aa r X i v : . [ m a t h . S G ] J u l CONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY
KO HONDA AND YANG HUANGA
BSTRACT . We lay the foundations of convex hypersurface theory (CHT) incontact topology, extending the work of Giroux in dimension three. Specifi-cally, we prove that any closed hypersurface in a contact manifold can be C -approximated by a convex one. We also prove that a C -generic family of mu-tually disjoint closed hypersurfaces parametrized by t ∈ [0 , is convex exceptat finitely many times t , . . . , t N , and that crossing each t i corresponds to abypass attachment. As applications of CHT, we prove the existence of compati-ble (relative) open book decompositions for contact manifolds and an existence h -principle for codimension contact submanifolds. C ONTENTS
1. Introduction 21.1. Convex contact structures 21.2. Main results 41.3. Applications 52. A convexity criterion 72.1. Gradient-like vector fields 82.2. A convexity criterion 93. Construction of C -folds in dimension Z PL Z PL -dimensional plugs 164.2. Installing and uninstalling plugs 174.3. Proof of Theorem 1.2.3 in dimension 3 184.4. Proof of Theorem 1.2.4 in dimension 3 194.5. Further remarks 215. Construction of C -folds in dimension > C -folds 235.2. Construction of partial C -folds 296. Quantitative stabilization of open book decompositions 336.1. Strongly adapted contact forms 346.2. Quantitative stabilization of OBD 366.3. Shifting the binding 46 YH is partially supported by the grant KAW 2016.0198 from the Knut and Alice WallenbergFoundation. -dimensional case 557.2. A Peter-Paul contactomorphism 557.3. The higher-dimensional case 568. Construction of the plug 608.1. ǫ -convex hypersurfaces 618.2. Construction of the Y -shaped plug 619. Three definitions of the bypass attachment 639.1. Definitions and examples 639.2. Normalization of contact structure near a folded Weinsteinhypersurface 659.3. Bypass attachment as a bifurcation 679.4. Bypass attachment as a partial open book 7510. C -approximation by convex hypersurfaces 7711. The existence of (partial) open book decompositions 7812. Applications to contact submanifolds 8012.1. Some Morse-theoretic lemmas 8112.2. Existence h -principle for contact submanifolds 84Appendix A. Wrinkled and folded embeddings 86A.1. Wrinkled and cuspidal embeddings 86A.2. Cuspidal embeddings of a disk 88A.3. Folding hypersurfaces 90References 911. I NTRODUCTION
Convex contact structures.
Morse theory is a topologist’s favorite tool forexploring the structure of manifolds. The significance of Morse theory — herewe mean the traditional finite-dimensional version, not Floer theory — in contactand symplectic topology was advocated by Eliashberg and Gromov in [EG91].In particular, according to [EG91, Definition 3.5.A], a contact manifold ( M, ξ ) is convex if there exists a Morse function, called a contact Morse function , whichadmits a gradient-like vector field whose flow preserves ξ . Just as a manifold canbe reconstructed from its Morse function by a sequence of handle attachments intraditional Morse theory, a contact manifold can be reconstructed from a contactMorse function by a sequence of contact handle attachments. The analogous theoryin symplectic topology is known as the theory of Weinstein manifolds.Eliashberg and Gromov asked in [EG91] whether there exist non-convex con-tact manifolds. Around 2000 Giroux gave a negative answer to the question (cf.[Gir02]) by showing that every closed contact manifold is convex. This can also beformulated as his celebrated correspondence between contact structures and open ONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY 3 book decompositions. This is in sharp contrast to the theory of Weinstein mani-folds, where it is relatively easy to see that any compatible Morse function cannothave critical points of index greater than half of the dimension of the manifold.
Remark . It might be the case that “convexity” is one of the most abusedterminologies in mathematics. We will not use the term “convex contact manifold”in the sense of Eliashberg and Gromov for the rest of the paper.At this point, the question bifurcates into two:
Question 1.1.2.
How do we establish Morse theory on contact manifolds?
Question 1.1.3.
How do we use Morse theory to better understand contact mani-folds?
Let us first address Question 1.1.2, which was first answered by Giroux in bothdimension and in higher dimensions. Giroux used two completely different setsof techniques to treat the -dimensional and higher-dimensional cases.We first discuss the -dimensional case. In his thesis [Gir91], Giroux introducedwhat is now known as convex surface theory into -dimensional contact topology.It is an extremely powerful and efficient way of studying embedded surfaces incontact -manifolds, and can recover most of the pioneering results of Bennequin[Ben83] and Eliashberg [Eli92]. Using convex surface theory, Giroux showed thatfor closed contact -manifolds, there is a one-to-one correspondence between iso-topy classes of contact structures and compatible open book decompositions (cf.Section 6) up to positive stabilization.Before moving onto higher dimensions, let us recall the definition of a convexhypersurface following [Gir91]: Definition 1.1.4.
A hypersurface Σ ⊂ ( M, ξ ) is convex if there exists a contactvector field v , i.e., a vector field whose flow preserves ξ , which is transverse to Σ everywhere. Observe that regular level sets of a contact Morse function are convex hypersur-faces.The situation in dimensions > is quite different. Besides the fact that convexhypersurfaces can be defined in any dimension, until now there has been no system-atic convex hypersurface theory . Giroux’s proof [Gir02] that every closed contactmanifold is convex involves a completely different technology, i.e., Donaldson’s[Don96] technique of approximately holomorphic sections , transplanted into con-tact topology by Ibort, Mart´ınez-Torres, and Presas [IMTP00]. Donaldson usedthe approximate holomorphic technology to construct real codimension sym-plectic hypersurfaces of a closed symplectic manifold as the zero locus of an ap-proximately holomorphic section of a complex line bundle, while Ibort, Mart´ınez-Torres, and Presas constructed certain codimension contact submanifolds of aclosed contact manifold. What Giroux realized is that [Don96] and [IMTP00]could be used to produce compatible open book decompositions. Roughly speak-ing, given a closed contact manifold ( M, ξ = ker α ) , one considers the trivial linebundle C on M equipped with a suitable Hermitian connection determined by α . KO HONDA AND YANG HUANG
Then there exists a section s : M → C whose zero locus B := s − (0) is a closedcodimension contact submanifold called the binding , and s | s | : M \ B → S is a smooth fibration defining the compatible open book decomposition of ( M, ξ ) .As a consequence of using the approximate holomorphic technology, the higher-dimensional Giroux correspondence (cf. Corollary 1.3.1) is a much weaker state-ment compared to its -dimensional counterpart.1.2. Main results.
The main goal of this paper is to systematically generalizeGiroux’s convex surface theory to all dimensions. The main results of convex hy-persurface theory (CHT) are Theorem 1.2.3 and Theorem 1.2.4. In fact, even indimension , our method (cf. Section 4) somewhat differs from Giroux’s originalapproach, is simpler, and is consistent with our more general approach in higherdimensions.We first introduce some more terminology describing the anatomy of a convexhypersurface. Definition 1.2.1.
Let Σ ⊂ ( M, ξ = ker α ) be a convex hypersurface with respectto a transverse contact vector field v . Define the dividing set Γ(Σ) := { α ( v ) = 0 } and R ± (Σ) := {± α ( v ) > } as subsets of Σ . It turns out that
Γ(Σ) ⊂ ( M, ξ ) is a codimension contact submanifold, and R ± (Σ) are (complete) Liouville manifolds with Liouville form given by a suitablerescaling of α | R ± (Σ) , respectively. Moreover, the isotopy classes of Γ(Σ) , R ± (Σ) are independent of the choices of v and α .In dimensions ≥ , there exist Liouville manifolds that are not Weinstein by Mc-Duff [McD91], Geiges [Gei94, Gei95], Mitsumatsu [Mit95], and Massot, Nieder-kr¨uger, and Wendl [MNW13]. While these “exotic” Liouville manifolds are greatfor constructing (counter-)examples, there currently is no systematic understand-ing of such non-Weinstein Liouville manifolds, partially because of the lack of anappropriate Morse theory on such manifolds. This motivates the main assumption(and maybe also conclusion) of this paper. Assumption 1.2.2.
All Liouville manifolds are assumed to be Weinstein, unlessotherwise stated.
We emphasize that, in what follows, Assumption 1.2.2 may appear both as acondition and as a conclusion. Namely, whenever a Liouville manifold naturallyappears e.g., R ± of a convex hypersurface or pages of a compatible open bookdecomposition, we either assume it is Weinstein if it is a condition, or we prove itis Weinstein if it is a conclusion.Now we are ready to state the foundational theorems of CHT. Theorem 1.2.3.
Any closed hypersurface in a contact manifold can be C -approxi-mated by a convex one. ONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY 5
Theorem 1.2.4.
