈ Y } , where we recall that L t denotes the Liouville flow for time t . We have a contactomorphism φ g : Y → Y g defined by the Liouville flow. Moreprecisely, φ ∗ g ( θ | Y g ) = e g α . Obviously, there is a one to one correspondence { g : Y → R } ⇐⇒ (cid:40) Y g hypersurface of restricted contacttype diffeomorphic to Y under L t (cid:41) HE CONTACT BANACH-MAZUR DISTANCE 15
The definitions for the distances between hypersurfaces of such type is providedbelow.
Definition 2.24. d CBM ( Y g , Y g ) := d CBM ( e g α , e g α ) Definition 2.25. ∆ CBM ( Y g , Y g ) := ∆ CBM ( e g α , e g α ) Since we appeal to the definitions between contact forms, all the results we haveproved so far hold also in this case. In particular, d CBM ≤ ∆ CBM . The followingexample uses symplectic folding to create embeddings compatible with the definitionof d CBM , yet not allowed in ∆ CBM . It provides a smaller upper bound for d CBM thanwe have for ∆ CBM . It is not clear though if this example shows that the inequalityis strict as the best upper bound for ∆ CBM comes from inclusions and may not beoptimal.We work in C with coordinates ρ j = | z j | , θ j ∈ Z / π Z , j = 1 , . Let λ = ρ dθ + ρ dθ be the canonical primitive for the standard symplectic structure. Definition 2.26.
We define the ellipsoid E ( a, b ) = (cid:110) ( ρ , θ , ρ , θ ) ∈ C | π (cid:16) ρ a + ρ b (cid:17) ≤ , ≤ ρ ≤ a, ≤ ρ ≤ b (cid:111) Example 2.27.
We normalize S as the boundary of E ( π, π ) . Let α = λ | S = ∂E ( π,π ) .We can view the hypersurfaces of restricted contact type in C , ∂E (1 , and ∂B ( ) = ∂E ( , ) , as copies of S equipped with different contact forms as follows. There isa diffeomorphism Φ : S → ∂E ( a, b ) defined essentially by the Liouville flow. Itsformula is Φ( ρ , θ , ρ , θ ) = (cid:16) ρ π ( ρ a + ρ b ) , θ , ρ π ( ρ a + ρ b ) , θ (cid:17) Pulling back λ | ∂E ( a,b ) under Φ we get the form π ( ρ a + ρ b ) α on S . Thus, if a = 1 and b = 3 we get that ( S , π ( ρ + ρ ) α ) is the corresponding copy to ( ∂E (1 , , λ | ∂E (1 , ) .Similarly, for a = b = , ( S , π α ) is the corresponding copy to ( ∂B ( ) , λ | ∂B ( ) ) .Now using proposition 2.7 we get the upper bound ln(6) for the distance between thehypersurfaces of restricted contact type ∂E (1 , and ∂B ( ) .This essentially comes from the inclusions · B (cid:16) (cid:17) ⊆ E (1 , ⊆ · B (cid:16) (cid:17) As we know, inclusion is not always optimal and the next theorem, which is aparticular case of theorem 2 in [Sch05] helps us provide a smaller upper bound for d CBM . Theorem 2.28.
Assume a > a . Then there exists a symplectic embedding of E ( a , a ) into B ( a − δ ) , ∀ δ ∈ (0 , a − a ) . The proof amounts to the symplectic folding construction which is a compositionof Hamiltonian diffeomorphisms. In order to be precise, ∀ (cid:15) > we have symplecticembeddings E ( a, b ) (cid:44) → T ( a + (cid:15), b + (cid:15) ) and T ( a, a ) (cid:44) → B ( a + (cid:15) ) where T ( a, b ) is the open trapezoid and what is being folded is actually the trapezoid.We won’t need to talk about the proof further in here so we avoid excess definitions. According to the previous theorem we have the embeddings − δ B (cid:16) (cid:17) ⊆ E (1 , fold (cid:44) −−→ (6 − δ ) B (cid:16) (cid:17) , ∀ δ ∈ (cid:16) , (cid:17) Remark 1.5 ensures that this embedding restricted on ∂E (1 , provides a cs-embeddingas defined earlier and thus compatible with the definition of d CBM . This shows thatan upper bound for d CBM is ln(6 − δ ) , with δ ∈ (0 , / clearly less than ln(6) whichwas obtained using proposition 2.7. Moreover, the folding works in such a way that B ( ) remains fixed under the folding embedding map. An upper bound for ∆ CBM isonly ln(6) coming from the standard inclusions.Theorem 2.21 induces the next natural question. What is the relationship between d c /d SBM and d CBM ? Proposition 2.29.
For two starshaped domains
U, V inside a Liouville manifold ( (cid:99) W , ω, L ) we have d c ( U, V ) ≤ d CBM ( ∂U, ∂V ) . Remark 2.30.
If the isotopies produced in the proof satisfy the unknottedness con-dition, then we can also show that d SBM ( U, V ) ≤ d CBM ( ∂U, ∂V ) . This condition willbe explained after the proof of proposition 2.29, in remark 2.35.The extra difficulty in proving this theorem arises due to the extra freedom wehave when embedding ∂U and ∂V in the symplectization in order to calculate theirdistance. As explained before, the embeddings are only required to be transverse tosome locally modified Liouville vector field associated to the form θ + η . Thus, theproof amounts to be able to extend the embedding of the boundary ∂U (or ∂V ) toan embedding of the whole domain U (or V ). The first strategy one might think isto follow the flowlines of the Liouville vector field L θ + η , yet the vector field can nowvanish disallowing the definition of a full symplectomorphism from U (or V ) to thedomain bounded by φ ( U ) (or ψ ( V ) ). The way to define such a symplectomorphism isusing Liouville homotopies and a helpful proposition, proposition 11.8 from [CE12],which we now state. Proposition 2.31.
Let ( (cid:99) W , ω s , L s ) , s ∈ [0 , , be a homotopy of Liouville manifoldswith Liouville forms λ s . Then there exists a diffeotopy h s : (cid:99) W → (cid:99) W with h = Id such that h ∗ s λ s − λ = df s for all s ∈ [0 , . If moreover (cid:83) s ∈ [0 , Core ( (cid:99) W , ω s , L s ) iscompact (e.g. for the completion of a homotopy of Liouville domains), then we canachieve h ∗ s λ s − λ = 0 outside of a compact set. We will be interested in the case of Liouville domains and the completion of ahomotopy between them.The following lemma is essentially exercise 10.2.6 in [MS17] which is a generaliza-tion of their proposition 9.3.1 in the non-compact manifolds case. It is used to detectwhen a compactly supported symplectomorphism is a Hamiltonian one.
Lemma 2.32.
Let ( M, ω = − dλ ) be a non-compact symplectic manifold. φ ∈ Symp c ( M, ω ) belongs to Ham c ( M, ω ) ⇐⇒ φ ∈ Symp c, and there exists a com-pactly supported smooth function F : M → R such that φ ∗ λ − λ = dF If one prefers to use the flux homomorphism, the following holds.
Lemma 2.33. If ω = − dλ and ψ t : M → M is a compactly supported symplecticisotopy, then F lux ( { ψ t } ) = [ λ − ψ ∗ λ ] . HE CONTACT BANACH-MAZUR DISTANCE 17
The following lemma, lemma 2.10 from [Ush18], will be useful in order to controlthe support of Hamiltonian diffeomorphisms.
Lemma 2.34.
Let X be a manifold without boundary equipped with a smooth familyof 1-forms λ t , ≤ t ≤ such that dλ t is symplectic and independent of t . Assumefurthermore that the Liouville vector fields L λ t of λ t are each complete. Let W be acompact codimension-zero submanifold of X with boundary ∂W , having the propertiesthat each L λ t is positively transverse to ∂W and that every point of X lies on a flowlineof L λ t that intersect W. Then, there exists a smooth family of symplectomorphisms F t : X → X such that F = Id and the support of F t is contained in the interior of W for all t . It is worth noting that the definition of d c /d SBM dictates that if we want to makeany meaningful comparison between these distances and d CBM , we have better torestrict to the case where the cs-embeddings of the contact manifolds ∂U and ∂V (equipped with the contact form induced by the transverse Liouville vector field)in the symplectization are isotopic through contact hypersurfaces to the canonicalembeddings as the boundaries of U and V via the restriction of a Hamiltonian isotopy.This is because the maps φ and ψ interleaving U and V in the definition of d c areHamiltonian isotopic to the identity and the goal is the cs-embeddings to be therestrictions of these Hamiltonian isotopies to the respective boundaries. With this inmind we now prove proposition 2.29. Proof.
We denote by ( Y, α ) and by ( Y, β ) the hypersurfaces of restricted contact type ( ∂U, θ | ∂U ) and ( ∂V, θ | ∂V ) . Also, let U = W α and V = W β their fillings. The goal isto show that the cs-embeddings required to define d CBM ( α, β ) induce a Hamiltonianisotopy defined on (cid:99) W which is used to calculate d c ( W α , W β ) . This will yield thedesired inequality d c ( U, V ) = d c ( W α , W β ) ≤ d CBM ( α, β ) = d CBM ( ∂U, ∂V ) sincewhen calculating d c , we calculate the infimum over a possibly larger set of functionsthan d CBM .Let C ≥ . Let φ : ( Y, α ) → ( SY, dθ ) and ψ : ( Y, β ) → ( SY, dθ ) be cs-embeddings, i.e. φ ∗ ( θ + η ) = α and ψ ∗ ( θ + η ) = β , for two compactly supportedexact one forms η , η , with φ ( Y ) ⊂ W Cβ and ψ ( Y ) ⊂ W Cα .Since we make the assumption that φ and ψ are isotopic to the standard em-beddings s α and s β through cs-embeddings, there are families φ s : ( Y, α ) → ( SY, dθ ) and φ s : ( Y, α ) → ( SY, dθ ) , s ∈ [0 , , of cs-embeddings with φ = s α and ψ = s β respectively. This means that φ ∗ s ( θ + η ,s ) = α and ψ ∗ s ( θ + η ,s ) = β for two fam-ilies of compactly supported exact 1-forms η ,s and η ,s with η , = η , = 0 and η , = η , η , = η . It is enough to extend these families to families of Hamiltonianisotopies of (cid:99) W denoted by Φ s and Ψ s that are compactly supported in neighbour-hoods of s α ( Y ) and s β ( Y ) . The idea is that we want Φ s | s α ( Y ) = φ s and Ψ s | s β ( Y ) = ψ s This is where we we need to use proposition 2.31.For s ∈ [0 , , we have a homotopy of Liouville manifolds ( (cid:99) W , d ( θ + η ,s ) , L s ) withLiouville forms θ + η ,s . Then according to proposition 2.31 there is a diffeotopy h s : (cid:99) W → (cid:99) W , h = Id , h ∗ s ( θ + η ,s ) − θ = df s where f s compactly supported in aneighbourhood of s α . Proposition 2.32 yields that h s is indeed a Hamiltonian isotopy.In this case, λ = − θ , φ = h and F = − f + g , where dg = η . Alternatively, one canuse lemma 2.33 and observe that θ − h ∗ θ is exact. Moreover, we have h ( W α ) ⊂ W Cβ .So, this yields a map U Φ (cid:44) −→ CV as required in the definition of d c . Running the same argument for the homotopy of Liouville manifolds ( (cid:99) W , d ( θ + η ,s ) , L (cid:48) s ) gives the secondmap V Ψ (cid:44) −→ CU required in the definition of d c .Hence, what we achieved is that given any two cs-embeddings φ , ψ used to calcu-late d CBM , we get maps Φ , Ψ used to calculate d c . This yields the desired inequality d c ( U, V ) ≤ d CBM ( ∂U, ∂V ) . (cid:3) We now explain the condition under which we can show the inequality d SBM ( U, V ) ≤ d CBM ( ∂U, ∂V ) mentioned in remark 2.30. Remark 2.35. If Ψ ◦ Φ : C U → CU is Hamiltonian isotopic to the identitymap of (cid:99) W within CU , then we say that the isotopies satisfy the unknottednesscondition. In such favorable case, the proof can be extended to show that also d SBM ( U, V ) ≤ d CBM ( ∂U, ∂V ) . Lemma 2.34 helps with controlling the support ofthe relevant Hamiltonian diffeomorphisms. In particular, if X = (cid:100) CU and λ t is thepullback of θ under Ψ t ◦ Φ t , then we see that the unknottedness condition is satisfiedand we effectively showed that d SBM ( U, V ) ≤ d CBM ( ∂U, ∂V ) .The next question is to what extent the inequality in proposition 2.29 is strict. Ifone wants to show the reverse inequality, namely d c ≥ d CBM the only problem thatthey encounter is whether the Hamiltonian isotopies used in the definition of d c arecollapsing parts of the boundaries ∂U and ∂V into the core of (cid:99) W as such types ofHamiltonians do not yield cs-embeddings in the symplectization which are needed tocalculate d CBM . Let us explain this more thoroughly in the next remark.
Remark 2.36.
Recall that d CBM is defined only using the symplectization part andnot part of the core as it is designed to measure the distance even between non-fillablecontact manifolds. This means that we can relate d CBM and d SBM only if we assumethat the Hamiltonian isotopies in the definition of d c , by which the infimum in thedefinition is achieved, are compactly supported outside of a neighbourhood of thecore we can show that equality holds.We have the following proposition. Proposition 2.37.
Under the assumption discussed in remark 2.36, we have d c ( U, V ) = d CBM ( ∂U, ∂V ) .Proof. By proposition 2.29, it is enough to show d CBM ≤ d c . Hence, if U, V arestarshaped domains, it is enough to prove that Hamiltonian isotopies of (cid:99) W achieving U Φ (cid:44) −→ CV and V Ψ (cid:44) −→ CU induce cs-embeddings φ and ψ which can be used to calculate d CBM ( ∂U, ∂V ) . Pick a hypersurface Y as before and fix the contact form α = θ | Y .Then ∂U = Y g and ∂V = Y g , for two smooth functions g , g : Y → R . Then thecorresponding forms are α = e g α and β = e g α . Under this formulation the goalis to show d CBM ( Y g , Y g ) ≤ d c ( U, V ) = d c ( W α , W β ) .Having cs-embeddings as allowed in the definition of d CBM means that there exist φ : ( Y, α ) → ( SY, dθ ) and ψ : ( Y, β ) → ( SY, dθ ) with φ ∗ ( θ + η ) = α and ψ ∗ ( θ + η ) = β for some exact compactly supported 1-forms η , η and moreover φ ( Y ) ⊂ W ( Ce g α ) and ψ ( Y ) ⊂ W ( Ce g α ) . Given the Hamiltonian isotopies Φ = Φ t and Ψ = Ψ t ,we will show that Φ | s α ( Y ) and Ψ | s β ( Y ) are such embeddings. Hence the infimum iscalculated using a possibly larger set of functions and the inequality d CBM ≤ d c istrue. HE CONTACT BANACH-MAZUR DISTANCE 19
The only thing left to check is that indeed φ = Φ ◦ s α and ψ = Ψ ◦ s β have therequired properties. First we need to check that (Φ ) ∗ (( θ + η ) | Φ ( s α ( Y )) ) = θ | s α ( Y ) and (Ψ ) ∗ (( θ + η ) | Ψ ( s β ( Y )) ) = θ | s β ( Y ) for two exact compactly supported 1-forms η , η .Then it is straightforward to observe that φ ∗ θ = α and ψ ∗ θ = β as s ∗ α θ = α and s ∗ β θ = β . We actually show this by showing something stronger, namely that thisholds for the families Φ t , Ψ t and the families η ,t , η ,t of exact compactly supported1-forms.We know that Hamiltonian flows are exact symplectomorphisms, i.e. in this case Φ ∗ t θ − θ = df ,t and Ψ ∗ t θ − θ = df ,t . Furthermore, by assumption, the supportof the functions f ,t and f ,t is compact as the Hamiltonian isotopy is assumed tobe compactly supported outside of a neighbourhood of the core. Then we have Φ ∗ t ( θ − η ,t ) = θ and Ψ ∗ t ( θ − η ,t ) = θ for the exact compactly supported 1-forms η ,t = − (Φ ∗ ) − ( df ,t ) and η ,t = − (Ψ ∗ ) − ( df ,t ) .The last thing to check is that φ ( Y ) ⊂ W ( Cβ ) and ψ ( Y ) ⊂ W ( Cα ) . This isstraightforward as this holds for both Φ and Ψ . (cid:3) Remark 2.38.
If the isotopies satisfy the unknottedness condition, namely Ψ ◦ Φ in the proof is isotopic to the identity map of (cid:99) W through Hamiltonian isotopiessupported inside CU , then d CBM ( ∂U, ∂V ) = d SBM ( U, V ) .What we showed is that in the restricted case where the Hamiltonians are com-pactly supported outside of a neighbourhood of the core, d CBM calculates exactly d c (or d SBM under the unknottedness condition). So, one can view d CBM as thegeneralization of d c /d SBM in the non-fillable case.3.
Review of contact homology
The first degree of freedom in theorem 2.5, namely one of the two directions of R will be the volume of the contact manifold ( Y, λ ) . The second degree of freedomwill come from the persistence module of filtered contact homology. It is going to bethe filtration level for which the unit of the algebra becomes exact. Recall that forovertwisted contact structures the unit is always exact as contact homology vanishes.A reference for this is [Yau06]. Equivalently, it is always the case that the bar corre-sponding to the empty word of orbits is finite, or in other words the empty word is anexact chain. The class of the empty word in various homology theories is known asthe contact invariant, yet this terminology is not standard for the contact homologyalgebra. It will be helpful for the reader to briefly recollect the notions of contacthomology here. We mostly follow [Bou03] as this is a concise treatment of the subject.Contact homology mimics the idea of Morse homology working with the actionfunctional A : C ∞ ( S , Y ) → R defined as A ( γ ) := (cid:90) γ α Definition 3.1.
Let α be a contact form on Y. The unique vector field characterizedby dα ( R α , · ) = 0 and α ( R α ) = 1 is called the Reeb vector field of α Definition 3.2.
A map γ : R /T Z → Y is called a closed Reeb orbit of α if γ (cid:48) ( t ) = R α ( γ ( t )) .The following lemma, whose proof can be found in [Bou03], shows that the criticalpoints of the functional A are precisely the closed Reeb orbits of α . Lemma 3.3. γ ∈ Crit ( A ) iff γ closed Reeb orbit of α with period (cid:82) γ α In a complete analogy with Morse theory, critical points have to be non-degenerate.The non vanishing condition for the Hessian in Morse theory translates in contacthomology to the fact that the linearized Reeb flow over a periodic orbit does not havean eigenvalue equal to 1.
Definition 3.4.
Let γ be a closed Reeb orbit and γ (0) = p . dφ t | γ := Φ γ : ξ p → ξ p iscalled the linearized return map or Poincaré return map of γ .Note that the contact condition on α says that dα is non-degenerate, i.e. ( ξ, dα ) is a symplectic vector bundle. Since the deRham differential commutes with the Liederivative L R α we have that the map Φ γ : ξ p → ξ p is symplectic. Definition 3.5.
The closed Reeb orbit γ is non-degenerate if the map Φ γ : ξ p → ξ p has no eigenvalue equal to 1.Note that γ is non-degenerate if and only if γ is a non-degenerate critical point of A , modulo reparametrization. Thanks to Bourgeois [Bou02], we have the followinglemma. Lemma 3.6.
Fix a period threshold
T > . The space of contact forms C T supporting ξ with all orbits of period ≤ T being non-degenerate is open and dense in the spaceof contact forms supporting ξ . Using Baire’s theorem we have in particular (cid:92) T ≥ C T (cid:54) = ∅ which yields the followinglemma. Lemma 3.7.
For any contact structure ξ on Y, there exists a contact form α for ξ such that all closed orbits of R α are non-degenerate. The next natural question is what is the grading of each orbit γ . Contact homol-ogy has a relative grading by Z / c ( ξ ) · H ( Y ) which is absolute on null-homologousorbits. We now introduce the grading which will also help us select the generators ofthe algebra. The grading is highly dependent on the notion of the Conley-Zehnderindex. More details about its definition and properties can be found in [Bou03].The Conley-Zehnder index depends on the chosen symplectic trivialization T for ξ . If γ is a closed Reeb orbit in Y n − its grading is given by | γ | := CZ T ( γ ) + n − where T is a trivialization of ξ along γ and CZ T ( γ ) the Conley-Zehnder index of γ with respect to T .This dependence becomes more explicit when γ is null-homologous and we restrictto trivializations which extend over the corresponding spanning surfaces. Let Σ γ bea spanning surface for γ and τ a trivialization for ξ over Σ γ which agrees with T over the Reeb orbit γ . It is easy to see that if A ∈ H ( Y ) then we can connect sum HE CONTACT BANACH-MAZUR DISTANCE 21 an already chosen spanning surface Σ γ for γ with A and this will alter the Conley-Zehnder index as follows. If τ (cid:48) is a trivialization for ξ over Σ γ A which agrees with T over γ then CZ τ (cid:48) ( γ, Σ γ A ) = CZ τ ( γ, Σ γ ) + 2 (cid:104) c ( ξ ) , A (cid:105) Thus it becomes helpful to also introduce a grading on H ( M, Z ) . The grading of A ∈ H ( M, Z ) is | A | = − (cid:104) c ( ξ ) , A (cid:105) . CH ( Y, ξ ) is a DG-algebra which is generated by closed, non-degenerate Reeb orbitswhich are good. The reason for excluding the collection of bad orbits is orientationissues when gluing pseudoholomorphic curves. Let γ = γ be a simple Reeb orbitand for k ∈ N , let γ k be the k-fold cover of γ , namely if f k : S → S is given by f k ( e πit ) = e πikt , then γ k is the map γ k := γ ◦ f k : S → Y . We partition the set ofclosed Reeb orbits into two subsets according to their grading behavior when multiplycovered. The grading of γ can behave in two distinct ways. Either the parity of | γ k | is equal to the parity of | γ | for all k or the parity of | γ k | is not equal to the parity of | γ k − | for all k . Definition 3.8.
Orbits in the first category are called good and orbits in the secondcategory are called bad.The reason for this is the behavior of the Conley-Zehnder index under iterations.The index can also be seen as an integer valued winding number which counts thewinding of any push-off of γ using the chosen trivialization for ξ or equivalently therotation of the eigenspaces of the linearized flow over γ .The grading in monomials is defined as in any DG-algebra by | γ · · · γ k | = | γ | + · · · + | γ k | .In the case that the contact form is degenerate, the proof of lemma 3.7 suggeststhat we have to perturb the contact form by first choosing an action threshold andafter the perturbation all orbits of action ≤ T are non-degenerate. The formula forthe grading changes according to | γ p | = CZ ( S T ) + 12 dim( S T ) + index p ( f T ) + n − where f T the perturbing function to make α less symmetric. Also, S T := { q ∈ M | φ T ( q )= q }∼ ,where ∼ the equivalence relation under the circle action induced by the Reeb flow φ t and p the critical point corresponding to one of the Reeb orbits γ p created afterperturbation by f T corresponding to p.The last step in defining the chain complex is to define the differential ∂ : CC ∗ ( Y, ξ ) → CC ∗− ( Y, ξ ) . As in the Morse case, we have to count certain “trajectories” between our criticalpoints which in this case are Reeb orbits. As it is the case with all field theories,these trajectories will be special surfaces in the symplectization asymptotic to ourorbits. In symplectic field theory, these trajectories are pseudoholomorphic curves ofgenus zero with one positive end and finitely many negative ends. A Stokes’ theorem argument shows that ∂ reduces the action, so we get a restriction on how many neg-ative ends can exist and how large their periods can be.The difficulty that arises here is to define a proper count of pseudoholomorphiccurves and thus obtain coefficients for the differential. For this to work, we need theassociated moduli space to be cut out transversely, which means that the relevantlinearized operator is everywhere surjective, i.e. we have the regularity property. Wewill describe the situation when we work under this rare favorable situation, but werefer to [Par19] for the full picture. In this work, the only count we need to make isfor the lowest action orbit which needs to be bounding a unique pseudoholomorphicplane. We will explain why we can obtain a meaningful count in this special case,using the main theorem from [Par19] when we describe how to obtain the pseudo-holomorphic plane.Contact homology turns out to be invariant under different choices of the support-ing contact form α on Y and of the chosen compatible almost complex structure on SY . Thus it is both permitted and easier for us to implicitly make a choice of formin order to describe the theory concretely. In words, we have picked a contact form α on Y which induces a splitting ( SY = R + × Y, dθ = d ( rα )) .We consider an almost complex structure on ( SY, d ( rα )) cylindrical in the followingsense. • J is R -invariant. • J ( ∂ r ) = R α . • J ( ξ ) = ξ is compatible with α , i.e. g ( · , · ) = dα ( · , J · ) is an inner product. Definition 3.9.
A map u : ( S , j ) → ( SY, J ) is called pseudoholomorphic if du ◦ j = J ◦ du , i.e. du is complex linear.Note that up to isomorphism S admits a unique complex structure j . In otherwords, any complex surface of genus zero is biholomorphic to the Riemann sphere C P . In the following, we consider punctured holomorphic spheres inside the sym-plectization SY . By this we mean that we consider curves Σ = S − { x, y , ..., y k } such that if ( r, θ ) are polar coordinates centered at each puncture and u ( r, θ ) =( a ( r, θ ) , f ( r, θ )) ∈ SY = R × Y we have lim r → a ( r, θ ) = (cid:40) + ∞ , for the puncture x −∞ , for the punctures y i lim r → f ( r, θ ) = (cid:40) γ ( − T π θ ) , for the puncture xγ i ( T i π θ ) , for the punctures y i Despite the fact that pseudoholomorphic curves were an effective tool when study-ing closed symplectic manifolds (Gromov-Witten theory), for quite a while it was notunderstood how pseudoholomorphic curves behave in non compact symplectic mani-folds. Hofer in [Hof93] in a successful effort to prove the Weinstein conjecture for S ,showed that bounds on the energy of such object, yield and interesting behavior andforce such curves to be asymptotic to Reeb orbits γ , γ i and this asymptotic behavior HE CONTACT BANACH-MAZUR DISTANCE 23 is the requirement we ask for the curves we will be counting. Their periods will bedenoted T for the orbit corresponding to the positive puncture x and T i for the restof the punctures corresponding to y i .Recall that in Morse homology we have to count trajectories from critical pointsof index s to critical points of index s − . This is also what we do here. The factthat we work with orbits and not points gives rise to an extra difficulty. Orbits canbe multiply covered, so that raises new issues when a complex curve is asymptoticto them. This is of course not an issue in Morse homology. We denote by m ( γ j ) themultiplicity of γ j over its underlying simple Reeb orbit. The coefficients involved inthe differential are related to the moduli spaces of such punctured pseudoholomorphiccurves.Let (cid:99) M ( γ ; γ , ..., γ k ) be the set pseudoholomorphic curves satisfying the asymptoticconditions above. We define an equivalence relation ∼ on this set as follows. Definition 3.10.
The maps u : ( S − { x, y , ..., y k } , j ) → ( SY, J ) and u (cid:48) : ( S −{ x (cid:48) , y (cid:48) , ..., y (cid:48) k } , j (cid:48) ) → ( SY, J ) are equivalent if and only if there exists a biholomor-phism h : ( S , j ) → ( S , j (cid:48) ) so that h ( x ) = x (cid:48) , h ( y i ) = y (cid:48) i for i = 1 , ..., k , and u = u (cid:48) ◦ h .We denote M ( γ ; γ , ..., γ k ) = (cid:99) M ( γ ; γ , ..., γ k ) / ∼ . The set of equivalence classes M ( γ ; γ , ..., γ k ) has a natural R -action induced by the translation in the R -coordinate.We will define the differential by counting certain generalized flowlines with ap-propriate associated weights. In order for this count to be finite, the fact that M ( γ ; γ , ..., γ k ) / R has a nice topological structure, i.e. that it is a compact 0-dimensional manifold is required. In general, this is not the case. A way to overcomethis is the main theorem from [Par19]. As stated previously though, in this presen-tation we assume that the favorable assumption of transversality holds. We have thefollowing. Lemma 3.11. M ( γ ; γ , ..., γ k ) / R is a union of compact manifolds with corners alonga codimension 1 branching locus. Each such manifold has a rational weight, so thatnear each branching point, the sum of all entering weights equals the sum of all exitingweights. Moreover, each manifold with corner in this union has dimension ( n − k −
1) + CZ ( γ ) + k (cid:88) i =1 CZ ( γ i ) + 2 c rel ( ξ, Σ) − where c rel ( ξ, Σ) is the first Chern class of ξ on Σ , relative to the fixed trivializationsof ξ along the closed Reeb orbits at the punctures. The rational weights take into account the group of automorphisms of the holo-morphic curves. If u is a rigid element in M ( γ ; γ , ..., γ k ) / R of dimension 0, then theweight of u is k , where k is the order of the automorphism group of u .In order to be able to show that the operator ∂ is indeed a differential, we needto understand the boundary of M ( γ ; γ , ..., γ k ) , i.e. possible degenerations of suchpseudoholomorphic curves. Definition 3.12.