Let ξ be a contact structure on Σ × [0 , such that the hypersur-faces Σ × { , } are convex. Then, up to a boundary-relative contact isotopy, thereexists a finite sequence < t < · · · < t N < such that the following hold: • Σ × { t } is convex if t = t i for any ≤ i ≤ N . • For each i , there exists small ǫ > such that ξ restricted to Σ × [ t i − ǫ, t i + ǫ ] is contactomorphic to a bypass attachment. For an initial study of bypass attachments in higher dimensions the reader isreferred to [HHa].
Remark . Theorem 1.2.4 was conjectured by Paolo Ghiggini in the afternoonof April 10, 2015 in Paris.1.3.
Applications.
As an immediate application of Theorem 1.2.3 and Theorem 1.2.4,we can extend Giroux’s -dimensional approach to constructing compatible openbook decompositions to higher dimensions. This is the content of the following twocorollaries. Note, however, that we do not address the stabilization equivalence ofthe compatible open book decompositions in this paper. We plan to investigate thisin future work. Corollary 1.3.1 ([Gir02]) . Any closed contact manifold admits a compatible openbook decomposition.
Corollary 1.3.2.
Any compact contact manifold with convex boundary admits acompatible partial open book decomposition.
Corollary 1.3.3.
Given a possibly disconnected closed Legendrian submanifold Λ in a closed contact manifold, there exists a compatible open book decompositionwith a page containing Λ . This completes our exploration of Question 1.1.2 for the time being.Next we turn to Question 1.1.3, which is a much harder question. For exam-ple, we would like to obtain classification results for contact structures on higher-dimensional manifolds (e.g., the spheres) besides the “flexible” ones due to Bor-man, Eliashberg, and Murphy [BEM15]. Unfortunately, our current understandingof contact Morse theory is not good enough for us to classify anything in higher di-mensions. Instead, we will use the (mostly dynamical) techniques developed in thispaper to address the existence problems of contact manifolds and submanifolds.The existence problems of contact manifolds and submanifolds were first ad-dressed by Gromov [Gro86] using his magnificent zoo of h -principles. In par-ticular, he proved a full h -principle for contact structures on open manifolds (cf.[EM02, 10.3.2]) and an existence h -principle for isocontact embeddings Y ⊂ ( M, ξ ) under the assumptions that either Y has codim Y ≥ or is open with codim Y = 2 .The existence problem turned out to be much harder for closed manifolds. Indimension , an existence h -principle for contact structures was proved by Martinet[Mar71] and Lutz [Lut77]. For overtwisted contact -manifolds, a full h -principlewas proved by Eliashberg [Eli89]. In dimension , there is a rich literature of KO HONDA AND YANG HUANG partial results: the existence of contact structures on certain classes of -manifoldswas established by Geiges [Gei91, Gei97], Geiges-Thomas [GT98, GT01], andBourgeois [Bou02]. Afterwards, a complete existence h -principle for contact -manifolds were established by Casals, Pancholi, and Presas [CPP15] and Etnyre[Etn], independently. Finally, the existence h -principle for contact manifolds ofany dimension, as well as the full h -principle for overtwisted contact manifolds ofany dimension, was established by Borman, Eliashberg, and Murphy [BEM15].So far the story is mostly about contact manifolds themselves. Now we turn tothe existence problem of contact submanifolds or (iso-)contact embeddings. Re-sults in this direction are surprisingly rare. Besides the aforementioned h -principleof Gromov, there exist constructions of contact submanifolds by Ibort, Mart´ınez-Torres, and Presas [IMTP00], mentioned earlier. In low dimensions, there alsoexist works by Kasuya [Kas16], Etnyre-Furukawa [EF17], and Etnyre-Lekili [EL]on embedding contact -manifolds into certain contact -manifolds.In the rest of the introduction we will explain the existence h -principle for codi-mension contact submanifolds. Since the case of open submanifolds has alreadybeen settled by Gromov, we may assume that all the submanifolds involved areclosed. Definition 1.3.4.
Let ( M, ξ ) be a contact manifold. A submanifold Y ⊂ M is an almost contact submanifold if there exists a homotopy ( η t , ω t ) , t ∈ [0 , , where η t ⊂ T M | Y is a codimension distribution of T M along Y and ω t is a conformalsymplectic structure on η t , such that:(1) η = ξ | Y and ω is induced from ξ | Y ; and(2) T Y ⋔ η and the normal bundle T Y M ⊂ η is ω -symplectic. A straightforward calculation (cf. [BCS14, Lemma 2.17]) shows that any even-codimensional submanifold with trivial normal bundle is almost contact.
Corollary 1.3.5.
Any almost contact submanifold can be C -approximated by agenuine contact submanifold. Corollary 1.3.6.
Any contact submanifold can be C -approximated by anothercontact submanifold with the opposite orientation. Wrapping up the introduction, we remark that by combining Corollary 1.3.5with Gromov’s h -principle for contact structures on open manifolds, one can easilydeduce the existence h -principle for contact structures, giving an alternate proofof a result of Borman-Eliashberg-Murphy [BEM15]. This is left to the reader asan exercise. Finally, note that in contrast to the contact structures constructed in[BEM15], the contact submanifolds constructed by Corollary 1.3.5 are not a priori overtwisted. The readers are referred to the recent work of Pancholi-Pandit [PP]for more discussions on iso-contact embeddings. Acknowledgments.
KH is grateful to Yi Ni and the Caltech Mathematics Depart-ment for their hospitality during his sabbatical. YH thanks the Geometry groupat Uppsala: Georgios Dimitroglou Rizell, Luis Diogo (his officemate), TobiasEkholm, Agn`es Gadbled, Thomas Kragh, Wanmin Liu, Maksim Maydanskiy and
ONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY 7
Jian Qiu for conversations about Everything in the last two years (2017–2019). Wethank Cheuk Yu Mak for pointing out some typos.2. A
CONVEXITY CRITERION
Let Σ ⊂ ( M n +1 , ξ ) be a closed cooriented hypersurface. The goal of thissection is to give a sufficient condition for the characteristic foliation Σ ξ on Σ (seeDefinition 2.0.1) which guarantees the convexity of Σ .Let α be a contact form for ξ . Let ( − ǫ, ǫ ) × Σ be a collar neighborhood of Σ = { } × Σ ⊂ M . Fix an orientation on Σ such that the induced orientation on ( − ǫ, ǫ ) × Σ agrees with the orientation determined by α ∧ ( dα ) n . We now introducethe characteristic foliation Σ ξ on Σ . Definition 2.0.1.
The characteristic foliation Σ ξ is an oriented singular line fieldon Σ defined by Σ ξ = ker dβ | ker β , where β := α | Σ ∈ Ω (Σ) . The orientation of Σ ξ is determined by the requirementthat the decomposition T Σ = Σ ξ ⊕ Σ ⊥ ξ respect the orientation, where the orthog-onal complement Σ ⊥ ξ , taken with respect to an auxiliary Riemannian metric on Σ ,is oriented by β ∧ ( dβ ) n − | Σ ⊥ ξ .Remark . The characteristic foliation depends only on the contact structureand the orientation of Σ , and not on the choice of the contact form.Note that x ∈ Σ is a singular point of Σ ξ if T x Σ = ξ x as unoriented spaces. Wesay x is positive (resp. negative ) if T x Σ = ± ξ x as oriented spaces, respectively.The significance of the characteristic foliation in -dimensional contact topol-ogy is that it uniquely determines the germ of contact structures on any embeddedsurface. The corresponding statement for hypersurfaces in contact manifolds ofdimension > is unlikely to hold, i.e., the characteristic foliation by itself is not enough to determine the contact germ. Instead we have the following characteriza-tion of contact germs on hypersurfaces in any dimension. The proof is a standardapplication of the Moser technique and is omitted here. Lemma 2.0.3.
Suppose ξ i = ker α i , i = 0 , , are contact structures on M suchthat β = gβ ∈ Ω (Σ) for some g : Σ → R + , where β i = α i | Σ . Then thereexists an isotopy φ s : M ∼ → M, s ∈ [0 , , such that φ = id M , φ s (Σ) = Σ and ( φ ) ∗ ( ξ ) = ξ on a neighborhood of Σ . Generally speaking, Σ ξ can be rather complicated, even when Σ is convex withLiouville R ± (Σ) . For our purposes of this paper, it is more convenient to regard Σ ξ as a vector field rather than an oriented line field. Of course there is no naturalway to specify the magnitude of Σ ξ as a vector field, which motivates the followingdefinition: Two vector fields v , v on Σ are conformally equivalent if there existsa positive function h : Σ → R + such that v = hv . This is clearly an equivalencerelation among all vector fields, and we will not distinguish conformally equivalentvector fields in the rest of the paper unless otherwise stated. KO HONDA AND YANG HUANG
In order to state the convexity criterion, we need to prepare some generalitieson gradient-like vector fields in the following subsection. Our treatment on thissubject will be kept to the minimum. The reader is referred to the classical worksof Cerf [Cer70] and Hatcher-Wagoner [HW73] for more thorough discussions. In-deed the adaption of the techniques of Cerf and Hatcher-Wagoner to CHT will becarried out in [HHb]. Note that similar techniques in symplectic topology havebeen developed by Cieliebak-Eliashberg in [CE12].2.1.