A broken pseudoholomorphic curve is a set C , ..., C N of finitecollections of punctured pseudoholomorphic curves C i = { u i , ..., u li } such that thenegative punctures/orbits of C i coincide with the positive punctures/orbits of C i − . Moreover, the only positive puncture of C corresponds to γ and the negative orbitsof C N correspond to γ , ..., γ k .The boundary of M ( γ ; γ , ..., γ k ) is made up of broken pseudoholomorphic curves.Energy bounds on punctured pseudoholomorphic curves help to control the degen-eration behavior and show that M ( γ ; γ , ..., γ k ) together with its boundary brokenpseudoholomorphic curves is a compact manifold. See [BEH +
03] section 10.A class A ∈ H ( Y, Z ) is associated to each curve in M ( γ ; γ , ..., γ k ) . This has theeffect that we can decompose M ( γ ; γ , ..., γ k ) into the connected components corre-sponding to A . Then, we denote by M A ( γ ; γ , ..., γ k ) the corresponding connectedcomponent of the moduli space. This association is essential in order to define thedifferential of the chain complex, yet the only case we explain here is the case of nullhomologous orbits. This is important for this work as the most essential orbit, i.e.the one providing the l -invariant, is contractible. We do not need to explain all caseshere as there is no direct reference to it in this text. A thorough explanation of thiscan be found in [Bou03], lecture 2.Assuming that the orbit γ is null-homologous, we pick a spanning surface Σ γ andwe use it to trivialize ξ over γ . Now, to a pseudoholomorphic curve in M ( γ ; γ , ..., γ k ) ,we glue the surfaces we obtained for each of the orbits in { γ, γ , ..., γ k } and we obtaina closed surface. We let A ∈ H ( Y, Z ) be its homology class.Using the trivializations discussed before, the formula for the dimension of the cor-responding connected component of M ( γ ; γ , ..., γ k ) , which we denote M A ( γ ; γ , ..., γ k ) ,is dim M A ( γ ; γ , ..., γ k ) = | γ | − k (cid:88) i =1 | γ i | + 2 (cid:104) c ( ξ ) , A (cid:105) We define the coefficients of the differential. Let
Γ := γ γ ...γ k . First, define thenumbers n Aγ, Γ = , if dim M A ( γ ; γ , ..., γ k ) (cid:54) = 1 (cid:88) c ∈M A ( γ ; γ ,...,γ k ) / R | Aut ( c ) | , if dim M A ( γ ; γ , ..., γ k ) = 1 where | Aut ( c ) | is the order of the automorphism group of the rigid element c of themoduli space. These numbers count rigid pseudoholomorphic curves positively as-ymptotic to γ and negatively asymptotic to Γ in the homology class A ∈ H ( M, Z ) .These numbers are finite and nonzero for finitely many classes A due to the fact thatthe moduli spaces involved are compact.The coefficients of the differential are now defined to be n γ, Γ = (cid:88) A ∈ H ( M, Z ) n Aγ, Γ e π ( A ) ∈ Q [ H ( M, Z ) / R ] HE CONTACT BANACH-MAZUR DISTANCE 25 where R a submodule of H ( M, Z ) with zero grading and π : H ( M, Z ) → H ( M, Z ) / R the natural projection. The usual choice is R = H ( M ; Z ) which gives coefficients in Q .The differential is defined by ∂γ := m ( γ ) (cid:88) Γ=( γ , ..., γ (cid:124) (cid:123)(cid:122) (cid:125) i ,..., γ k , ..., γ k (cid:124) (cid:123)(cid:122) (cid:125) ik ) n γ, Γ γ · · γ (cid:124) (cid:123)(cid:122) (cid:125) i · · · γ k · · γ k (cid:124) (cid:123)(cid:122) (cid:125) i k where we recall that m ( γ ) is the multiplicity of the orbit at + ∞ , γ .We extend the differential to monomials using the graded Leibniz’s rule and to anyelement of the DG-algebra by linearity. The unit of this DG-algebra is Γ for which k = 0 . Contact homology is the homology of this complex.Contact homology is a functor from the category with objects contact manifoldsand morphisms deformation classes exact symplectic cobordisms to the category withobjects supercommutative Z / -graded unital Q -algebras and morphisms graded uni-tal Q -algebra homomorphisms. In the proof of lemma 4.14, we will be interestedin a refinement which keeps track of the action of orbits. Since the differential de-creases action, we are allowed to consider the subcomplex CC ∗ ( Y, α ) ≤ t ⊆ CC ∗ ( Y, α ) consisting of all orbits of action ≤ t . Notice that this refinement depends on thechosen contact form. Whenever αt ≥ α (cid:48) t (cid:48) pointwise, we get an induced functorial map CH ∗ ( α ) ≤ t → CH ∗ ( α (cid:48) ) ≤ t (cid:48) , see for example [Par19] section 1.7. We will be mostlyinterested in the case t = t (cid:48) .4. Proof of the bi-Lipschitz embedding
The way to prove theorem 2.5 in 3 dimensions will be to construct a 2-parameterfamily of overtwisted contact forms on Y modifying the Lutz twist construction asfound in [Wen08]. The first parameter will be related to the volume of Y and thesecond one to the l -invariant whose definition is immediately provided as it will beof often use in the rest of this section. Let α be a contact form supporting an over-twisted contact structure on Y . Since CH ( Y, ξ ) vanishes, there is some filtrationlevel for which a primitive x for the unit element appears. This filtration is basicallythe action whose definition we now recall. Note that by definition, talking about theaction always assumes a choice of a contact form.An element of CH ( Y, ξ ) has the form y = γ a · · · γ a k k + · · · + γ a m m · · · γ a n n , i.e. it is apolynomial in good orbits. In the beginning of section 3, we only defined the actionof an orbit and not the action of an element of CH ( Y, ξ ) . This definition is providedbelow. Definition 4.1.
The action of an element y ∈ CH ( Y, ξ ) is defined as follows. If y isa monomial in good orbits, i.e y = γ a · · · γ a k k , then A ( y ) := k (cid:88) i =1 a i A ( γ i ) If y = Γ + · · · + Γ n , where Γ i = γ i a i · · · γ i am i m , i.e y is a polynomial in good orbits,then A ( y ) := max j (cid:8) A (Γ j ) (cid:9) We are now ready to provide the definition of the l -invariant of a contact form α . Definition 4.2.
We call the lowest action A ( x ) of such a primitive x the l -invariant. Remark 4.3.
The l -invariant is an invariant of a contact form α supporting a contactstructure ξ . In the cases that we need to emphasize this, we will use the notation l ( α ) .The section is organized as follows. In subsection 4.1 we recall the notion of a Lutztwist in 3 dimensions. In 4.2 we recall some of Wendl’s work and adapt results toour case. In 4.3 we construct the 2-parameter family needed in the proof and finallyin 4.4 we prove the main theorem, i.e. theorem 2.5, in the 3-dimensional case. Thehigher dimensional case is discussed in section 5. Remark 4.4.
We provide a brief list of the parameters involved in the constructionsof this section and brief explanations. This will not make sense unless the readerarrives to the part of this work where they are needed. We only mention them herein order to state explicitly that the constructions below will only make sense forsufficiently small choice of these parameters. • δ : Used in the perturbation of the horizontal Morse-Bott torus. Defined inpage 29. • δ (cid:48) : Used in the perturbation in order to set up the contact homology chaincomplex up to a certain action threshold A . Defined in page 30. • A : Lowest action of Reeb orbit before the Lutz twist. Discussed in page 35. • B : Action of the family of orbits in the neighbourhood of a transverse knot K . Discussed in page 35. • (cid:15) : It is min { ln( A ) , ln( B ) } and will define the parameter domain of the 2-parameter family of 1-forms to be defined. • ε : Radius of specific Lutz tube we use. See section 4.1. • ε : First time when behavior of h ( r ) , h ( r ) changes. Defined in subsection4.4.2. • δ : Defines smoothing interval. Defined in subsection 4.4.2. • δ : Chosen so as the path of the specific Lutz tube to be continuous (relatedto h ( r ) ). See subsection 4.4.2 • δ : Chosen so as the path of the specific Lutz tube to be continuous (relatedto h ( r ) ). See subsection 4.4.2.4.1. The Lutz twist.
The first step towards the construction of the 2-parameter family is to perform a fullLutz twist. Although by now these constructions are standard and [Gei08],[Wen08]are excellent references, we include it here for completeness.Let Y be a co-oriented contact 3-manifold and P ⊂ Y an embedded S positivelytransverse to ξ . Let S × D be a tubular neighbourhood of P in Y (i.e. P = S ×{ } )such that ξ = ker ( dθ + r dφ ) where ∂ θ agrees with the positive orientation on P . Thisis possible to consider due to the fact that in a neighborhood of a transverse knot, HE CONTACT BANACH-MAZUR DISTANCE 27 the local model for ξ looks like the one described. See for instance [Gei08], example2.5.16. The radius of the disk factor is assumed to be ε which will be sufficientlysmall. Performing a Lutz twist along P means that we replace the contact structure ξ = ker ( dθ + r dφ ) inside S × D by the structure ξ (cid:48) = ker ( h ( r ) dθ + h ( r ) dφ ) . Thefunctions h , h are only required to satisfy the following 3 properties • h ( r ) = ± and h ( r ) = ± r near r = 0 . • h ( r ) = 1 and h ( r ) = r near r = ε , where the radius of D is ε . • ( h ( r ) , h ( r )) is never parallel to ( h (cid:48) ( r ) , h (cid:48) ( r )) for r (cid:54) = 0 .The “ + ” version is called a full Lutz twist, whereas the “ − ” version is called a halfLutz twist. The result of a Lutz twist is always an overtwisted contact structureas it is straightforward to see that we create an overtwisted disk inside the tubularneighbourhood of P where for the smaller of the two radii, r such that h ( r ) = 0 the Legendrian/meridian { (0 , r , φ ) | φ ∈ [0 , π ) } is the boundary of an overtwisteddisk. h ( r ) h ( r ) Figure 2.
Description of a general full Lutz twist
Remark 4.5.
A Lutz twist is primarily a contact topological alteration of the struc-ture. In this work we would like to view it as a dynamical one. To this end, althoughit is not always true that in a local neighborhood of a transverse knot the contactform is precisely the standard dθ + r dφ , since it is true that two contact forms withthe same kernel differ by multiplication by a smooth positive function, we can canmake the form locally to look like the standard one by multiplying by an appropriatepositive smooth function. In this work, using the above justification, anytime we areinterested in working with a transverse knot, we assume that in a neighborhood of itthe form looks like the standard one.The advantage of a full twist is that it does not alter the homotopy class of ξ as a 2-plane field thus giving us, according to Eliashberg [Eli89], the unique up toisotopy overtwisted contact representative of the original plane distribution. There isa generalization of the full Lutz twist in higher dimensions according to [EP11],[EP16]and as it is shown there, the contact structure obtained after the generalized Lutztwist is homotopic to the original one through almost contact structures, i.e. the twocontact structures are formally homotopic. This, combined with corollary 1.3 from[BEM15] implies that after the twist, we still get up to isotopy, the unique overtwistedrepresentative of the homotopy class of ξ that we started with. A half Lutz twistgeneralization is also discussed, yet it seems that this is not a natural way to define ahalf Lutz twist in higher dimensions as it not only changes the contact structure, but the original smooth manifold itself. Although using this approach of thinking aboutLutz twists in higher dimensions seems very natural, negatively stabilized open bookshelp us more. There is a very helpful pseudoholomorphic curves analysis in [BvK10]which we use to generalize the result to higher dimensions.4.2. Recollection of Wendl’s work and adaptation to our case.
Our goal is to control the volume and the l -invariant of contact forms supportingthe overtwisted contact structure ξ on Y . These are the two degrees of freedom of R . Although the first is easy to do just by multiplying the contact form, for thesecond one we have to work more. We have to control the l -invariant as defined indefinition 4.2. In order to do this, we have to adapt the Lutz twist construction ina way that allows us to control the action of the lowest action orbit which bounds aunique pseudoholomorphic plane.In this we mainly follow [Wen08], subsection 4.2. Our first goal will be to un-derstand the Reeb dynamics within the Lutz tube and then the foliation by pseu-doholomorphic curves of the part of the symplectization of our overtwisted contactmanifold Y that corresponds to the Lutz tube S × D . In what follows, we work in S × int ( D ) assuming that the radius of D is equal to ε .Let’s let X = S × int ( D ) equipped with a contact form described as above, h ( r ) dθ + h ( r ) dφ . We pick a suitable basis { v , v } for ξ on the complement ofthe core circle P = S , where v = ∂ r , v = D ( r ) ( − h ( r ) ∂ θ + h ( r ) ∂ φ ) and D ( r ) = h ( r ) h (cid:48) ( r ) − h (cid:48) ( r ) h ( r ) .We define a complex structure J : ξ → ξ , given by v (cid:55)→ β ( r ) v and v (cid:55)→ − β ( r ) v ,where β a smooth function chosen so as to ensure smoothness of J near the core P .So we can assume that is different than 1 only sufficiently close to P .Naturally, we let ˜ J be the unique R -invariant compatible almost complex structureon R × X determined by J and α = h ( r ) dθ + h ( r ) dφ .The following proposition describes the local Reeb dynamics within S × int ( D ) . Proposition 4.6.
Let r > and h (cid:48) ( r )2 πh (cid:48) ( r ) = pq ∈ Q ∪ {∞} , where p, q ∈ Z relativelyprime, sign( p ) = sign( h (cid:48) ( r )) and sign( q ) = sign( h (cid:48) ( r )) . Then the torus L r := { r = r } is foliated by orbits of the form x ( t ) = (cid:16) θ + h (cid:48) ( r ) D ( r ) t, r , φ h (cid:48) ( r ) D ( r ) t (cid:17) = (cid:16) θ + qT t, r , φ − πpT t (cid:17) all having minimal period T = q D ( r ) h (cid:48) ( r ) = 2 πp D ( r ) h (cid:48) ( r ) If h (cid:48) ( r ) = p = 0 or h (cid:48) ( r ) = q = 0 pick whichever formula makes sense. The torus L r is Morse-Bott iff h (cid:48) ( r ) h (cid:48) ( r ) (or its reciprocal if needed) has non vanishing derivative at HE CONTACT BANACH-MAZUR DISTANCE 29 r = r . Moreover, P := S × { } is a closed orbit f minimal period T = | h (0) | . For k ∈ Z its k-fold cover P k is degenerate iff kh (cid:48)(cid:48) (0)2 πh (cid:48)(cid:48) (0) ∈ Z and otherwise it has CZ Φ ( P k ) = 2 (cid:22) − kh (cid:48)(cid:48) (0)2 πh (cid:48)(cid:48) (0) (cid:23) + 1 where Φ the natural symplectic trivialization of ξ along P provided by the coordinates. Observe that a formula for the action of the orbits in terms of the coordinate r isgiven by either A ( r ) = 2 πp D ( r ) h (cid:48) ( r ) or A ( r ) = q D ( r ) h (cid:48) ( r ) depending on where the derivativeof either h ( r ) or h ( r ) vanishes. A quick analysis checking the signs of the derivativeof A ( r ) with respect to r and the possible values for p, q yields two minima for theaction precisely when r is such that h ( r ) = 0 . Figure 3 may help the reader withthis argument. According to figure 2 there are two such values for r , which we denoteby r + and r (cid:48) + . The first of the two will be used to control the l -invariant while for thesecond one, when we pick the specific h ( r ) , we will require that h ( r + ) < h ( r (cid:48) + ) soas for the orbits corresponding to r + to have the least action among all orbits. Thisis explained in claim 4.8. h ( r ) h ( r ) h > h > h (cid:48) < h (cid:48) > h (cid:48)(cid:48) < h (cid:48)(cid:48) < h < h > h (cid:48) < h (cid:48) < h (cid:48)(cid:48) > h (cid:48)(cid:48) < h < h < h > h > h (cid:48) > h (cid:48) < h (cid:48)(cid:48) > h (cid:48)(cid:48) > h (cid:48) > h (cid:48) > h (cid:48)(cid:48) < h (cid:48)(cid:48) > Figure 3.
Analysis for the signs of A (cid:48) ( r ) Let us know proceed with understanding the foliation. Let r − , r + ∈ (0 , ε ) . Inhis thesis, [Wen05] section 3.1, Wendl proves (by solving a system of ODEs using areasonable geometric ansatz) that the region in the symplectization { ( a, θ, r, φ ) ∈ R × X | r ∈ ( r − , r + ) ⊂ (0 , ε ) } is foliated by finite energy ˜ J -holomorphic cylinders asymptotic to { r = r − } and { r = r + } .Next, also within section 3.1, he studies the case when r − → . There are twocases to consider, namely q = 0 and q (cid:54) = 0 . In this work we are interested only in thesetting where our function h ( r ) so that h (cid:48) ( r + ) = 0 , i.e. q = 0 . This ensures that thepositive ends of the pseudoholomorphic planes foliating the solid tube do not wind around the θ direction.Then, using Gromov’s removable singularity theorem, he shows that the previousfoliation can be extended smoothly to a finite energy foliation of the region { r < r + } by ˜ J -holomorphic disks each positively asymptotic to some simply covered Reeb or-bit on the torus { r = r + } and transverse to the core P. As explained previously,these simply covered Reeb orbits have period T = 2 π · h ( r + ) . These Reeb orbits,which form a 2-torus, can be arranged, by suitable choices to be explained, to bethe lowest action orbits bounding a holomorphic plane. As stated before, the proofof claim 4.8 provides more insight. Hence, we end up having at this point that π · h ( r + ) is the l -invariant. Strictly speaking, we need a perturbation so as to havedegenerate orbits, yet the action is not far from π · h ( r + ) as will be explained below.The upshot is that modification of h ( r ) provides the second degree of freedom inthe construction of the 2-parameter family. This is because, for any choice of h ( r ) providing a Lutz twist, Wendl’s construction is feasible, thus we get holomorphicplanes bounded by orbits of certain chosen action. The key idea is that this modifica-tion is Lipschitz. This will be proved in subsection 4.4 where we will get bounds usingGray’s stability theorem. Although not relevant to our work, for the reader interestedin contact structures, we also remark that the pseudoholomorphic disk is “smaller”than the overtwisted disk as the overtwisted disk has its Legendrian boundary at thespecific r between r + and r (cid:48) + for which h ( r ) = 0 .What was not mentioned explicitly above, is that actually the asymptotes of thepseudoholomorphic curves forming the stable finite energy foliation are foliating inturn the tori { r = r + } and { r = r − } (recall that in our case r − = 0 and h (cid:48) ( r + ) = 0 sothere is only one interesting torus corresponding to r + ). This turns out to be helpfulin the following manner. If we need to set up the contact homology chain complexwe have to perturb our form so that all Reeb orbits are isolated. This can be done asin [Wen05] following [Bou02]. We have to be extra careful about the orbits boundingpseudoholomorphic planes. Our perturbation will be performed in two steps.The first which agrees with [Wen05] section 3.3 guarantees that the resulting holo-morphic planes generically are positively asymptotic to the elliptic and hyperbolicorbits created after perturbation of the form in a small neighbourhood of the torus { r = r + } . Recall that our goal is of course to exhibit existence and uniqueness of thepseudoholomorphic plane bounded by an orbit of least action l and that means thatthe contact homology class of the identity vanishes precisely at action level l .The perturbation is performed as follows. We first choose a smooth cut off function b ( r ) supported in a neighbourhood of the torus L + = { r = r + } and is equal to 1 near L + . Moreover, choose a small number δ and a Morse function µ : S → R with twocritical points at θ = θ + and θ = θ − . The perturbed contact form is then α δ := (cid:40) (1 + δb ( r ) µ ( θ )) α, r ∈ supp ( b ) α, otherwise HE CONTACT BANACH-MAZUR DISTANCE 31
Then one can see that the corresponding Reeb vector field in the solid tube incoordinates ( θ, r, φ ) is given by R δ = ( h (cid:48) + δµ ( b (cid:48) h + bh (cid:48) ) , − δµ (cid:48) bh , h (cid:48) + δµ ( b (cid:48) h + bh (cid:48) )) D (1 + δbµ ) has two orbits at r = r + corresponding to the critical points of µ (since h (cid:48) ( r + ) = 0 , b (cid:48) ( r + ) = 0 and b ( r + ) = 1 ). One of them is the elliptic O e and the other one is thehyperbolic O h .What has been achieved so far is that the orbits on the positive torus L r + arenon-degenerate, thus we have two non-degenerate, isolated ones O h , O e . Moreover,those are simply covered so they are not bad. Since δ can be chosen arbitrarily small,their action is arbitrarily close to the action of the orbits of the original Morse-Botttorus. The foliation consists of a family of pseudoholomorphic planes. One rigidplane positively asymptotic to O h and a family of planes parametrized by the openinterval (0 , positively asymptotic to O e . For more details the interested reader canconsult [Wen05], section 3.3.The next step in the perturbation process is the one described by Bourgeois in[Bou02]. This allows all the remaining orbits (not lying on L + which is already per-turbed) to become non-degenerate, thus isolated so as to be able to set up the CHcomplex. The initial worry is whether this perturbation will affect the foliation andpossibly make the rigid plane we are interested in disappear. Thanks to [Wen05]section 4.5, the necessary Fredholm analysis shows that for sufficiently small defor-mation parameter δ (cid:48) , the foliation is stable under deformations of the form α δ and ofthe almost complex structure J .The perturbed contact form is α δ,δ (cid:48) := (1 + δ (cid:48) f T ) α δ , where f T a smooth Morsefunction with support a small neighbourhood of the set consisting of points on non-isolated orbits of action ≤ T . The new Reeb vector field is given by R α δ,δ (cid:48) = R α δ + X where X is the vector field with the properties i ( X ) dα δ = δ (cid:48) df T (1 + δ (cid:48) f T ) α δ ( X ) = − δ (cid:48) f T δ (cid:48) f T Due to the fact that δ (cid:48) is chosen sufficiently small, this new perturbation will createno new orbits below a sufficiently large action threshold A .So, we are thus now able to set up the contact homology chain complex. The restof this section will be a discussion on how to control the l -invariant. We remark thatafter the perturbations above, we get A ( O h ) = 2 πh ( r + )(1 + δµ ( θ − )) . This will bethe precise l -invariant. Claim 4.7. O h has degree 1. Proof.
The grading of any null-homologous orbit γ in contact homology algebra isgiven by the formula | γ | = CZ τ ( γ ) + n − (cid:104) c ( ξ, τ ) , A (cid:105) ∈ Z /c ( ξ ) · H ( Y ) for any trivialization τ and any null-homology A of γ . In our case, n = 2 and as weshow CZ τ ( O h ) = 0 and (cid:104) c ( ξ, τ ) , A (cid:105) = 2 .Let’s start with the calculation of CZ τ ( O h ) . Since initially our orbits are degener-ate, the formula in order to calculate Conley-Zehnder indices of simply covered orbitsis, according to lemma 2.4 in [Bou02], CZ ( γ pT (cid:48) ) = µ ( S T (cid:48) ) − dim ( S T (cid:48) ) + index p ( f T (cid:48) ) where to recall things, φ T is the Reeb flow for time T , N T = { p ∈ Y | φ T ( p ) = p } and S T the quotient of N T under the Reeb flow. µ ( S T (cid:48) ) is the generalized Conley-Zehnder(see [Gut14]). T (cid:48) is an action level less or equal to T . Recall that perturbation ismade by fixing an action level T and then the result of this perturbation process isthat all orbits of action ≤ T become non-degenerate. In particular, here N T (cid:48) is thetorus of radius r = r + foliated by degenerate horizontal orbits, S T (cid:48) = S , f T (cid:48) a Morsefunction on S T (cid:48) and p a critical point of f T (cid:48) corresponding to some degenerate orbit.We have dim ( S T (cid:48) ) = 1 , index p ( f T (cid:48) ) ∈ { , } and as we now show µ ( S T (cid:48) ) = . Thus,the elliptic orbit corresponding to the critical point of index 1 will have CZ ( O e ) = 1 and the hyperbolic orbit CZ ( O h ) = 0 .In coordinates ( θ, r, φ ) , the flow in general is given by φ t ( θ, r, φ ) = ( θ + h (cid:48) ( r ) D ( r ) t, r, φ − h (cid:48) ( r ) D ( r ) t ) So we get, dφ t = (cid:16) h (cid:48) ( r ) D ( r ) (cid:17) (cid:48) t
00 1 00 (cid:16) − h (cid:48) ( r ) D ( r ) (cid:17) (cid:48) t We will perform a symplectic change of basis so as to calculate the index using somehelpful axiom. It is also helpful to work in this basis for the relative Chern class term.The new basis for the linearized flow will be (cid:104) ∂ θ , ∂ r , − h ( r ) ∂ θ + h ( r ) ∂ φ (cid:105) .So now the matrix is given by dφ t = f ( r ) t where f ( r ) = − h (cid:48)(cid:48) ( r ) h (cid:48) ( r ) − h (cid:48) ( r ) h (cid:48)(cid:48) ( r ) D ( r ) . We restrict the linearized flow on ξ and we lookat t = 1 . This restriction looks like dφ t = (cid:18) f ( r ) t (cid:19) We aim to use the symplectic shear axiom. In our case, h (cid:48) ( r + ) = 0 hence sgn ( f ) > . Making the obvious last symplectic change of basis, the matrix for dφ t whenrestricted to ξ looks like (cid:18) − f ( r ) t (cid:19) HE CONTACT BANACH-MAZUR DISTANCE 33 with sgn ( − f ) < . Thus, by the symplectic shear axiom µ ( dφ ) = − sgn ( − f )2 = .Hence, as expected µ ( S T (cid:48) ) = .Let’s now focus on (cid:104) c ( ξ, τ ) , A (cid:105) = 2 . We have to pick a section of ξ , constant along O h with respect to our given trivialization for ξ . This is (cid:104) ∂ r , h ( r ) ∂ θ − h ( r ) ∂ φ (cid:105) . Weextend it over A and count its zeroes. We choose the second basis vector here. Thisonly vanishes at the origin of the disk A positively once. Thus, as it is expected (cid:104) c ( ξ, τ ) , A (cid:105) = 2 . (cid:3) This means that O h has degree one more than the empty word. Claim 4.8.
Any other orbit bounding a holomorphic plane must have action morethan that of O h . As mentioned earlier, this depends on the Lutz twist modification parameters andespecially on the function h ( r ) which can be chosen accordingly in order to ensurethis. We now provide a rigorous proof of this statement. Proof.