Gradient-like vector fields.
Let Y be a closed manifold of dimension n . Asmooth function f : Y → R is Morse if all the critical points of f (i.e., points p ∈ Y such that df ( p ) = 0 ) are nondegenerate , i.e., there exists a coordinate chartaround p such that locally f takes the form(2.1.1) − x − · · · − x k + x k +1 + · · · + x n . Here k is called the Morse index , or just index , of the critical point p .Following the terminology from [CE12], a smooth function f : Y → R is generalized Morse if the critical points of f are either nondegenerate or of birth-death type. Here a critical point p ∈ Y of f is of birth-death type if there existlocal coordinates around p such that f takes the form − x − · · · − x k + x k +1 + · · · + x n − + x n . Similarly, k is defined to be the (Morse) index of p . The birth-death type criticalpoint fits into a -parameter family of (generalized) Morse functions − x − · · · − x k + x k +1 + · · · + x n − + tx n + x n , such that for t < , there exist two nondegenerate critical points of indices k and k + 1 ; for t = 0 , there exists a birth-death type critical points; and for t > , thereare no critical points.It is a well-known fact (due to Morse) that any smooth function can be C ∞ -approximated by a Morse function. Moreover, Cerf proved that any -parameterfamily of smooth functions can be C ∞ -approximated by a family of generalizedMorse functions, where the birth-death type critical points appear as describedabove, only at isolated moments.Given a generalized Morse function f : Y → R , we say a vector field v on Y is gradient-like for f if the following two conditions are satisfied:(GL1) Near each critical point of f , v = ∇ f with respect to some Riemannianmetric; and(GL2) f is strictly increasing along (non-constant) flow lines of v . Definition 2.1.1.
A vector field v on Y is Morse (resp. generalized Morse ) if thereexists a Morse (resp. generalized Morse) function f : Y → R such that v isgradient-like for f .Remark . The terminology “generalized Morse function”, imported from [CE12],will be sufficient for the purposes of this paper since we will only encounter -parameter families of functions. In [HHb], we will need to deal with -parameter ONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY 9 families of functions where new singularities, i.e., the swallowtails, will appear. Inthat case, the terminology “generalized Morse function” will be too vague.In the rest of this subsection, we present a simple criterion for a vector field to beMorse which will be useful for our later applications. The corresponding versionfor generalized Morse vector fields is left to the reader as an exercise.
Proposition 2.1.3.
A vector field v on a closed manifold Y is Morse if and only ifthe following conditions are satisfied:(M1) For any point x ∈ Y with v ( x ) = 0 , there exists a neighborhood of x and a locally defined function f of the form given by Eq. (2.1.1) such that v = ∇ f .(M2) For any point x ∈ Y with v ( x ) = 0 , the unique flow line of v passingthrough x converges to zeros of v in both forward and backward time.(M3) There exists no broken loops , where a broken loop is a nonconstant map c : R / Z → Y such that there exists a sequence a < a < · · · < a N = 1 such that c maps a i to a zero of v and ( a i , a i +1 ) to an oriented flow line of v from c ( a i ) to c ( a i +1 ) for each i .Proof. The “only if” direction is obvious. To prove the “if” direction, let Z ( v ) = { x , . . . , x k } be the finite set of zeros of v , where the finiteness is guaranteed by(M1) and the compactness of Y . Then we define a partial order on Z ( v ) such that x i ≺ x j if there exists a flow line of v from x i to x j . The fact that ≺ is a partialorder follows from (M3).We then construct a handle decomposition of Y starting from the minimal ele-ments Z of Z ( v ) (note that a minimal element of Z ( v ) has index by (M2)) andinductively attaching handles as follows: Starting with a standard neighborhood of Z , suppose we have already attached the handles corresponding to Z j . Then weattach the handles corresponding to the minimal elements of Z ( v ) − Z j , and thenlet Z j +1 be the union of Z j and the minimal elements of Z ( v ) − Z j . (cid:3) To avoid using the adjective “generalized” everywhere in this paper, we willadopt the following convention:
Convention 2.1.4.
When we say a -parameter family of vector fields is Morse, weallow birth-death type singularities at isolated moments. Birth-death type singu-larities however are not allowed for a single Morse vector field.2.2. A convexity criterion.
The goal of this subsection is to give a sufficient con-dition for a hypersurface to be convex. To this end, we introduce the notions of
Morse and
Morse + hypersurfaces where the characteristic foliations have simpledynamics. Definition 2.2.1.
A hypersurface Σ ⊂ ( M, ξ ) is Morse if Σ ξ is a Morse vectorfield on Σ . We say Σ is Morse + if, in addition, there exist no flow trajectories froma negative singular point of Σ ξ to a positive one. From now on we say Σ ξ is Morse if there exists a representative in the conformalequivalence class of Σ ξ which is Morse. As in Convention 2.1.4, when we say a -parameter family of hypersurfaces Σ t , t ∈ R , is Morse, we allow (Σ t ) ξ to be generalized Morse at isolated t -values. Lemma 2.2.2. If Σ is a Morse hypersurface, then a C ∞ -small perturbation of Σ is Morse + .Proof. Choose a contact form ξ = ker α . It suffices to observe that dα | Σ is non-degenerate on a neighborhood of the singular points of Σ ξ . It is a standard fact (seee.g. [CE12, Proposition 11.9]) that the Morse index ind ( x ) ≤ n if x is a positivesingular point of Σ ξ , and ind ( x ) ≥ n if x is negative. The claim therefore followsfrom the usual transversality argument. (cid:3) The following proposition gives a sufficient condition for convexity:
Proposition 2.2.3.
Any Morse + hypersurface Σ is convex. Moreover R ± (Σ) nat-urally has the structure of a Weinstein manifold.Proof. This is a straightforward generalization of the usual proof for surfaces dueto Giroux that Σ is convex if it has a Morse + characteristic foliation.Let x = { x , . . . , x m } (resp. y = { y , . . . , y ℓ } ) be the positive (resp. negative)singular points of Σ ξ . Then β := α | Σ is nondegenerate on an open neighborhood U ( x ) of x . Suppose the indexing of the finite set x is such that the stable manifoldof x i +1 (here the stable manifold is with respect to the gradient flow of the Morsefunction), viewed as the core disk of a handle attachment, intersects the boundaryof a tubular neighborhood of the i th skeleton Sk x i in a sphere. Here Sk x i is theunion of the stable manifolds of x i := { x , . . . , x i } . In particular we necessarilyhave ind ( x ) = 0 , where ind is the Morse index, but we do not require ind ( x i ) ≥ ind ( x j ) for i > j . Such an arrangement is possible thanks to the assumption thatthere is no trajectory of Σ ξ going from y to x .Our first step is to make a conformal modification β e g β (still calling theresult β ) so that it becomes Liouville on a tubular neighborhood U (Sk x m ) of Sk x m .Arguing by induction, suppose that β is Liouville on U (Sk x i ) such that ∂U (Sk x i ) is naturally a contact manifold. Let D i +1 be the stable manifold of x i +1 suchthat D i +1 ∩ ∂U (Sk x i ) is a Legendrian sphere Λ ⊂ ∂U (Sk x i ) . Using the flow of Σ ξ , we may identify a tubular neighborhood of D i +1 \ ( U (Sk x i ) ∪ U ( x i +1 )) with [0 , r × Y , where Y is an open neighborhood of the -section in J (Λ) such that: • { } × Y ⊂ ∂U (Sk x i ) ; • { } × Y ⊂ ∂U ( x i +1 ) ; • ∂ r is identified with Σ ξ up to the multiplication by a positive function.It follows that one can write β = gλ on [0 , × Y , where λ is a contact form on Y and g is a positive function on [0 , × Y . Note that dβ = ∂ r gdr ∧ λ + d Y g ∧ λ + gdλ is symplectic if ∂ r g > . By assumption we have ∂ r g > when r is close to 0 or 1.Rescaling β | U (Sk x i ) by a large constant K ≫ , we can extend β | U (Sk x i ) ∪ U ( x i +1 ) to a Liouville form on U (Sk x i +1 ) . Moreover, we can assume ∂U (Sk x i +1 ) is trans-verse to Σ ξ by slightly shrinking U (Sk x i +1 ) . Hence by induction we can arrangeso that β is a Liouville form on U (Sk x m ) . ONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY 11