Let A be the action of the lowest action orbit bounding a unique pseudo-holomorphic plane before the Lutz twist. If there is no such orbit we let A = + ∞ .We need to choose h ( r ) , h ( r ) satisfying the conditions described in section 4.1 inorder to be able to perform the Lutz twist. We recall that r + < r (cid:48) + are the twovalues of r for which the function h ( r ) vanishes. There, the tori L r + and L r (cid:48) + arefoliated by horizontal Reeb orbits and the minima for the action occur. We have toperturb the Morse-Bott torus L r + = { r = r + } in order to get an isolated orbit ofleast action O h which bounds a unique pseudoholomorphic plane. Recall that then A ( O h ) = 2 πh ( r + )(1 + δµ ( θ − )) .We have to impose some additional requirements in order to know what the l -invariant after the Lutz twist is. The first one is πh ( r + ) < A or equivalently h ( r + ) < A π and the second one is | h ( r + )(1 + δµ ( θ − ) | < | h ( r (cid:48) + ) | . The combinationof both guarantees first that the hyperbolic orbit at r = r + is the one with the leastpossible action among all Reeb orbits of the contact manifold. (cid:3) We denote the empty word by . In order to obtain the needed count and thusget the coefficient (cid:104) ∂O h , (cid:105) (which we need to it be equal to 1), one needs to ensuretransversality at the holomorphic plane u , or in other words to ensure that the lin-earized Cauchy-Riemann operator at u is surjective. Of course, if one wants to usetechniques from [Par19] and thicken the moduli spaces in order to obtain a propercount of curves, i.e. coefficients for the differential and thus a well defined contacthomology algebra, they have to make sure that this thickening process does not alterthe coefficient (cid:104) ∂O h , (cid:105) , which geometrically at least was calculated to be equal to 1.Problems that arise when compatifying the moduli space are multiply coveredcurves or breaking along Reeb orbits. Intuitively, since O h is embedded, we do nothave to worry about multiple covers and since O h has the lowest action, no breakingof the pseudoholomorphic plane into buildings can occur as it would have to breakalong an orbit of lower action than that of O h and there are no such orbits. Thisintuition is backed up by part ( iv ) of Theorem 1.1 in [Par19]. In words, it states thatwhen the moduli space is of dimension 0 and regular, then the algebraic count weobtain by thickening agrees with the geometric count we already have. Concretely, inour case, the geometric count is 1 and the formal algebraic count we get after setting up CH ( Y, λ ) is also 1. Thus, the coefficient (cid:104) ∂O h , (cid:105) is indeed 1 as needed.As mentioned above, u is a leaf of a stable finite energy foliation so transversalityholds and moreover all neighboring finite energy surfaces obtained by the implicitfunction theorem are also leaves of the foliation. For more on this, the interestedreader should consult [Wen05], section 4.5. Claim 4.9. O h is not the positive end of any pseudoholomorphic cylinder.Proof. The differential decreases action and O h is designed to be the orbit of leastaction among all Reeb orbits. (cid:3) The fact that this is the unique holomorphic plane bounded by O h comes froman argument regarding positivity of intersections of pseudoholomorphic curves in 4dimensions as presented by Bourgeois and Van Koert in [BvK10] adapted to our case. Claim 4.10. u is the unique holomorphic plane bounded by O h , so ∂O h = 1 .Proof. A general plane in the symplectization looks like(4.1) u ( s, t ) = ( u a ( s, t ) , u θ ( s, t ) , u r ( s, t ) , u φ ( s, t )) We split the proof into two cases.Case 1: If u θ is constant, then u is equivalent to u .Take r a regular value of u r ( s, t ) and a circle γ in the preimage u − r ( r ) . Consider ∂ ˜ ψ , ˜ ψ ∈ [0 , π ) the tangent vector to γ on the Riemann surface Σ . We note that ∂ ˜ ψ u has components only in the φ direction. We prove that in the end of the prooffor case 1. Now, u passes through { r = r } with constant a and θ directions, hencetranslating u in the a direction, u turns out to intersect u in at least the circle at { r = r } . This yields a contradiction to positivity of intersections and thus any othersuch u is equivalent to u .Let us now see why ∂ ˜ ψ u has components only in the φ direction. We decompose thetangent space of the symplectization as (cid:104) v , v , ∂ a , R α (cid:105) where v , v the trivializationof the contact structure as defined in the beginning of the section. We write ∂ ˜ ψ u = A ( s, t ) v + B ( s, t ) v + Γ( s, t ) ∂ a + ∆( s, t ) R α = A ( s, t ) ∂ r + − B ( s, t ) h ( r ) + ∆( s, t ) h (cid:48) ( r ) D ( r ) ∂ θ + Γ( s, t ) ∂ a + B ( s, t ) h ( r ) − ∆( s, t ) h (cid:48) ( r ) D ( r ) ∂ φ Since we are restricted at a regular value r = r we get A = 0 . Moreover, u θ isconstant so − Bh + ∆ h (cid:48) = 0 . Now, applying J to ∂ ˜ ψ u we get J ∂ ˜ ψ u = A ( s, t ) β ( r ) v − B ( s, t ) β ( r ) v + Γ( s, t ) R α − ∆( s, t ) ∂ a = A ( s, t ) β ( r )( − h ( r ) D ( r ) ∂ θ + h ( r ) ∂ φ ) + − B ( s, t ) β ( r ) ∂ r + Γ( s, t ) h (cid:48) ( r ) ∂ θ − h (cid:48) ( r ) ∂ φ D ( r ) − ∆( s, t ) ∂ a Since u is J -holomorphic, J has to preserve the tangent space and the θ -componentis constant. Thus the coefficient of ∂ θ is equal to 0, i.e. − Aβ ( r ) h ( r ) + Γ h (cid:48) ( r ) D ( r ) = 0 HE CONTACT BANACH-MAZUR DISTANCE 35
We already have A = 0 , thus Γ = 0 . So the plane u passes through { r = r } withconstant a and θ coordinates.Case 2: If u θ is not constant, then positivity of intersections for pseudoholomorphiccurves is contradicted.If u is a solution to the Cauchy-Riemann equations of the general form 4.1, then u c ( s, t ) = ( u a ( s, t ) , u θ ( s, t ) + c, u r ( s, t ) , u φ ( s, t )) is also one. This is asymptotic to some other orbit γ c . Concretely γ ( t ) = ( θ , r , − h (cid:48) ( r ) D ( r ) t ) and γ c ( t ) = ( θ + c, r , − h (cid:48) ( r ) D ( r ) t ) . They are obviously not linked to each other so lk ( γ , γ c ) = 0 . Calculating the linking number in another way we get a contradiction.Since u θ is not constant, there is a constant c s.t. u c ∩ u (cid:54) = ∅ . By positivity ofintersections, lk ( γ , γ c ) > , which yields the desired contradiction. (cid:3) What we showed is that the contact element becomes exact at most at filtration level A ( O h ) , thus its class vanishes for action less or equal than A ( O h ) . This means thatthe l -invariant is less or equal than action of O h which is πh ( r + ) · (1 + δµ ( θ − )) .The fact that this is actually the l -invariant comes from the fact that any other orbitbounding a holomorphic plane must have action more than that of O h , i.e. claim 4.8.Summarizing what has been achieved, we have shown that using Wendl’s construc-tion, we are in the position to know precisely what the l -invariant is, i.e. what isthe action level for which the identity of contact homology algebra becomes exact orequivalently what is the right endpoint of the bar corresponding to in the barcodeof the persistence module CH ≤ t ( Y, λ ) . More concisely, Proposition 4.11.
The l -invariant of the contact form α δ,δ (cid:48) which is an appropriateperturbation of h ( r ) dθ + h ( r ) dφ in the Lutz tube is l ( α ) = A ( O h ) = 2 πh ( r + ) · (1 + δµ ( θ − )) Construction of the 2-parameter family.
The parameter domain of this family is H (cid:15) = { (ln( √ x ) , ln( y )) ∈ R | ln( y ) < (cid:15), x, y > } It is depicted in figure 4. ln( √ x )ln( y ) εH ε Figure 4.
The source H (cid:15) The (cid:15) involved in the definition of H (cid:15) depends on the contact form α . It is less thanthe minimum of two quantities. The first one is the lowest action of a Reeb orbit of ( Y, ξ ) before we perform the Lutz twist. The second one is the action of a specialReeb orbit which helps us control volume. These are more thoroughly explained be-low.For ε > and for a knot K, we denote by nbd ( K, ε ) a tubular neighbourhoodof the knot K of radius ε . Fix two transverse knots K , K to ξ in Y . Fix ε > sufficiently small such that nbd ( K , ε ) ∩ nbd ( K , ε ) = ∅ and a contact form λ on Ywith ker( λ ) = ξ which in tubular neighborhoods of K , K looks like the standardform dθ + r dφ . We perform a Lutz twist along K and obtain an overtwisted contactform on Y. We work with the full twist here as we need to preserve the homotopytype of the plane field ξ . In order to be able to control the l -invariant, we impose therestriction that πh ( r + )(1 + δµ ( θ − )) < (cid:15) . This yields a differential form of the form λ ot = (cid:40) λ on Y \ ( S × D ) h ( r ) dθ + h ( r ) dφ on S × D where in the above formula K is identified with S . In order to avoid confusion,we emphasize that (cid:15) is related to the l -invariant and ε to the radius of the tubularneighbourhood of K .The coordinates on the tubular neighborhood nbd ( K , ε ) are ( θ, r, φ ) ∈ S × (0 , ε ) × [0 , π ) . We normalize by requiring V ol ( Y, λ ot ) = (cid:82) Y λ ot ∧ dλ ot = 1 . Let L be the l -invariant, namely the action of the horizontal orbits { r = r + } in S × D or equiv-alently the lowest filtration level for which a primitive for the unit of the contacthomology algebra appears. This normalization has the effect that (0 , ln( L )) ∈ H (cid:15) maps under the bi-Lipschitz embedding to λ ot .We have that the l -invariant is the least action of an orbit bounding a uniquepseudoholomorphic plane. After the Lutz twist we performed, and the subsequentperturbations, it is equal to πh ( r + )(1 + δµ ( θ − )) , where r + is the smallest real num-ber such that h (cid:48) ( r + ) = 0 .In order to affect the volume and make it equal to k, it is enough to multiply λ ot by √ k . Moreover, we can modify the l -invariant just by modifying the choice of h ( r ) . We have to be careful though as we can only obtain information about the l -invariant as long as πh ( r + )(1 + δµ ( θ − )) is less than the next filtration level forwhich a primitive for the empty word appears. This, as already explained, is relatedto the quantity (cid:15) in the definition of H (cid:15) before theorem 2.5. For this modification werequire that πh ( r + )(1 + δµ ( θ − )) = l . Pictorially, figure 5 suggests that the modi-fication has the result that the y-intercept of the path ( h ( r ) , h ,l ( r )) is l π (1+ δµ ( θ − )) .Note that the only restrictions we have for the functions h ( r ) , h ( r ) is the behaviorof this path close to the endpoints of [0, ε ] and that the vector ( h ( r ) , h ,l ( r )) has towind around the origin of R once without ever being parallel to ( h (cid:48) ( r ) , h (cid:48) ,l ( r )) . Noother requirement in the interior of [0 , ε ] (in particular close to r + ∈ (0 , ε ) ) is assumed.Note that this last alteration of the contact form does not have a significant impacton volume, as it is enough for the change to take place only in a tube of small radius HE CONTACT BANACH-MAZUR DISTANCE 37 ε . Yet, in order for the modification of the l -invariant to have no impact on volume,we compensate by multiplying the form by a bump function ν l , supported outside ofthe solid tube in question. In particular, ν l is supported in the tubular neighborhoodaround K . This creates no new orbits of action less than that of the horizontal orbits { r = r + } as will be explained below.The next question that arises is about the (cid:15) in the definition of H (cid:15) . In short, (cid:15) isthe logarithm of the largest controlled l -invariant one can have. The word controlhere means both being able to leave the volume of the contact manifold unchangedas we modify the l -invariant (this is where the number B is needed in what follows)and determine what is the lowest action of a primitive for the unit of the contacthomology algebra (and this is where the number A comes from). The value of (cid:15) isdetermined as follows.First, let as before A be the lowest action of a Reeb orbit of ( Y, ξ ) before we per-form the Lutz twist. Also, let B be the action of the D -parametrized family of Reeborbits in the tubular neighborhood of K . Since the contact form locally is dθ + r dφ ,these orbits are specified by the local Reeb vector field ∂ θ . Thus, we have to set (cid:15) := min { ln( A ) , ln( B ) } .After briefly describing the construction, we are now able to provide it concretelyand thus define the embedding F . This embedding sends (ln( √ k ) , ln( l )) ∈ H (cid:15) to theform(4.2) α k,l = √ k · λ ot = (cid:40) √ k · ( ν l + 1) · λ on Y \ ( S × D ) √ k · ( h ( r ) dθ + h ,l ( r ) dφ ) on S × D where h ( r ) and h ,l ( r ) are given in figure 5 and ν l will be explained shortly. Later,it will be helpful for our calculations to give an explicit parametrization for this path ( h ( r ) , h ,l ( r )) and this is what we will do. It will be mostly part of two ellipsoidalarcs. The reason for considering (ln( √ k ) , ln( l )) instead of ( k, l ) is that multiplicationof forms by constants, i.e. flowing uniformly using the Liouville vector field in thesymplectization is translated to linear movement in the source space H (cid:15) . h ( r ) h ( r ) h ,u ( r + ) Figure 5.
The path defining the particular Lutz twist rθ δε δ − ε A ( K ) S × D δ supp ( ν l ) Figure 6.
The support of the compensating function ν l We now explain the compensating function ν l . Figure 6 helps on this task. Inorder to describe ν l we work in the tubular neighborhood of K . This is equippedwith the contact form λ = dθ + r dφ . The coordinates are given by ( θ, r, φ ) ∈ [0 , π ] × [0 , δ ] × [0 , π ] . Pick (cid:15) << min { δ, π } . Then ν l is a bump function supported within [0 , (cid:15) ] × [0 , δ − (cid:15) ] × [0 , π ] . The form locally now becomes λ l = ( ν l + 1) · ( dθ + r dφ ) .The function ν l is additionally required to have the property that if modifying h ,l ( r ) leads to a change in volume of Y by adding V ∈ R > , ν l controls the volume of thetubular neighborhood of L and yields V ol ( K × D , λ l ) = V ol ( K × D , λ ) − V .Note that this process creates no new orbits of action less than B , so in particularno orbits of action less than A after this modification could serve as a primitive of theunit of contact homology. Recall A is the least action for a primitive of the emptyword and since the volume compensating process does not reduce the action of anyorbit, it cannot yield primitives for the empty word of action less than A .One can easily check that α k,l is a contact form and less importantly see that mul-tiplying our form by √ k corresponds to flowing using the Liouville vector field in thesymplectization for ln( √ k ) “seconds”. This has exactly the effect that the contactform gets multiplied by √ k .As expected, it turns out that (cid:15) can be directly related to the systolic ratio of theinitial contact form λ . We have ρ sys ( α k,l ) = T min ( α k,l ) V ol ( Y,α k,l ) = l k , as far as O h discussed inthe previous section is the orbit with the least action. If before the twist, we can find atransverse K with its surrounding orbits having action ≥ A ( O h ) , we have ρ sys ( α k,l ) = l k . In particular, in our examples we have (cid:112) k · ρ sys ( α k,l ) = l = ln( (cid:15) ) , which in otherwords says (cid:15) = e √ k · ρ sys ( α k,l ) . Moreover, since we have ρ sys ( α k,l ) ≤ ρ sys ( λ ) essentiallyby construction, any (cid:15) ≤ e √ k · ρ sys ( λ ) works for the construction. We can possiblychoose an even larger (cid:15) if the orbit bounding a unique pseudoholomorphic plane for λ is of large enough action.4.4. Proof of the bi-Lipschitz embedding theorem.
Our goal for this section is to prove the following inequalities d ∞ ( (cid:126)x, (cid:126)y ) ≤ d CBM ( F ( (cid:126)x ) , F ( (cid:126)y )) ≤ d ∞ ( (cid:126)x, (cid:126)y ) HE CONTACT BANACH-MAZUR DISTANCE 39 where F : ( H , d ∞ ) → ( C Y,ξot , d
CBM ) the bi-Lipschitz embedding in question.To be more precise, the parameter domain of the 2-parameter family of forms α k,l is H (cid:15) = { (ln( √ x ) , ln( y )) ∈ R | ln( y ) < (cid:15), x, y > } , so the embedding will actually be F : ( H (cid:15) , d ∞ ) → ( C Y,ξot , d
CBM ) . This is not an issue though since ( H , d ∞ ) and ( H (cid:15) , d ∞ ) are isometric. The left inequality will be proved in subsection 4.4.1 and the right onewhich is the more involved one in subsection 4.4.2.This map will be defined shortly. Recall that (cid:15) is a number chosen to be less thanthe action of a primitive for the empty word of orbits before we perform any Lutztwist. We only have to worry about picking a sufficiently small (cid:15) in the case when Y is already algebraically overtwisted. What we will actually prove will be slightlystronger, yet less symmetric.The following lemmas will be helpful below. They describe the behavior of volumeand l -invariant under dilation of the contact form. Lemma 4.12. If α ≺ β , then V ol (( Y, α )) ≤ V ol (( Y, β )) . Also, V ol (( Y, C · α )) = C · V ol (( Y, α )) .Proof. First, we prove that
V ol ( φ ( Y )) = V ol ( Y, α )) , or in other words that we canonly map ( Y, α ) into SY in a volume preserving manner. Recall that α ≺ β meansthat there exist a cs-embedding φ : ( Y, α ) → W ( β ) ⊂ ( SY, d ( rβ )) with φ ∗ ( rβ + η ) = α ,for some exact, compactly supported in a neighbourhood of φ ( Y ) , one-form η = df .Recall also that the way we measure the volume of a hypersurface in SY is first byconsidering the form α = rβ | φ ( Y ) and then V ol ( φ ( Y )) = (cid:90) φ ( Y ) α ∧ dα .We thus have, V ol ( φ ( Y )) = (cid:90) φ ( Y ) α ∧ dα = (cid:90) Y φ ∗ α ∧ d ( φ ∗ ( α )) = (cid:90) Y ( α − φ ∗ ( η )) ∧ d ( α − φ ∗ ( η ))= (cid:90) Y α ∧ dα − (cid:90) Y α ∧ d ( φ ∗ η ) − (cid:90) Y φ ∗ η ∧ dα + (cid:90) Y φ ∗ η ∧ d ( φ ∗ η )= (cid:90) Y α ∧ dα + (cid:90) Y φ ∗ η ∧ dα = (cid:90) Y α ∧ dα + (cid:90) ∂Y = ∅ φ ∗ f ∧ α = (cid:90) Y α ∧ dα = V ol ( Y, α ) where the th equality above follows from the fact that η is exact and the th one byStokes’ theorem.The embedding φ ( Y ) yields that there exists a Liouville cobordism between φ ( Y ) and { } × Y in ( SY, d ( rβ )) . Stokes’ theorem implies the first claim. The second claimis obvious from the definition of the volume of Y . (cid:3) Remark 4.13.
The proof of this shows in particular that the allowed cs-embeddingspreserve the volume.
Lemma 4.14. If α ≺ β , then l ( α ) ≤ l ( β ) . Also, l ( C · α ) = C · l ( α ) Proof.
The assumption of the lemma means that there is an embedding φ : ( Y, α ) → W ( β ) which implies the existence of a trivial Liouville cobordism between φ ( Y ) and Y β . So, we get a map CH ∗ ( β ) ≤ t → CH ∗ ( α ) ≤ t which maps 0 to 0. In particular,the filtration level for which the contact invariant for β vanishes, i.e. l ( β ) , has to be larger or equal to the filtration level for which the contact invariant for α vanishes,i.e. l ( α ) . This proves l ( α ) ≤ l ( β ) .The second claim follows by the definition of the action of an orbit and the factthat multiplying the contact form by some number C just rescales the dynamics. (cid:3) Left Inequality.
Let (cid:126)x = (ln( √ k ) , ln( l )) and (cid:126)y = (ln( √ k ) , ln( l )) so that F ( (cid:126)x ) = α k ,l = α and F ( (cid:126)y ) = α k ,l = β . By the definition of d CBM and the previous lemma, as far asvolume is concerned, we have that
V ol ( F ( (cid:126)y )) C ≤ V ol ( F ( (cid:126)x )) ≤ C · V ol ( F ( (cid:126)y )) for any C such that α k ,l ≺ C · α k ,l and α k ,l ≺ C · α k ,l .The above inequalities are equivalent to(4.3) ln 1 C ≤
12 ln
V ol ( F ( (cid:126)x )) V ol ( F ( (cid:126)y )) = 12 ln (cid:16) k k (cid:17) = ln( (cid:112) k ) − ln( (cid:112) k ) ≤ ln C Because of the symmetry in the definition of d CBM , we also get
V ol ( F ( (cid:126)x )) C ≤ V ol ( F ( (cid:126)y )) ≤ C · V ol ( F ( (cid:126)x )) for any C such that α k ,l ≺ C · α k ,l and α k ,l ≺ C · α k ,l .These inequalities are again equivalent to(4.4) ln 1 C ≤
12 ln
V ol ( F ( (cid:126)y )) V ol ( F ( (cid:126)x )) = 12 ln (cid:16) k k (cid:17) = ln( (cid:112) k ) − ln( (cid:112) k ) ≤ ln C Combining now 4.3 and 4.4 and taking infimum over such C we obtain(4.5) | ln( (cid:112) k ) − ln( (cid:112) k ) | ≤ d CBM ( α, β ) Next, for any C such that α k,l ≺ C · α k,l and α k,l ≺ C · α k,l or equivalently α k,l ≺ C · α k,l and C α k,l ≺ α k,l , we get C · α k,l ≺ α k,l ≺ C · α k,l which implies by the previous lemma that C l ≤ l ≤ Cl Dividing by l and taking logarithms yields ln (cid:16) C (cid:17) ≤ ln (cid:16) l l (cid:17) ≤ ln( C ) Again by the symmetry in the definition we obtain a similar inequality, ln (cid:16) C (cid:17) ≤ ln (cid:16) l l (cid:17) ≤ ln( C ) Then, using both inequalities and taking infimum over all such C we obtain HE CONTACT BANACH-MAZUR DISTANCE 41 (4.6) | ln( l ) − ln( l ) | ≤ d CBM ( α, β ) Hence (4.6) and (4.5) yield(4.7) d ∞ ( (cid:126)x, (cid:126)y ) ≤ d CBM ( α, β ) Right Inequality.
Obtaining the right hand inequality of the bi-Lipschitz condition above follows bythe fact that changing the parameters in the 2-parameter family of contact forms iscompatible with the following triangle inequality. In order to keep things clear, weabuse notation and instead of d CBM ( α k ,l , α k ,l ) we write d CBM (( √ k , l ) , ( √ k , l )) . d CBM ( F ( (cid:126)x ) , F ( (cid:126)y )) = d CBM (( (cid:112) k , l ) , ( (cid:112) k , l )) ≤ d CBM (cid:16)(cid:16)(cid:112) k , l (cid:17) , (cid:16)(cid:112) k , (cid:114) k k l (cid:17)(cid:17) + d CBM (cid:16)(cid:16)(cid:112) k , (cid:114) k k l (cid:17) , (cid:16)(cid:112) k , l (cid:17)(cid:17) (4.8)We are going to study the behavior after modifying each parameter separately,though in the case of volume this has a small effect on the l -invariant and thus wehave to travel through the point (cid:16) ln( √ k ) , ln (cid:16)(cid:113) k k l (cid:17)(cid:17) in H (cid:15) . This is explainedbelow.First, we modify volume. As mentioned, in contrast with the modification of the l -invariant, our modification procedure does not allow us to modify solely the volume, orin other words we cannot only move horizontally in H (cid:15) . For this reason, we will workwith a view towards the triangle inequality 4.8. Let α = F ( (cid:126)x ) = F (ln( √ k ) , ln( l )) and γ = F ( (cid:126)y ) = F (ln( √ k ) , ln (cid:0)(cid:113) k k l (cid:1) ) with k ≤ k . We have d CBM ( α, γ ) = { ln C | C · γ ≺ α ≺ C · γ } ≤ ln (cid:114) k k = ln (cid:112) k − ln (cid:112) k because in this case we have that α = (cid:113) k k · γ , so one such C that obviously works is max (cid:110)(cid:113) k k , (cid:113) k k (cid:111) . If we don’t assume k ≤ k we just have to use an absolute valuein the above inequality. Hence, we obtain(4.9) d CBM ( α, γ ) ≤ (cid:12)(cid:12) ln (cid:112) k − ln (cid:112) k (cid:12)(cid:12) Note again that changing volume modifies the l -invariant precisely turning l into (cid:113) k k l . This explains the form of the middle term appearing in 4.8.Let’s now discuss the way to modify h ( r ) , i.e. effectively the l -invariant. Re-call that h ( r ) is a function from [0 , ε ) to R satisfying the properties for a Lutztwist from section 4.1. Recall also that in order to alter it, we additionally re-quire πh ,l ( r + )(1 + δµ ( θ + )) = l and that ∀ l , ( h ( r ) , h ,l ( r )) is never parallel to ( h (cid:48) ( r ) , h (cid:48) ,l ( r )) in order for the Lutz twist to yield a contact form.We consider a smooth family h ,t : [0 , ε ) × [ l , l ] → R interpolating between h ,l and h ,l . This family is assumed ∀ t ∈ [ l , l ] to satisfy the 3 properties required for a Lutz twist as in section 4.1. Namely, ∀ t ∈ [ l , l ] a Lutz twist can be performedusing h ,t instead of h . It will also be helpful to assume either ∂h ,t ∂t ≥ or ∂h ,t ∂t ≤ depending on whether we increase or decrease the value of the l -invariant. This familyof functions yields a family of contact forms on the manifold Y given by α t = λ tot = (cid:40) ( ν t + 1) · λ on Y \ ( S × D )( h ( r ) dθ + h ,t ( r ) dφ ) on S × D Where ν t as described previously, is designed to compensate for the small changein volume when modifying h ,l or stated differently, to allow us to move verticallyin H (cid:15) . We have already shown that the effect of this in dynamics is controlled wellenough for our computations. Namely, no new orbits of less action are created whenmultiplying by the compensating function. We denote by ξ t the corresponding con-tact structures and we let γ = α , β = α .Since we have a smooth family of contact structures, Gray’s stability theorem pro-vides a function f t such that ψ ∗ t γ t = f t α = f t γ , where ψ t a smooth isotopy of Y . Wewill specify f t using Moser’s trick as in the proof of Gray’s theorem.First, we have(4.10) d CBM ( γ, β ) ≤ || ln f − ln f || ∞ = || ln f || ∞ This can be seen as follows. We have that f ( y ) = 1 , ∀ y ∈ Y . Moreover, β = f γ . This implies that s β ( Y ) ⊆ W ( β ) ⊆ W ( || f || ∞ · γ ) . It is also clear that s γ ( Y ) ⊆ W ( β ) ⊆ W ( || f || ∞ · β ) as β (cid:54) = γ .Now, working as in the proof of Gray’s stability theorem, we have µ t = ˙ α t ( R t ) = ddt (ln f t ) ◦ ψ − t which in this case translates to(4.11) µ t = ddt ( h ,t ( r )) · (cid:16) − h (cid:48) ( r ) D t ( r ) (cid:17) where D t ( r ) = h ( r ) h (cid:48) ,t ( r ) − h (cid:48) ( r ) h ,t ( r ) .If we consider the 1-parameter family of contact forms { α ,l } l = tl = s , we have || ln f t − ln f s || ∞ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) ts µ u ◦ ψ u du (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ Replacing µ by its formula as in (4.11) and using the triangle inequality for integralswe get(4.12) || ln f t − ln f s || ∞ ≤ (cid:90) ts (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ddu ( h ,u ( r )) · (cid:16) − h (cid:48) ( r ) D u ( r ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ du So, it all boils down then to finding an upper bound for the right hand side ofequation (4.12). Although the analysis can be done in general, we are going to workwith the concrete case where ( h ( r ) , h ( r )) is mostly part of arcs of two ellipses. Thisis both concrete and sufficient. HE CONTACT BANACH-MAZUR DISTANCE 43
To this end, we parametrize ( h ( r ) , h ,l ( r )) as follows. Recall that we denote by ε the radius of the Lutz tube. Pick some small ε such that ε >> ε > . This ε willbe the first time when the path ( h ( r ) , h ,l ( r )) changes behavior. Moreover, pick somesmall δ > and δ > such that (1 + δ ) cos(2 πε ) = 1 and ε = (1 + δ ) u sin(2 πε )2 π (1+ δµ ( θ − )) .These δ , δ ensure continuity of the path ( h ( r ) , h ,l ( r )) since for ≤ r ≤ ε the pathis ( h ( r ) , h ,l ( r )) = (1 , r ) and for r slightly larger that ε the path is ( h ( r ) , h ,l ( r )) = (cid:16) (1 + δ ) cos(2 πr ) , (1 + δ ) u sin(2 πr )2 π (1+ δµ ( θ − )) (cid:17) . So, to summarize, we let h ( r ) := (cid:40) , ≤ r ≤ ε (1 + δ ) cos(2 πr ) , ε < r ≤ ε and moreover we let h ,u ( r ) := (cid:40) r , ≤ r ≤ ε (1 + δ ) u sin(2 πr )2 π (1+ δµ ( θ − )) , ε < r ≤ ε The paths can be extended to ( ε , ε ) in a similar way. This way needs to respectthat for any time parameter u , the absolute value of the second y -intercept is largerthan the one of the first y -intercept h ,u ( r + ) , it is independent of u and close to ε thepaths become ( h ( r ) , h ,u ( r )) = (1 , r ) . The first requirement is in order to controlthe l -invariant, the second to control an upper bound in the proof and the last re-quirement in order for the path to be compatible with the definition of a path usedto perform a Lutz twist.As it is obvious from their definition, h ( r ) , h ,u ( r ) are not even differentiable at r = ε . We will use a standard mollifier called the truncated Gaussian distributionin order to make both functions smooth. For some δ << ε , we consider the interval ( ε − δ , ε + δ ) over which the smoothing will take place. The basic idea in whatfollows is to convolute our functions h ( r ) and h ,u ( r ) with a truncated Gaussiansupported in ( ε − δ , ε + δ ) in order to define the respective smoothings H ( r ) and H ,u ( r ) . As it is evident here, capitalizing the notation means smoothing.In what follows, we will be using the truncated Gaussian g ( r ; µ, σ, a, b ) = 2 σ √ π exp( − ( r − µσ ) )( erf ( b − µ √ σ ) − erf ( a − µ √ σ )) where the error function is given by erf ( x ) = 2 √ π (cid:90) x e − t dt The first truncated Gaussian we will use in our case is the following g ( x ) := exp (cid:0) − x − ε δ (cid:1) δ √ (cid:16) (cid:90) √ − √ e − t dt (cid:17) , ε − δ ≤ x ≤ ε + δ , otherwise This is basically the truncated Gaussian with carefully chosen parameters g ( x ; µ = ε , σ = δ , a = ε − δ , b = ε + δ ) Note that the non trivial part of this Gaussian can be written as a constant O ( δ ) oforder δ times the function exp (cid:0) − x − ε ) δ (cid:1) . Moreover, its derivative is of order O ( δ ) .We will use the notation Gf which for any function f stands for the convolution of f with the specific truncated Gaussian chosen above. Gf is smooth and is supportedin ( ε − δ , ε + δ ) .We now define the smoothing of any function f ( r ) supported in [ ε − δ , ε + δ ] .Let B = ( ε − δ , ε − δ ) and B = ( ε + δ , ε + δ ) . Moreover, let g B = g ( r ; ε − δ , δ , ε − δ , ε − δ ) and g B = g ( r ; ε + δ , δ , ε + δ , ε + δ ) . Using g B , g B we define the smooth cutoff functions χ B i ( r ) = (cid:90) R χ B i ( r − y ) g B i ( y ) dy, i = 1 , where χ B i the characteristic function of the interval B i , i = 1 , .Now the smoothing of any function f supported in ( ε − δ , ε + δ ) is defined as F ( r ) := ( χ B ( r ) + χ B ( r )) f ( r ) + Gf ( r )= ( χ B ( r ) + χ B ( r )) f ( r ) + (cid:90) ε + δ ε − δ g ( r − t ) f ( t ) dt It is easy to check that if the inputs of the smoothing process are h ( r ) , h ,u ( r ) andthe outputs are H ( r ) , H ,u ( r ) then the outputs are smooth functions that agree with h ( r ) and h ,u ( r ) at the endpoints of the interval [ ε − δ , ε + δ ] . So, in [ ε − δ , ε + δ ] we replace the continuous path ( h ( r ) , h ,u ( r )) by the smooth path ( H ( r ) , H ,u ( r )) .The following picture is illuminating. ε δ − ε δ + ε g ( r ; ε , δ , ε − δ , ε + δ ) χ B χ B Figure 7.