The treatment of the negative singular points of Σ ξ is similar. Let Sk ′ y ℓ be theunion of the unstable manifolds of y (with respect to the Morse function). Then bythe same argument we can assume that β is a Liouville form on − U (Sk ′ y ℓ ) , wherethe minus sign indicates the opposite orientation.Using the flow of Σ ξ , we can identify Σ \ ( U (Sk x m ) ∪ U (Sk ′ y ℓ )) with Γ × [ − , s such that: • Γ × {− } is identified with ∂U (Sk x m ) ; • Γ × { } is identified with ∂U (Sk ′ y ℓ ) ; and • R h ∂ s i = Σ ξ .We can write β = hη near Γ × {− , } , where η is a contact form on Γ and h = h ( s ) is a positive function such that h ′ ( s ) > near Γ × {− } and h ′ ( s ) < near Γ × { } . Extend h to a positive function Γ × [ − , → R such that h ′ ( s ) > for s < , h ′ (0) = 0 , and h ′ ( s ) < for s > . Let f = f ( s ) : Γ × [ − , → R be a strictly decreasing function with respect to s such that f ( − ǫ ) = 1 , f (0) = 0 ,and f ( ǫ ) = − . Then define ρ = f dt + hη on R t × Γ × [ − , , ρ = dt + β on R × U (Sk x m ) , and ρ = − dt + β on R × U (Sk ′ y ℓ ) . We leave it to the reader tocheck that ρ is contact and that ρ | { }× Σ agrees with α | Σ up to an overall positivefunction. The proposition now follows from Lemma 2.0.3. (cid:3)
3. C
ONSTRUCTION OF C - FOLDS IN DIMENSION In order to make a hypersurface Σ ⊂ ( M, ξ ) convex, we would like to modifythe characteristic foliation Σ ξ so it is directed by a Morse vector field and thenapply Proposition 2.2.3. (Note that going from Morse to Morse + is a C ∞ -genericcondition.) This will be achieved by certain C -small perturbations of Σ which wecall C -folds , where C stands for “contact” as opposed to the topological foldingdiscussed in Appendix A. The C -folds are most easily described in dimension 3and the general case will be constructed in Section 5 using -dimensional C -folds.It turns out that C -folds alone are enough to make any Σ ξ Morse if dim Σ = 2 .If dim Σ > , then C -folds are not quite sufficient and we will need an additionaltechnical construction in Section 8.The standard model of a C -fold will be constructed in a Darboux chart ( R z,s,t , ξ = ker α ) , α = dz + e s dt. Let
Σ = { z = 0 } be the surface under consideration with normal orientation ∂ z and characteristic foliation Σ ξ directed by ∂ s . The goal of this section is to “fold” Σ to obtain another surface Z which coincides with Σ outside of a compact set,and analyze the change in the dynamics of the characteristic foliations.In § Z PL and then in § Z PL to obtain a suitably generic smooth surface Z such thatthe characteristic foliation Z ξ has the desired properties. The letter Z is chosen tomimic the shape of the fold.3.1. Construction of Z PL . Choose a rectangle (cid:3) = [0 , s ] × [0 , t ] ⊂ Σ , where s , t > . We define Z PL to coincide with Σ outside of (cid:3) . Remark . We observe that the a priori more general case [ s − , s ] × [ t − , t ] can be reduced to (cid:3) by applying a diffeomorphism to R .Choose z > and a small constant ǫ > . We construct three rectangles P , P , P and two trapezoids P , P in R , which, together with R s,t \ (cid:3) , glue togive Z PL , i.e., we define • P := [0 , s ] × [ − e ǫ z , − e − s − ǫ z + t ] ⊂ { z = z } ; • P i , i = 1 , . . . , , are the faces ( = P , (cid:3) ) of the convex hull of P ∪ (cid:3) ,ordered counterclockwise so that P ⊂ { s = 0 } . (cid:3) P st z F IGURE Z PL . Definition 3.1.2. A piecewise linear C -fold of Σ is a PL surface defined by Z PL := (Σ \ (cid:3) ) ∪ ∪ ≤ i ≤ P i . The rectangle (cid:3) ⊂ Σ is called the base of the C -fold. We will also write Z ǫ PL tohighlight the dependence on ǫ > . Observe that, away from the corners, the characteristic foliation ( Z PL ) ξ on Z PL satisfies • ( Z PL ) ξ = R h ∂ s i on Σ \ (cid:3) , P , P and P ; • ( Z PL ) ξ is the linear foliation on P and P with “slopes” − and − e − s ,respectively, where “slope” refers to the value of dt/dz = − e − s . SeeFigure 3.1.2.F IGURE P (left)and P (right).We now analyze the dynamics of the PL flow on Z PL . Here a flow line of ( Z PL ) ξ is by definition a PL curve on Z PL such that each linear piece is tangentto ( Z PL ) ξ . Note that the flow lines are not necessarily uniquely determined by theinitial conditions due to the presence of corners. ONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY 13
We begin by introducing a few quantities which characterize the various sizesof the fold.
Definition 3.1.3.
Given Z PL as above, its z -height , s -width , and t -width are givenby: Z ( Z PL ) := z , S ( Z PL ) := s , T ( Z PL ) := t + (1 − e − s ) z . The following proposition characterizes a key feature of ( Z PL ) ξ when the pa-rameters of the fold are appropriately adjusted. Lemma 3.1.4.
Fix s , z > . If t < (1 − e − s ) z , then there exists ǫ > such that any (necessarily unique) flow line of ( Z ǫ PL ) ξ passing through the point ( − , a ) ∈ R s,t , where a ∈ (0 , t ) , converges to the interval P ∩ P in forwardtime. Similarly, any flow line of ( Z ǫ PL ) ξ passing through ( s + 1 , a ) , a ∈ (0 , t ) ,converges to the interval P ∩ P in backward time.Proof. We prove the first statement; the second is similar. Since t − z + z e − s < , it follows that any flow line passing through the point ( − , a ) ∈ Σ with a ∈ (0 , t ) lands on the edge P ∩ P after traversing once over the faces P , P , P and P in forward time. Hence the flow line in forward time spirals around andlimits to P ∩ P . (cid:3) Assumption 3.1.5.
From now on we assume that all Z P L satisfy t < (1 − e − s ) z < (1 . t . Blocking ratio.
We now introduce the blocking ratio which, roughly speaking,measures the ratio between the amount of flow line of ( Z PL ) ξ trapped by the foldand the t -width of the fold itself.Consider an embedded piecewise smooth surface S ⊂ R z,s,t which is homeo-morphic to R and coincides with R s,t outside of a compact set. Let π : R z,s,t → R z,t be the projection map. Let R be the (nonempty) set of rectangles R ⊂ R z,t such that: • the edges are parallel to the z - and t -axes; and • π ( S ) coincides with the t -axis outside of R .Let t min ( S ) be the infimum over R ∈ R of the length of the edges of R parallelto the t -axis and let t b ( S ) be the length of π ( R s,t \ S ) . Definition 3.1.6.
The blocking ratio of S is (3.1.1) ρ ( S ) := t b ( S ) /t min ( S ) . Applying the above definition to Z PL , we have t min ( Z PL ) = T ( Z PL ) + o ( ǫ ) and t b ( Z PL ) = t . It follows that ρ ( Z PL ) → t / T ( Z PL ) as ǫ → . Moreover, according to Lemma 3.1.4, for any fixed s , z , if t is sufficientlyclose to (1 − e − s ) z as in Assumption 3.1.5, we obtain a surface Z PL such that ρ ( Z PL ) > . Smoothing of Z PL . In this subsection we smooth the piecewise linear Z PL constructed in § C -fold Z of Σ .Choose a parameter δ > that is much smaller than the parameter ǫ > chosenin Definition 3.1.2. For each z ′ ∈ (0 , z ) , the slices R z ′ := Z PL ∩ { z = z ′ } arerectangles. (When z ′ = 0 or z , we take R z ′ = ∂Z PL ∩ { z = z ′ } .) For each < δ ′ < δ we construct R δ ′ z ′ on the plane z = z ′ by first extending all the sidesof the rectangle by δ ′ and then rounding the corners using a fixed model scaledby δ ′ . Choose a function φ : [0 , z ] → [0 , δ ] which is smooth on (0 , z ) and has“derivative −∞ ” at z = 0 , z .To obtain the φ -smoothing Z of Z PL , also called a C -fold of Σ , we make thefollowing modifications to Z PL :(1) replace R z ′ by R φ ( z ′ ) z ′ for z ∈ (0 , z ) ;(2) remove the bounded component of { z = 0 } − R φ (0)0 ; and(3) adjoin the bounded component of { z = z } − R φ ( z ) z .The base e (cid:3) ⊂ Σ of Z is the closure of Σ \ Z . By construction e (cid:3) converges to (cid:3) when all the parameters tend to zero.The following proposition describes the key dynamical properties of Z ξ . Theproof follows immediately from Lemma 3.1.4. Proposition 3.2.1.
The exists a smoothing Z of Z P L whose vector field Z ξ satisfiesthe following:(TZ1) Z ξ has four nondegenerate singularities: – a positive source e + near the midpoint of P ∩ P , – a positive saddle h + near the midpoint of R s,t ∩ P , – a negative sink e − near the midpoint of P ∩ P , and – a negative saddle h − near the midpoint of R s,t ∩ P .(TZ2) Fix s , z > . For any t < (1 − e − s ) z (we’re assuming Z P L satisfiesthis) and ǫ > small, – the stable manifolds of h ± intersect the line { s = − } at two points { ( − , κ ) , ( − , t + κ ) } , and – the unstable manifolds of h ± intersect the line { s = s + 1 } at twopoints { ( s + 1 , − κ ) , ( s + 1 , t − κ ) } ,where κ → as ǫ → . Moreover, any flow line of Z ξ passing throughthe interval {− } × [ κ, t + κ ] ⊂ R s,t converges to the negative sink inforward time. Similarly, any flow line of Z ξ passing through the interval { s + 1 } × [ − κ, t − κ ] converges to the positive source in backward time.(TZ3) For any fixed s , z > , there exists ǫ > sufficiently small and t suffi-ciently close to (1 − e − s ) z such that the blocking ratio ρ ( Z ) > . See Figure 3.2.1 for an illustration of the effect of a C -fold on the characteristicfoliation. Remark . We had some freedom in choosing the intervals [ κ, t + κ ] and [ − κ, t − κ ] in (TZ2). We could have chosen [ κ , t + κ ] and [ κ , t + κ ] subject ONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY 15 C -fold h + h − e + e − F IGURE C -fold.to κ > κ and κ > κ which are required since the surface Z was pushed in thepositive z -direction.4. C ONVEX SURFACE THEORY REVISITED
The goal of this section is to prove Theorem 1.2.3 and Theorem 1.2.4 in di-mension using the folding techniques developed in Section 3. In dimension ,Theorem 1.2.3 was proved by Giroux in [Gir91] in a stronger form where C isreplaced by C ∞ . Theorem 1.2.4 can be inferred from Giroux’s work on bifurca-tions [Gir00] and the bypass-bifurcation correspondence. The technical heart ofGiroux’s work is based on the study of dynamical systems of vector fields on sur-faces, a.k.a., Poincar´e-Bendixson theory. In particular, one invokes a deep theoremof Peixoto [Pei62] to prove the C ∞ -version of Theorem 1.2.3 and much more workto establish Theorem 1.2.4.In this section we give elementary, Morse-theoretic proofs of Theorem 1.2.3and Theorem 1.2.4 in dimension . Our strategy is the following: First apply a C ∞ -small perturbation of Σ ⊂ ( M , ξ ) such that the singularities of Σ ξ becomeMorse. There exists a finite collection of pairwise disjoint transverse arcs γ i , i ∈ I ,in Σ such that any flow line of Σ ξ passes through some γ i . We will construct a -dimensional plug supported on a collar neighborhood of each γ i in § γ i , i.e., they all necessarily converge to singulari-ties in the plug. Each plug consists of a large number of C -folds constructed inSection 3. This proves Theorem 1.2.3. To prove Theorem 1.2.4, we slice Σ × [0 , into thin layers using Σ i := Σ × { iN } , ≤ i ≤ N , for large N such that thedifference between (Σ i ) ξ and (Σ i +1 ) ξ is small. (By “small” we mean the vectorfields in question are C -close to each other. The global dynamics of (Σ i ) ξ maystill drastically differ from that of (Σ i +1 ) ξ .) Within each layer we insert plugs on Σ i as in the case of a single surface so that the isotopy from Σ i to Σ i +1 is throughMorse surfaces, i.e., (Σ t ) ξ is Morse for all iN ≤ t ≤ i +1 N . For technical reasons,it is desirable to eliminate the plugs created on Σ i when we reach Σ i +1 , replacingthem by new plugs on Σ i +1 , so that one can inductively run from i = 0 to i = N and make all intermediate surfaces Morse. Then the only obstructions to convex-ity occur at finitely many instances where the surface is Morse but not Morse + ,corresponding to bypass attachments. This section is organized as follows: In § -dimensional plugsand in § § § -dimensional plugs. The construction of a plug is local. Consider M =[0 , z ] × [0 , s ] × [0 , t ] with coordinates ( z, s, t ) and contact form α = dz + e s dt .Here z , s , t > are arbitrary, but for most of our applications, we should thinkof z , s as being much smaller than t . In other words, (TZ3) in Proposition 3.2.1,i.e., t ≈ (1 − e − s ) z , will not be satisfied.Consider the surface B = { } × [0 , s ] × [0 , t ] with B ξ = R h ∂ s i . Let ∂ − B = { } × { } × [0 , t ] and ∂ + B = { } × { s } × [0 , t ] be the bottom and top sides of B , respectively. Pick a large integer N ≫ . Let (cid:3) k,l ⊂ B be boxes defined by (cid:3) k,l := (cid:20) l − s , l s (cid:21) × (cid:20) k + lN t , k + l + 2 N t (cid:21) , where ≤ k < ⌊ N/ ⌋ , ≤ l ≤ . See Figure 4.1.1. t s F IGURE C -folds on B . Here N ≈ .Applying the constructions from Section 3, we can create pairwise disjoint C -folds Z k,l on B such that the following hold: Recalling the definition of the z -height Z ( Z k,l ) from Definition 3.1.3, • the base of each Z k,l approximately equals (cid:3) k,l ; • Z ( Z k,l ) < z and the triple ( Z ( Z k,l ) , s , t N ) satisfies (TZ3).The resulting surface B ∨ ⊂ M is called a plug on B . Clearly B ∨ agrees with B near the boundary and hence can be implemented on any surface which contains B as a subdomain.The key property of the dynamics of B ∨ ξ is as follows: Given any ǫ > small,if we choose N ≫ /ǫ , then for any point x ∈ ∂ − B with t ( x ) ∈ ( ǫ, t − ǫ ) , thepossibly broken flow line of B ∨ ξ passing through x converges to a sink in forwardtime. Here a broken flow line refers to a piecewise smooth map c : [0 , → B ∨ such that c (0) = x , c ( ) is a saddle, c (1) is a sink, and c | [0 , / and c | (1 / , areboth orientation-preserving smooth trajectories of B ∨ ξ . ONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY 17
Remark . In the above construction, the maximal value of l is called the depth of the plug. Hence the depth of B ∨ is . In view of Proposition 3.2.1, one canconstruct a plug of depth by reducing the overlap between the t -projections of (cid:3) k,l and (cid:3) k,l +1 . However the only thing that matters that the depth of a plug isa finite number independent of z , s , t . In fact it is merely a consequence ofthe finite blocking rate stated in Proposition 3.2.1. On the other hand, the number N → ∞ necessarily as z and/or s tend to .4.2. Installing and uninstalling plugs.
The construction of a plug B ∨ is suffi-cient to prove Theorem 1.2.3 in dimension . In order to prove Theorem 1.2.4, wealso need to interpolate between B and B ∨ with some control of the intermediatedynamics. We now explain this procedure.Let ( M, ξ = ker α ) be as before with the exception that the parameters z , s , t are all reset. Let B z := { z } × [0 , s ] × [0 , t ] . Replace B z / by a plug B ∨ z / ,where Z ( B ∨ z / ) ≪ z so that in particular B ∨ z / is still contained in M . B B ∨ z / B z F IGURE B , B ∨ z / and B z .For the moment consider the PL model of the plug B ∨ z / , i.e., the C -folds Z involved in the construction are replaced by the corresponding Z PL . It is fairlystraightforward to foliate the regions bounded between B and B ∨ z / and between B ∨ z / and B z by a family of PL surfaces; see Figure 4.2.1 for a schematic picture.Then one can apply the smoothing scheme from § M = ∪ ≤ a ≤ z e B a , where e B = B , e B z = B z , and e B z / is the smoothed version of B ∨ z / .To analyze the dynamics of ( e B a ) ξ for each a ∈ [0 , z ] , we introduce the partiallydefined and possibly multiply-valued holonomy map ρ a : ∂ − e B a ∂ + e B a , where ∂ ± e B a are the top and bottom sides of e B a as before: Given x ∈ ∂ − e B a , if there existsa possibly broken flow line of ( e B a ) ξ starting from x and ending at y ∈ ∂ + e B a , then y ∈ ρ a ( x ) . Note that such y may not be unique. If there is no such flow line, then ρ a ( x ) is not defined.Define the norm k ρ a k := sup x ∈ ∂ − e B a | t ( x ) − t ( ρ a ( x )) | . Here | t ( x ) − t ( ρ a ( x )) | = 0 if ρ a ( x ) is not defined, and the supremum is taken overall possible ρ a ( x ) if ρ a is not single-valued at x .The following lemma is obvious but will be important for our applications. Lemma 4.2.1.