The functions used for smoothingOur task is to bound (4.12). First, the maximum of the integrand in (4.12) cannotoccur if r ∈ [0 , ε ] ∪ [ ε , ε ) since there we have either ddu ( h ,u ( r )) = 0 or h (cid:48) ( r ) = 0 .Next, if ε + δ ≤ r ≤ ε then we get h (cid:48) ( r ) = (1 + δ )2 π sin(2 πr ) and D u ( r ) = u (1 + δ )(1 + δ )1 + δµ ( θ − ) HE CONTACT BANACH-MAZUR DISTANCE 45 so this yields − h (cid:48) ( r ) D u ( r ) = (1 + δµ ( θ − ))2 π sin(2 πr )(1 + δ ) u and thus (4.12) becomes (cid:90) ts (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ddu ( h ,u ( r )) · (cid:16) (1 + δµ ( θ − ))2 π sin(2 πr )(1 + δ ) u (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ du ≤ | (1 + δµ ( θ − ))2 π (1 + δ ) | (cid:90) ts (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ddu ( h ,u ( r )) · (cid:16) u (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ du = | (1 + δµ ( θ − ))2 π (1 + δ ) | (cid:90) ts (cid:12)(cid:12)(cid:12) ddu ( h ,u ( r + )) · (cid:16) δµ ( θ − ))2 π (1 + δ ) h ,u ( r + ) (cid:17)(cid:12)(cid:12)(cid:12) du = (cid:90) ts (cid:12)(cid:12)(cid:12) ddu ( h ,u ( r + )) · (cid:16) h ,u ( r + ) (cid:17)(cid:12)(cid:12)(cid:12) du = | ln( h ,t ( r + )) − ln( h ,s ( r + )) | = | ln( l t ) − ln( l s ) | The first equality above comes from the fact that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ddu ( h ,u ( r )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ occurs at r = r + .Recall also that the l -invariant is designed to be h ,u ( r + ) = (1+ δ ) u sin(2 πr )2 π (1+ δµ ( θ − )) . So altering u has the effect of altering the l -invariant. For brevity, we denote the l -invariant attime t by l t .The only thing to show now is that the smoothing does not have the effect that (cid:12)(cid:12) − H (cid:48) ( r ) D u ( r ) (cid:12)(cid:12) > u in ( ε − δ , ε + δ ) , where D u ( r ) = H ( r ) H (cid:48) ,u ( r ) − H (cid:48) ( r ) H ,u ( r ) thesmooth version of D u ( r ) . What we will show is that δ can be chosen sufficientlysmall in order to ensure this as the quotient is of order δ .For r ∈ [ ε − δ , ε + δ ] we have H (cid:48) ( r ) = ( χ (cid:48) B ( r ) + χ (cid:48) B ( r )) h ( r ) + ( χ B ( r ) + χ B ( r )) h (cid:48) ( r ) + Gh (cid:48) ( r ) and H (cid:48) ,u ( r ) = ( χ (cid:48) B ( r ) + χ (cid:48) B ( r )) h ,u ( r ) + ( χ B ( r ) + χ B ( r )) h (cid:48) ,u ( r ) + Gh (cid:48) ,u ( r ) Now, by performing the long and necessary calculations we have that lim δ → (cid:12)(cid:12)(cid:12) − H (cid:48) ( r ) D u ( r ) (cid:12)(cid:12)(cid:12) = 0 so δ can be chosen sufficiently small in order to ensure that (cid:12)(cid:12)(cid:12) − H (cid:48) ( r ) D u ( r ) (cid:12)(cid:12)(cid:12) ≤ u , ∀ r ∈ [0 , ε ) .Summarizing our results after this analysis, we obtain(4.13) || ln f t − ln f s || ∞ ≤ | ln( l t ) − ln( l s ) | Evaluating at t = 2 , s = 1 , (i.e. considering the family { α ,l } l = l l = l ) and using 4.10we get(4.14) d CBM ( γ, β ) ≤ || ln f t − ln f s || ∞ ≤ | ln( l ) − ln( l ) | = | ln( l ( β )) − ln( l ( γ )) | as required.Using (4.7), (4.8), (4.9), (4.14) for the first contact form α = α k ,l , the middlecontact form γ = α k , (cid:113) k k l and the second contact form β = α k ,l we get d ∞ ( (cid:126)x, (cid:126)y ) = d ∞ (( (cid:112) k , l ) , ( (cid:112) k , l )) ≤ d CBM ( α, β ) ≤ d CBM ( α, γ ) + d CBM ( γ, β ) ≤ | ln( (cid:112) k ) − ln( (cid:112) k ) | + | ln( (cid:114) k k l ) − ln( l ) |≤ | ln( (cid:112) k ) − ln( (cid:112) k ) | + | ln( l ) − ln( l ) | ≤ d ∞ ( (cid:126)x, (cid:126)y ) (4.15)or in the weaker but more symmetric form(4.16) d ∞ ( (cid:126)x, (cid:126)y ) ≤ d CBM ( α, β ) ≤ d ∞ ( (cid:126)x, (cid:126)y ) (cid:3) Extension of the result to higher dimensions
It was not known for quite a while what is the natural generalization of the notionof overtwistedness in higher dimensions. For instance, the question of how a higherdimensional analogue of an overtwisted disk should look like was very recently an-swered in [BEM15], along the process of classifying and establishing an h-principle forovertwisted structures in higher dimensions. The most compatible, with the theoryknown so far (in the sense that one can show existence of D not , bLobs and plastikstufesafter performing the twist) version of generalized Lutz twists appeared recently in[Ada16]. In this work, Adachi describes the construction of the higher dimensionalanalogue of the Lutz tube and instead of considering a contact form which general-izes the 3-dimensional h ( r ) dθ + h ( r ) dφ , a confoliation 1-form on S × R n is picked,called ω tw , which in 3 dimensions is forced to be a contact form. This confoliation isshown to be conductive, thus thanks to [AW00], there is a contact form which is C ∞ -close to this confoliation form Although Adachi’s construction seems to link nicelywith the development of the theory so far (i.e. for instance he proves existence ofovertwisted disks as defined in [BEM15]), it is difficult to work with his local modelsand eventually calculate the dynamics of the Reeb vector field.Etnyre and Pancholi, in [EP11] and [EP16], provided a generalization of the Lutztwist having in mind the notion of a plastikstufe. Although their construction isvery explicit in terms of dynamics, it requires a very hard work understanding thepseudoholomorphic curves involved. The approach that best fits the scope of thiswork is to use the description of contact manifolds as open book decompositions andwhat we need in this work was developed in [BvK10]. We describe it and we relateit to our goal.5.1. Strategy for the extension of the result.
Bourgeois and Van Koert in [BvK10] proved that negatively stabilized open bookshave vanishing contact homology. Casals, Murphy and Presas recently showed in
HE CONTACT BANACH-MAZUR DISTANCE 47 [CMP19] that such contact manifolds are overtwisted. The notion of open bookswhich is easily generalized to higher dimensions will be the most fundamental tool inthis extension to the higher dimensional cases. This section is inspired and followsthe work from [BvK10].Stabilization means the following. Let P be the (2 n − -dimensional page of theopen book of the contact manifold ( M, ξ ) . Let L be a Lagrangian ( n − -disk with ∂L ⊂ ∂P . Suppose that the monodromy of the open book is the identity on aneighborhood of L . Attaching a Weinstein ( n − -handle to P along ∂L we get aLagrangian sphere in the new page (cid:101) P . Choosing as monodromy a right-handed Dehntwist along this Lagrangian sphere and composing with the original monodromy onthe rest of the page, we obtain a contact manifold. If we choose a boundary parallelLagrangian ball L we have that the resulting contact manifold is contactomorphic to ( M, ξ ) . It is conjectured that this holds more generally, i.e. removing the assumptionthat the Lagrangian ball is boundary parallel. In our case, we need an alternativedescription via contact connected sums (or Murasugi sums) and this is explained in[BvK10], section 7. We briefly describe the idea.Negative stabilization corresponds to contact connected summing our initial con-tact manifold ( M n − , η ) with a special contact manifold ( S n − , α L ) whose construc-tion will be reviewed. This latter manifold is viewed as an open book decompositionwith pages being T ∗ S n − and monodromy a left-handed Dehn-Seidel twist. The sit-uation highly resembles the 3-dimensional case as the contact form we consider is ageneralization of the 3-dimensional overtwisted contact form we used when perform-ing the Lutz twist. In fact, we still work locally and the 3-dimensional situation isrecovered quite naturally as it is explained in [BvK10], at the end of section 6. It willbe enough to work on ( S n − , α L ) . By this we mean that we will describe an analo-gous to the 3-dimensional case modification on ( S n − , α L ) and then we will connectsum with the contact manifold of interest ( M n − , η ) . It is important to mention thatthe modification and the connected sum processes commute.One might be worried that the process of connected summing has the effect that thehomotopy type of the original hyperplane field ker( η ) changes and thus the promisedresult cannot be extended to higher dimensions. This concern can be easily lifted bythe fact that the set of almost contact structures (and in particular of contact struc-tures) forms a group under connected sum. Thus, after performing the connected sumwith the special contact ( S n − , ξ − = ker( α L )) , we can invert by connected summingwith its inverse ( S n − , ξ + ) and cancel the possible alteration of the homotopy typeof ker( η ) . Note that this inversion will not affect our calculations as we will primarilybe interested in the behavior of the lowest action orbit. We can adjust for the lowestaction orbit of the inverting ( S n − , ξ + ) to have arbitrarily large action.This is also good point to explain how the contact connected sum procedure af-fects the dynamics. Orbits that do not pass through the connected sum region arenot affected. Furthermore, there are orbits that lie entirely in the connecting tubewhich are called tube orbits and there are also wandering orbits which start at themanifold M go through the tube to S n − and return to M . Their action can bemade arbitrarily large so they are of least importance for us. Moreover, Ustilovsky in | (cid:126)p |− π gp Figure 8.
The function g used in the definition of a ( − -fold Dehn twisthis thesis [Ust99] showed that we can choose a contact form on the tube so that alltube orbits lie within a contact sphere in the middle of the tube and all such orbitshave odd degree k ≥ n − . Thus in dimensions ≥ , we don’t have any orbits ofdegree 1 within the tube. Recall that the 3-dimensional case was handled previously.It turns out that our focus has to be purely on S n − .5.2. The Bourgeois-Van Koert open book construction.