The number sup ≤ a ≤ z k ρ a k → as N → ∞ . We call the foliation from B to B ∨ z / installing a plug and the foliation from B ∨ z / to B z uninstalling a plug . Then Lemma 4.2.1 basically says that neitherinstalling nor uninstalling a plug affects the local holonomy by much. For the restof Section 4, we assume that N ≫ without further mention.4.3. Proof of Theorem 1.2.3 in dimension 3.
Given any surface Σ ⊂ ( M, ξ ) , itis well-known (see e.g. [Gei08, Section 4.6]) that, up to a C ∞ -small perturbation,we can assume that Σ ξ is Morse near the (isolated) singularities.An embedded rectangle B = [0 , s ] × [0 , t ] ⊂ Σ with coordinates ( s, t ) is a foliated chart if Σ ξ | B = R h ∂ s i . Let us write B ǫ = [0 , s ] × [ ǫ, t − ǫ ] for a slightlysmaller foliated chart with ǫ > small. Lemma 4.3.1.
There exists a finite index set I , a small constant ǫ > , and acollection of pairwise disjoint foliated charts B i = [0 , s i ] × [0 , t i ] , i ∈ I , such thatthe following holds:(*) each flow line of Σ ξ intersects some B i and for any x ∈ Σ which is neithera singularity of Σ ξ nor contained in any B ǫi , the flow line of Σ ξ passingthrough x enters some B ǫi or limits to some Morse singularity in forwardtime (resp. in backward time). Such a collection B I = { B i } i ∈ I satisfying (*) is called a barricade on Σ or a barricade for Σ ξ . Proof of Lemma 4.3.1.
We first construct a finite collection of pairwise disjointopen transverse arcs γ i , i ∈ I , such that any flow line of Σ ξ passes through some γ i . Let N be the union of small open neighborhoods of the Morse singularities.Since Σ is compact, Σ − N can be covered by a finite number of small foliatedcharts B ǫi = (0 , s i ) × ( ǫ, t i − ǫ ) in Σ ; we set γ i = { s i } × ( ǫ, t i − ǫ ) .We inductively modify the γ i so that they are pairwise disjoint. If γ ∩ γ = ∅ ,then we consider the components γ ,j , j = 1 , . . . , ℓ , of γ − N δ ( γ ) , where δ > is much smaller than ǫ . We slightly extend γ ,j to E ( γ ,j ) such that:(1) all the flow lines intersected by γ are intersected by ∪ j E ( γ ,j ) , and(2) γ , E ( γ , ) , . . . , E ( γ ,ℓ ) are mutually disjoint.We refer to the procedure as “splitting γ along γ ”. Renaming the arcs γ , E ( γ , ) , . . . , E ( γ ,ℓ ) , γ , . . . so they are called γ i , i ∈ I , we may assume by induction that γ , . . . , γ k arepairwise disjoint and that γ k +1 nontrivially intersects some γ i , i ≤ k . As abovewe split γ k +1 along γ , . . . , γ k to obtain a collection { E ( γ k +1 ,j ) } j that is pairwisedisjoint and also disjoint from γ , . . . , γ k .The collection { γ i } i ∈ I can be thickened to a collection { B i } i ∈ I of pairwisedisjoint foliated charts such that any flow line of Σ ξ passes through some B ǫi ; herewe are taking γ i to be { s i } × ( ǫ, t i − ǫ ) and B i = [0 , s i ] × [0 , t i ] . Now let x ∈ Σ be a point which is neither a singularity of Σ ξ nor contained in any B ǫi . Assumingthe flow line of Σ ξ through x passes through some B ǫi in backward time and does ONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY 19 not limit to a Morse singularity in forward time. We claim that the forward flowline ℓ x starting at x passes through some B ǫi : Let Λ + x be the forward limit set,i.e., the set of points y ∈ Σ for which there exist x j ∈ ℓ x , j = 1 , , . . . , such that lim j →∞ x j = y and the distance along ℓ x from x to x j goes to ∞ . It is well-knownthat Λ + x is closed and is a union of flow lines of Σ ξ . By construction, there exists y ∈ Λ + x and a transverse arc γ i that passes through it. It follows that ℓ x also passesthrough γ i and hence through B ǫi . (cid:3) Remark . The proof of Lemma 4.3.1 was written so that it works in any di-mension.Suppose without loss of generality that each B i has a neighborhood [ − z i , z i ] × B i , where the contact form can be written as dz + e s dt and B i is identified with { } × B i . Now we construct a new surface Σ ∨ which C -approximates Σ , byreplacing every B i by the plug B ∨ i . Clearly the characteristic foliation Σ ∨ ξ satis-fies Conditions (M1)–(M3) of Proposition 2.1.3. Hence, up to a further C ∞ -smallperturbation if necessary, Σ ∨ is a convex approximation of Σ by Proposition 2.2.3.4.4. Proof of Theorem 1.2.4 in dimension 3.
Consider a contact structure ξ on Σ × [0 , such that Σ × { , } is Morse + in the sense of Definition 2.2.1. Thegoal is to show that up to an isotopy relative to the boundary, (Σ t ) ξ is Morse for all t ∈ [0 , . Here Σ t := Σ × { t } .Define L := { x ∈ Σ × [0 , | ξ x = T x Σ } . Up to a C ∞ -small perturbation of ξ ,we can assume L satisfies the following:(S1) L is a properly embedded 1-submanifold such that L ∩ Σ t is the singularset of (Σ t ) ξ ;(S2) the singularities of (Σ t ) ξ are Morse for all t ; and(S3) the restricted coordinate function t | L : L → [0 , is Morse and all itscritical points have distinct critical values.Suppose < a < · · · < a m < are the critical values of t | L , which weassume to be irrational. Fix ǫ > small. For each t ∈ [0 , there exists a barricade B I t for Σ t such that B I t is a barricade for any vector field that is ǫ -close to (Σ t ) ξ .Next choose an integer K ≫ such that, for i = 0 , , . . . , K , B I i , I i = I i/K , isa barricade for all Σ t , t ∈ [ i − K , i +1 K ] ∩ [0 , . In particular, for each a j , there existunique j + , j − such that j + = j − + 1 and j − K < a j < j + K .Let π : Σ × [0 , → Σ be the natural projection. By splitting π ( B I i +1 ) along π ( B I i ) if necessary (as in the proof of Lemma 4.3.1), we may assume that(4.4.1) π ( B I i +1 ) ∩ π ( B I i ) = ∅ , i = 0 , , . . . , K − . Moreover, we may choose the splitting so that the new B I i +1 remains a barricadefor all Σ t , t ∈ [ iK , i +2 K ] ∩ [0 , .We divide the proof into several steps.S TEP From Σ to Σ ∨ / ( N ′ K ) where N ′ > is a large integer. Recall we are allowing birth-death type singularities to be Morse.
Suppose for simplicity that B I consists of a single foliated chart, i.e., B I =[0 , s ] × [0 , t ] with ( B I ) ξ = R h ∂ s i . In the case where B I consists of more thanone component, we simply repeat the following construction for each component.Define the external holonomy b ρ : ∂ + B I ∂ − B I as follows: For any x ∈ ∂ + B I = { s } × [0 , t ] , a point y ∈ ∂ − B I = { } × [0 , t ] is in the image b ρ ( x ) ifthere exists a possibly broken flow line c : [0 , → Σ \ B I satisfying: • c (0) = x and c (1) = y ; • there exist b < b < · · · < b r = 1 , r ≥ , such that c ( b i ) is asingularity of (Σ ) ξ for all < i < r ; • c | ( b i ,b i +1 ) is an oriented flow line of (Σ ) ξ .Of course b ρ is not necessarily defined on all of ∂ + B I and when it is defined, it isnot necessarily single-valued.Since (Σ ) ξ is Morse by assumption, there exists δ > such that k b ρ k := sup x ∈ ∂ + B I | t ( x ) − t ( b ρ ( x )) | > δ. Otherwise, there is a sequence of points x i ∈ ∂ + B I such that | t ( x i ) − t ( b ρ ( x i )) | → and the compactness of the sequence of broken flow lines gives us x ∞ ∈ ∂ + B I such that | t ( x ∞ ) − t ( b ρ ( x ∞ )) | = 0 , which contradicts (M3) from Proposition 2.1.3.It follows from Lemma 4.2.1 that one can install a plug on B I as described in § Σ and Σ ∨ such that all the leaves are Morse.For convenience we pretend that Σ ∨ agrees with Σ on the complement of B I ,and that the difference is contained in a small invariant neighborhood of B I .In order to interpolate between Σ ∨ and Σ ∨ / ( N ′ K ) for a large integer N ′ > ,we use B I satisfying Eq. (4.4.1). If N ′ ≫ , then there is a -parameter familyof surfaces F s ⊂ Σ × [0 , N ′ K ] , s ∈ [0 , , such that F = Σ \ N ( B I ) , F ∩ Σ / ( N ′ K ) ⊃ N ( B I ) × { N ′ K } , ∂F s = ∂N ( B I ) × { } for all s ∈ [0 , , andthe ( F s ) ξ , s ∈ [0 , , are ǫ -close to one other so that B I is a barricade for all ( N ( B I ) × { } ) ∪ F s ; in particular, no new singularities are introduced in thisprocess. The barricading condition can be guaranteed by having chosen N ′ ≫ .See the upper-left corner of Figure 4.4.1 for an illustration of this procedure. By thebarricading condition the surfaces ( N ( B I ) ×{ } ) ∪ F s are Morse for all s ∈ [0 , .Next we install a plug on B I × { N ′ K } ⊂ ( N ( B I ) × { } ) ∪ F , uninstall theplug on B ∨ I , and lift the surface ( N ( B I ) × { δ } ) ∪ ( F \ ( N ( B I ) × { N ′ K } )) , < δ ≪ N ′ K , up to Σ / ( N ′ K ) , as shown in the upper-right, lower-right, and lower-left corners of Figure 4.4.1, respectively. Moreover all the intermediate surfaces areMorse by analogous reasons. This finishes our construction of the foliation from Σ to Σ ∨ / ( N ′ K ) .S TEP From Σ ∨ / ( N ′ K ) to Σ ∨ − /K , where − /K < a < + /K . Using B I and B I , we similarly construct the Morse foliation from Σ ∨ / ( N ′ K ) to Σ ∨ / ( N ′ K ) as in Step 1, and so on, until we get to Σ ∨ /K . Between Σ ∨ /K and Σ ∨ /K we use B I and B I , and so on. ONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY 21 N ′ K F IGURE Σ and Σ ∨ / ( N ′ K ) by Morsesurfaces. The blue parts represent B I i , i = 0 , .S TEP From Σ ∨ − /K to Σ ∨ + /K . The only modification needed in this step is due to the fact that the vector fields (Σ − /K ) ξ and (Σ + /K ) ξ are not C ∞ -close to each other in the usual sense. Rather,one observes either the birth or the death of a pair of nearby Morse singularities aswe go from (Σ − /K ) ξ to (Σ + /K ) ξ . In either case, we slightly modify the notionof barricades B I ± so that the unique (short) flow line connecting the pair of Morsesingularities is the only flow line that does not pass through B I ± . Similar remarksapply to all a i , ≤ i ≤ m .S TEP From Σ ∨ ( K − /K to Σ . In this final step, the only new ingredient is to uninstall the plugs as we go from Σ ∨ to Σ . By assumption Σ is Morse and in fact convex. Hence by the sameholonomy bound as in Step 1, all the intermediate surfaces are Morse.Finally we have foliated Σ × [0 , by surfaces of the form Σ t which are allMorse. The only obstruction to convexity occurs when (Σ t ) ξ is Morse but notMorse + and this corresponds to a bypass attachment (cf. Proposition 9.3.2). Thisconcludes the proof of Theorem 1.2.4 in dimension .4.5. Further remarks.
Compared to earlier groundbreaking works of Bennequin[Ben83] and Eliashberg [Eli92], convex surface theory is a more systematic frame-work for studying embedded surfaces in contact -manifolds. It is sufficientlypowerful that basically all known classification results of contact structures or Leg-endrian knots in this dimension follow from this theory.The only “drawback” of convex surface theory, at least in its original form[Gir91, Gir00], is that the monster of dynamical systems on surfaces is alwayslurking behind the story. More precisely, if one just wants to classify contact struc-tures or Legendrian knots up to isotopy, then the problem often reduces to a com-binatorial one by combining Giroux’s theory with, say, the bypass approach of[Hon00]. However, if one wants to obtain higher homotopical information of thespace of contact structures (say π n for n ≥ ), then some serious work on highercodimensional degenerations of Morse-Smale flows seems inevitable. As an example, in [Eli92] Eliashberg outlined the proof that the space of tightcontact structures on S is homotopy equivalent to S . This particular result isbased on the study of characteristic foliations on S ⊂ S , which is particularlysimple since we never have periodic orbits. In more general contact manifoldssuch as T , one cannot necessarily rule out periodic orbits from characteristic fo-liations, and hence the bifurcation theory quickly becomes unwieldy (the work[Ngo] probably comes close to the limit of what one can do). However, in lightof our reinterpretation/simplification of Giroux’s theory, it suffices to understandthe space of Morse gradient vector fields, instead of general Morse-Smale vectorfields.We hope our techniques can be applied to future studies of homotopy types ofthe space of contact structures. This topic however will not be pursued any furtherin this paper. 5. C ONSTRUCTION OF C - FOLDS IN DIMENSION > The goal of this section is to generalize the construction of C -folds in dimension in Section 3 to higher dimensions. We will construct two versions of C -folds indimension > and both will be used to construct the plug in Section 8. Notation.
Throughout this section, we will write Z ⊂ R for the C -fold con-structed in Section 3 and write Z for the higher-dimensional C -fold to be con-structed in this section. Sketch of the constructions. C -folds. These are strict generalizations of the construction in Section 3. In thiscase the ambient contact manifold is ( M = R z,s,t × W, ker α ) , α = dz + e s ( dt + λ ) , where W is a complete Weinstein manifold with Liouville form λ , and the hyper-surface on which we will construct the C -fold is Σ = { z = 0 } . If we take W to be a point, then we reduce to the situation in Section 3. The C -fold Z is con-structed by first taking the product hypersurface Z × W c , where W c ⊂ W is acompact Weinstein domain, and then rapidly damping out the Z -factor along thecylindrical Liouville vector field on W \ W c . Partial C -folds. These differ from C -folds in that W c is a Weinstein cobordismwith a nonempty negative boundary. Let ∂ ± W c be the positive and negative bound-aries of W c . As before we start with Z × W c and damp out the Z -factor on thecylindrical “ends” of W c . A subtle difference between ∂ + W c and ∂ − W c is thatwe rapidly damp out Z on [0 , ∞ ) × ∂ + W c as in the first case, but slowly dampout Z on ( − N, × ∂ − W c for N ≫ . The slow damping at the negative end ispossible in two scenarios:(1) either there is sufficient space in the negative end, e.g., we have a com-pleted negative end, or ONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY 23 (2) the size of Z is small with respect to a fixed collar neighborhood of ∂ − W c .We will be applying slow damping in the second scenario.The details of the above constructions will be given in the following three sub-sections.5.1. Construction of C -folds. Let ( W, λ ) be a complete Weinstein manifold and R t × W be the contactization of W with contact form β = dt + λ . Let W c ⊂ W be a compact subdomain such that W = W c ∪ ([0 , ∞ ) τ × Γ) , where Γ := ∂W c isthe contact boundary and τ is the direction of the (positive) symplectization of Γ .Define a contact handlebody H := ([0 , t ] × W c , dt + λ ) , where t > is the thickness of H . We emphasize that H is a compact contact manifold with a fixedcontact form such that all the Reeb orbits contained in H are chords of the samelength t .Consider the contact manifold ( M = R z,s,t × W, ξ = ker α ) , α = dz + e s β. We are interested in the hypersurface
Σ = { z = 0 } ⊂ M with characteristicfoliation Σ ξ = ∂ s . Roughly speaking, the goal is to fold Σ using H , so that theresulting characteristic foliation cannot pass through a region which approximates H . Remark . One can think of the constructions in Section 3 as a special casewhere H = [0 , t ] is equipped with the contact form dt .5.1.1. Product hypersurface.
Recall that in Section 3 we constructed the C -fold Z ⊂ ( R z,s,t , ker( dz + e s dt )) which agrees with R s,t outside of a rectangle (cid:3) = [0 , s ] × [0 , t ] . Let Z ξ be thecharacteristic foliation on Z .We will compute the characteristic foliation Z ′ ξ on the product hypersurface Z ′ := Z × W c ⊂ M . To simply the notation, we will not distinguish between thecharacteristic foliation (which is an oriented singular line field) and a trivializingvector field. Choose vector fields v on Z (away from the singularities of Z ξ ) suchthat α ( v ) = 1 and w on W c (away from the zero set of λ ) such that λ ( w ) = 1 . Lemma 5.1.2.