We now describe a special case of the construction from [BvK10] where an openbook decomposition for S n − is provided. This case is the most relevant to our work.Since we will need Dehn twists on ( T ∗ S n − , dλ ) with λ = (cid:126)pd(cid:126)q in order to describe theconstruction, we briefly recall them.We start with the auxiliary map σ t ( (cid:126)q, (cid:126)p ) = (cid:18) cos( t ) − | (cid:126)p | sin( t ) | (cid:126)p | sin( t ) cos( t ) (cid:19) (cid:18) (cid:126)q(cid:126)p (cid:19) Although the construction that appears in [BvK10] is more general, we will onlyneed the special case of a ( − -fold Dehn twist. We will define a function g using anauxiliary function (cid:101) g . Let (cid:101) g be a function satisfying • (cid:101) g (0) = − π . • Fix p > . For | (cid:126)p | > , it increases to 0 at (cid:101) g ( p ) . • For | (cid:126)p | ≥ p , (cid:101) g ( | (cid:126)p | ) = 0 .For a small (cid:101) ε > , we define g = (cid:101) g + (cid:101) ε | (cid:126)p | . A helpful figure illustrating g is figure 8.We define the self diffeomorphism of T ∗ S n − given by τ ( (cid:126)q, (cid:126)p ) = σ g ( | (cid:126)p | ) ( (cid:126)q, (cid:126)p ) This diffeomorphism is symplectic as τ ∗ dλ = dλ since we have τ ∗ λ = λ + | (cid:126)p | d ( g | (cid:126)p | ) .We observe that λ − τ ∗ λ is exact and a primitive for this difference is h ( | (cid:126)p | ) = 1 + (cid:90) | (cid:126)p | sg (cid:48) ( s ) ds We have the mapping torus for T ∗ S n − defined as follows. First, consider the map HE CONTACT BANACH-MAZUR DISTANCE 49 φ : T ∗ S n − × R → T ∗ S n − × R ( (cid:126)q, (cid:126)p ; ϕ ) (cid:55)→ ( τ ( (cid:126)q, (cid:126)p ); ϕ + h ( | (cid:126)p | )) This map preserves the contact form α = dϕ + (cid:126)pd(cid:126)q on T ∗ S n − × R , so we obtain acontact structure on the mapping torus A := T ∗ S n − × R /ϕ All we have to do to complete the construction is to glue in the binding. In fact, asit is more relevant to our work, it is better to change point of view and focus on aneighborhood of the binding rather than the mapping torus just described. Anotherdescription using a different mapping torus clarifies the situation.Consider (cid:101) A = (( T ∗ S n − − × R ) / ( x, ϕ ) ∼ ( x, ϕ + 1) (cid:39) ( T ∗ S n − − × S We have the map ψ : (cid:101) A → A (( (cid:126)q, (cid:126)p ); ϕ ) (cid:55)→ ( σ ϕg ( (cid:126)q, (cid:126)p ); h ( | (cid:126)p | ) ϕ ) which is a diffeomorphism onto its image and allows us to think of the constructionexplicitly “standing on the binding”. In fact, we can choose a neighborhood of thebinding so large that it covers the whole (cid:101) A , so B will describe the entire contactmanifold except for a set of positive codimension which is precisely the set in themapping torus corresponding to the zero section. Computations effectively only takeplace on B .We have a contact form (cid:101) α on (cid:101) A by pulling back α using ψ . Indeed, (cid:101) α = ψ ∗ α = (cid:101) h ( | (cid:126)p | ) dϕ + (cid:126)pd(cid:126)q where (cid:101) h ( | (cid:126)p | ) = 1 − (cid:90) | (cid:126)p | g ( s ) ds .Denote a neighborhood of the binding by B := ST ∗ S n − × D . As mentionedbefore, this neighborhood covers all of S n − except for the set in the mapping toruscorresponding to the zero section of the pages. We think of T ∗ S n − ⊂ R n describedby the equations. (cid:126)p · (cid:126)p = 1 , (cid:126)q · (cid:126)q = 0 , (cid:126)p · (cid:126)q = 0 On this neighborhood of the binding we have the contact form α = h ( r ) λ + h ( r ) dϕ where λ is the restriction of the canonical form (cid:126)pd(cid:126)q to ST ∗ S n − and ( r, ϕ ) the polarcoordinates on D . The functions h , h are chosen so that α is a contact form andmatches (cid:101) α in a collar neighborhood of the boundary, i.e. close to the set correspond-ing to the zero section of the pages. For α to be a contact form we need to assumethat h (cid:54) = 0 and h h (cid:48) − h (cid:48) h r = D ( r ) r (cid:54) = 0 . Such choice is depicted in the figure 9.It is both interesting and useful to digress a bit in order to write down explicitlyhow the Reeb orbits in B look like. We have that the Reeb field on the binding is R λ = (cid:126)p∂ (cid:126)q − (cid:126)q∂ (cid:126)p rr h ( r ) h ( r ) Zero SectionBinding
Figure 9.
The functions h ( r ) and h ( r ) and in a neighborhood of the binding is R α = 1 D ( r ) (cid:16) h (cid:48) ( r ) R λ − h (cid:48) ( r ) ∂ ϕ (cid:17) So the orbits are x ( t ) = (cid:16) θ + h (cid:48) ( r ) D ( r ) t, r, ϕ − h (cid:48) ( r ) D ( r ) t (cid:17) where θ the coordinate corresponding to the geodesic flow. We observe that wehave a closed orbit whenever h (cid:48) ( r )2 πh (cid:48) ( r ) = pq ∈ Q ∪ {∞} and the actions are T = q D ( r ) h (cid:48) ( r ) = 2 πp D ( r ) h (cid:48) ( r ) where sign( p ) = sign( h (cid:48) ( r ) ), sign( q ) = sign( h (cid:48) ( r ) ) and whenever h (cid:48) ( r ) = p = 0 or h (cid:48) ( r ) = q = 0 pick the one that makes sense.Taking a step back from describing the dynamics of the form, thus far a coorientedcontact structure on S n − given as the kernel of the 1-form α is constructed. Thiscontact sphere is denoted by ( S n − , α ) . As shown by Bourgeois and Van Koert, afterperturbation using a Morse function, there exists a unique closed orbit γ which haslinking number with the binding equal to 1. The choice of functions h , h and g canbe adjusted so that γ is the lowest action orbit, thus γ can only bound holomorphicplanes. What follows are statements proved in [BvK10] • γ is the lowest action orbit with action A ( γ ) (cid:39) πh ( r ) . • γ bounds a unique holomorphic plane. • γ has degree 1. Remark 5.1.
The fact that the action is approximately and not precisely πh ( r ) has to do with the fact that orbits are degenerate so we have to use a perturbationas in [Bou02], section 2.2.From the items above the next proposition quickly follows. Proposition 5.2.
The l -invariant of α is A ( γ ) (cid:39) πh ( r ) . HE CONTACT BANACH-MAZUR DISTANCE 51
Modification of contact form.
Before we start with any modification it is important to recall the general direction.Our goal is to begin with a contact manifold ( M, ξ ) , connect sum with a special mod-ifying contact S n − which makes it overtwisted and then compensate for possiblealteration of the homotopy type of the plane field by connected summing with theinverting S n − . As explained earlier, since almost contact structures form a groupunder connected sum, we can compensate for the possible change of the homotopytype of ker( η ) . After this, we bi-Lipschitz embed the space of contact forms support-ing this contact structure on M S n − S n − inside R . Since the modification onlytakes place away of the zero section of the pages of the open book for S n − and theDarboux ball we use to connect sum is close to the region corresponding to the zerosection of the pages, it is enough to describe the modification purely on S n − . Again,we have some restrictions to this modification. If we start with an algebraically over-twisted contact manifold we can make the l -invariant as large as the primitive forthe unit of the contact homology algebra before connected summing. Moreover, themodified l -invariant can be as large as the lowest action orbit on M and the modify-ing S n − . We can arrange for the actions of the orbits of the compensating S n − tobe arbitrarily large so as they do not affect our calculations. We remark that theserestrictions do not affect our result since it is a large-scale geometry result.Similarly to the 3-dimensional case we modify α on S n − as follows α k,l = F ((ln( n √ k ) , ln( l ))) = n √ k ( h ( r ) λ + h ,l ( r ) dφ ) where h ,l ( r ) is a smooth function which is identical to h ( r ) outside of a smallneighborhood of r and h ,l ( r ) = l π (1+ δµ ( θ − )) , where µ ( θ − ) is the value of the Morsefunction used for the perturbation at the critical point θ − which corresponds to thelowest action orbit γ . Following the proof from [BvK10], we have for the lowestaction orbit γ that • A ( γ ) = (1 + δµ ( θ − ))2 π n √ k · h ,l ( r ) , for < δ << which comes from theperturbation. • The degree of γ is equal to 1. • γ bounds a unique holomorphic plane.In summary, we have that the l -invariant l ( α k,l ) = A ( γ ) = (1 + δ )2 π n √ k · h ,l ( r ) .Note that when modifying it we can normalize in order for k to be equal to 1.Essentially the proof from subsection 4.4 carries over to the higher dimensionalcase with little modification. We briefly describe the argument. The following twolemmata, as in 3 dimensions, are essential for the proof. First, a quick calculationwhich is almost identical to the proof of lemma 4.12 shows Lemma 5.3. If α ≺ β , then V ol (( S n − , α )) ≤ V ol (( S n − , β )) . Also, V ol ( S n − , C · α ) = C n · V ol ( S n − , α ) . Moreover, since the proof of lemma 4.14 does not depend on the dimension, wehave that it still holds. We recall it for clarity.
Lemma 5.4. If α ≺ β , then l ( α ) ≤ l ( β ) . Also, l ( C · α ) = C · l ( α ) Left inequality.
Letting α = α k ,l , β = α k ,l and ( (cid:126)x, (cid:126)y ) = ((ln( n √ k ) , ln( l )) , (ln( n √ k ) , ln( l ))) and following the same calculations as in subsection 4.4.1 and using the previous 2adapted lemmata, we obtain | ln( n (cid:112) k ) − ln( n (cid:112) k ) | ≤ d CBM ( α, β ) and | ln( l ) − ln( l ) | ≤ d CBM ( α, β ) Thus, d ∞ ( (cid:126)x, (cid:126)y ) ≤ d CBM ( α, β ) The only difference comes from the dimension dependence of the volume so we haveto work with the n th root instead of a square root.5.3.2. Right inequality.
The spirit of the proof of the right-hand inequality is identicalto the 3-dimensional case. We still have to use Gray’s stability theorem in order toobtain the proper bounds, yet since we are not performing a Lutz twist the analysisof the modification is slightly different. Again we let (cid:126)x = ( n √ k , l ) and (cid:126)y = ( n √ k , l ) .We will use an analogous triangle inequality d CBM ( F ( (cid:126)x ) , F ( (cid:126)y )) = d CBM (( n (cid:112) k , l ) , ( n (cid:112) k , l )) ≤ d CBM (cid:16)(cid:16) n (cid:112) k , l (cid:17) , (cid:16) n (cid:112) k , n (cid:114) k k l (cid:17)(cid:17) + d CBM (cid:16)(cid:16) n (cid:112) k , n (cid:114) k k l (cid:17) , (cid:16) n (cid:112) k , l (cid:17)(cid:17) (5.1)Note that as in the 3-dimensional case we abuse the notation in order to avoid makingit too heavy. This inequality again comes from the fact that modification of volumehas a small effect on the l -invariant. As in the previous lower dimensional case, wecan use the trick of the compensating function in order to modify the l -invariant ofour contact form without modifying the volume. Similarly to the 3-dimensional case,letting α = F ((ln( n (cid:112) k ) , ln( l ))) β = F ((ln( n (cid:112) k ) , ln( l ))) γ = F (cid:16)(cid:16) ln( n (cid:112) k ) , n (cid:114) k k ln( l ) (cid:17)(cid:17) we obtain by purely using the volume modification as in the 3-dimensional case d CBM ( α, γ ) ≤ (cid:12)(cid:12)(cid:12) n (cid:112) k − n (cid:112) k (cid:12)(cid:12)(cid:12) (5.2)We now need to study how the l -invariant modification affects our embedding.Namely, obtain bounds when travelling from the form γ to the form β . This is againdone by using Gray’s stability theorem. Similarly to 3 dimensions, the quantity weneed to provide bounds for is(5.3) || ln f t − ln f s || ∞ ≤ (cid:90) ts (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ddu ( h ,u ( r )) · (cid:16) − h (cid:48) ( r ) D u ( r ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ du where f t the conformal factor coming from Gray’s theorem and D u ( r ) = h ( r ) h (cid:48) ,u ( r ) − h (cid:48) ( r ) h ,u ( r ) . We have that ddu ( h ,u ( r )) is zero outside of a neighborhood of r andattains its maximum at r . One can adjust the slope of h ( r ) near r in order to HE CONTACT BANACH-MAZUR DISTANCE 53 make || D u ( r ) || ∞ attain its minimum at r . This can be done by requiring h (cid:48) ( r ) (cid:39) and h (cid:48) ( r ) (cid:39) h (cid:48) ( r ) h ( r ) h ( r ) in a neighborhood of r . Then, we get that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ddu ( h ,u ( r )) · (cid:16) − h (cid:48) ( r ) D u ( r ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ = (cid:12)(cid:12)(cid:12) ddu ( h ,u ( r )) · (cid:16) − h ,u ( r ) (cid:17)(cid:12)(cid:12)(cid:12) Hence integrating (5.3) with respect to u we obtain || ln f t − ln f s || ∞ ≤ | ln( l t ) − ln( l s ) | Applying now this result to the forms γ and β we get more precisely that d CBM ( γ, β ) ≤ || ln f t − ln f s || ∞ ≤ | ln( l ( γ )) − ln( l ( β )) | Combining the triangle inequality with the two inequalities obtained above we get d CBM ( α, β ) ≤ d CBM ( α, γ ) + d CBM ( γ, β ) ≤ | n (cid:112) k − n (cid:112) k | + | ln( l ( β )) − ln( l ( γ )) |≤ | n (cid:112) k − n (cid:112) k | + (cid:12)(cid:12)(cid:12) ln (cid:16) n (cid:114) k k l (cid:17) − ln( l ) (cid:12)(cid:12)(cid:12) ≤ | n (cid:112) k − n (cid:112) k | + | ln( l ) − ln( l ) | ≤ d ∞ ( (cid:126)x, (cid:126)y ) As a concluding remark, let us summarize that what we achieved, combining theresults of the last and this subsection, is to extend the result to all odd dimensionsgreater than 3 and thus prove the theorem in its full generality.6.
Remark on the possibility of more degrees of freedom and use ofother homology theories
The first degree of freedom (volume) was quite natural to consider. Also, the classcorresponding to the empty word is a very special element of contact homology the-ories and in the overtwisted case, it is expected to vanish. It is therefore natural toask about the filtration level for which it vanishes. The empty word class (knownin most homology theories as the contact invariant, e.g. in ECH) appears at level0 and vanishes at some higher level. Working with the contact homology algebra ofovertwisted structures, all classes have to vanish at some filtration level, thus thereare no semi-infinite bars in the barcode of the persistence module CH ≤ t ∗ ( Y, λ ot ) . Onenatural generalization of the idea would be to check what are the vanishing levels orlengths for the other finite bars and thus bi-Lipschitz or quasi-isometrically embedpart of R n , where n − is the cardinality of the set of finite bars, into the space ofcontact forms. The algebra structure yields that most of the information is encodedto what we already have done. Namely, as explained in the introduction, the algebrastructure yields that the largest length of a finite bar in the barcode is the l -invariant.This shows that using contact homology at least, in the overtwisted case there isno other powerful enough (or large enough) bar in the barcode to help us distinguishbetween two contact forms and have more degrees of freedom. This is not the casewith ECH as one can see. ECH, in contrast with SFT theories, does not have a nat-ural algebra structure (i.e. a multiplication that behaves well under grading and ∂ ).This turns out to be a helpful thing, as in the absence of algebra structure, no finite bar in principle controls the length of any other finite bar. The question regardingthe l -invariant was about the freedom to alter the right endpoint of the correspondingvanishing bar (in this case the bar corresponding to the empty Reeb orbit) in thebarcode of contact homology. This directly translates to filtered ECH and questionsabout its finite bars.Another direction may be provided when looking at ECH capacities. ECH capac-ities capture the information emerging from semi-infinite bars and not finite bars,yet one might be able to show that some process analogous to the Lutz twist in oursetting helps us control them. Looking at the full ECH spectrum (see [Hut11]) willbe enough since parts of the symplectization are always trivial cobordisms. This is amatter of a future work direction. References [Ada16] Jiro Adachi. Generalizations of twists of contact structures to higher-dimensions via roundsurgery. arXiv e-prints , page arXiv:1610.09672, Oct 2016.[AW00] Steven J. Altschuler and Lani F. Wu. On deforming confoliations.
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