Away from the zeros of α | Z and λ , the characteristic foliation Z ′ ξ is given by (5.1.1) Z ′ ξ = R h Z ξ + dz ∧ ds ( Z ξ , v ) X λ i . Proof.
One can easily check that T ( Z × W c ) ∩ ξ = R h Z ξ , w − e s v, ker λ i . Basically the calculation of Z ′ ξ is reduced to computing the kernel K = aX + bY + cZ of the -dimensional vector space R h X, Y, Z i with a maximally nonde-generate -form h· , ·i . One can easily verify that K = h Y, Z i X + h Z, X i Y + h X, Y i Z works.Since e − s dα = ds ∧ dt + ds ∧ λ + dλ , the pairing h· , ·i := e − s dα ( · , · ) can becomputed as follows: h Z ξ , X λ i = 0 , h Z ξ , w − e s v i = ds ( Z ξ ) − e s ds ∧ dt ( Z ξ , v ) , h X λ , w − e s v i = 1 , where X λ is the Liouville vector field on ( W c , λ ) .We then set X = Z ξ , Y = X λ , Z = w − e s v to compute that Z ′ ξ = K = Z ξ − ( ds ( Z ξ ) − e s ds ∧ dt ( Z ξ , v )) X λ = Z ξ + dz ∧ ds ( Z ξ , v ) X λ , since α − ( dz + e s dt ) = 0 when evaluated on vectors on Z and hence ( ds ∧ α + dz ∧ ds − e s ds ∧ dt )( Z ξ , v ) = 0 . (cid:3) At the zeros of α | Z and λ , Eq. (5.1.1) can be interpreted as saying that Z ′ ξ contains the limit of the right-hand side as the points on Z × W c approach thezero. Remark . Lemma 5.1.2 is rather general and works with Z replaced by anysurface in R z,s,t .5.1.2. Dynamics of Z ′ ξ . We now investigate the dynamics of Z ′ ξ .Let us first consider the PL case Z ′ P L = Z P L × W c . Lemma 5.1.4.
The flow lines of ( Z ′ P L ) ξ passing through {− } s × (0 , t ) t × W c eventually limit to a negative singularity of ( Z ′ P L ) ξ and in particular do not leave Z ′ P L .Proof.
The lemma follows from two observations: (i) Since dz ∧ ds ( Z ξ , v ) is pos-itive on P , negative on P , and vanishes on P ∪ P ∪ P , the dz ∧ ds ( Z ξ , v ) X λ term in Eq. (5.1.1) is zero on P ∪ P ∪ P , a negative multiple of X λ on P , and apositive multiple of X λ on P . (ii) By Lemma 3.1.4, if a flow line of ( Z ′ P L ) ξ passesthrough {− } s × (0 , t ) t × W c , then its projection to Z P L only passes through P , P , P , and P . (cid:3) Next we describe the smoothed version Z ′ ξ . We identify the singular points of Z ′ ξ : Recall from Lemma 3.1.4 that Z ξ has four singular points e ± , h ± . It turnsout that dz ∧ ds ( Z ξ , v ) is nonvanishing at these singular points. Hence for eachsingular point x ∈ W c of the Liouville vector field X λ , there exist four singularpoints e x ± , h x ± of Z ′ ξ whose Morse indices are given by:ind ( e x + ) = ind W ( x ) , ind ( h x + ) = ind W ( x ) + 1 , ind ( e x − ) = 2 n − ind W ( x ) , ind ( h x − ) = 2 n − − ind W ( x ) , where ind W ( x ) denotes the Morse index of x ∈ W c ⊂ W . ONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY 25
Let
Sk( W ) be the isotropic skeleton of W c with respect to X λ . FollowingProposition 3.2.1, let us define I a := {− } × [ κ, t + κ − a ] ⊂ R s,t , a ≥ . Then I = I has the property that any flow line of Z ξ passing through I convergesto a singularity of Z ξ in forward time.The following is an immediate consequence of Lemma 5.1.4 and taking the limit Z → Z P L : Lemma 5.1.5.
There exists a C -small function σ : W c → R ≥ , which vanishesexactly on Sk( W ) such that: • each flow line of Z ′ ξ passing through I × Sk( W ) converges to a singularityof Z ′ ξ in forward time; • for x ∈ W c \ Sk( W ) , each flow line passing through I σ ( x ) ×{ x } convergesto a negative singularity of Z ′ ξ in forward time; and • for x ∈ W c \ Sk( W ) , each flow line passing through ( I \ I σ ( x ) ) × { x } exits Z × W c along Z × ∂W c in finite time.Moreover, as Z → Z P L , the sup norm | σ | C limits to . Technically, the function σ accounts for the speed of convergence of flow linesof ( Z ) ξ towards its singularities and those of X λ in W c towards Sk( W ) .5.1.3. Rapid damping.
In order for the C -fold to be the image of a continuous map Σ → M , one must damp out the Z -factor in the product hypersurface Z × W c as τ grows. Here recall W = W c ∪ ([0 , ∞ ) τ × Γ) and τ is the symplectizationdirection. We also define W cτ := W c ∪ ([0 , τ ] × Γ) . The damping procedure amounts to choosing a 1-parameter family of surfacesin R z,s,t interpolating between Z and the flat R s,t , parametrized by τ ∈ [0 , τ ] .For “rapid damping” we take τ > to be arbitrarily small. The damping will bedone in two steps corresponding to τ ∈ [0 , τ ] and τ ∈ [ τ , τ ] , where < τ < τ and τ − τ ≪ τ (i.e., τ is sufficiently close to τ ).S TEP From Z to a bump. Let η := λ | Γ be the contact form on Γ . We then write λ = e τ η on [0 , ∞ ) × Γ .We will construct a hypersurface I := ∪ ≤ τ ≤ τ ( S τ × { τ } ) ⊂ R z,s,t,τ as the trace of an isotopy of surfaces from S = Z to a bump surface S τ . Theactual hypersurface in M will be I × Γ .The PL model of the isotopy S τ , τ ∈ [0 , τ ] , is induced by the isotopy P τ := [0 , s ] × [ − e ǫ z + Kτ, − e − s − ǫ z + t − Kτ ] ⊂ { z = z } , τ ∈ [0 , τ ] of P , for a suitable K > . Here S τ is defined as in § P replaced by P τ ,and we take K > so that the new P τ is smaller than the base (cid:3) . We write P τi , ≤ i ≤ , for the faces of S τ . The actual smooth isotopy is obtained by the usualcorner rounding. In particular we have T I = R h T S τ , ∂ τ + f ∂ t i , where f is a τ -dependent function on R z,s,t which is ≥ on P τ and ≤ on P τ andvanishes when τ is close to { , τ } . See Figure 5.1.1 for an illustration of the linearfoliations on the front ( P τ ) and back ( P τ ) faces. Compare with Figure 3.1.2.F IGURE P τ (left)and P τ (right).We are now ready to compute the characteristic foliation ( I × Γ) ξ . Let S τ,ξ be the characteristic foliation on S τ , i.e., α | R ( S τ,ξ ) = 0 , and let v be a partiallydefined vector field on S τ such that α | R ( v ) = 1 . Lemma 5.1.6.
The characteristic foliation ( I × Γ) ξ is given by ( I × Γ) ξ = S τ,ξ + dz ∧ ds ( S τ,ξ , v ) ( ∂ τ + f ∂ t ) − e − τ f dz ∧ ds ( S τ,ξ , v ) R η , (5.1.2) for most points of I × Γ .Proof. This is similar to the calculation of Lemma 5.1.2. We compute T ( I × Γ) ∩ ξ = R h S τ,ξ , e τ + s v − R η , ∂ τ + f ∂ t − f e − τ R η , ker η i , where R η is the Reeb vector field of η . Next we have α = dz + e s ( dt + e τ η ) ,e − s dα = ds ∧ dt + e τ ds ∧ η + e τ ( dτ ∧ η + dη ) . Setting X = S τ,ξ , Y = e τ + s v − R η , Z = ∂ τ + f ∂ t − f e − τ R η , h X, Y i = e τ + s ds ∧ dt ( S τ,ξ , v ) − e τ ds ( S τ,ξ ) = e τ ( dz ∧ ds ( S τ,ξ , v )) , since(5.1.3) ( ds ∧ α + dz ∧ ds − e s ds ∧ dt )( S τ,ξ , v ) = 0 as before. h X, Z i = f ds ∧ dt ( S τ,ξ , ∂ t ) + e τ ds ( S τ,ξ )( − f e − τ ) = f ds ( S τ,ξ ) − f ds ( S τ,ξ ) = 0 , h Y, Z i = e τ + s ( ds ∧ dt ( v, f ∂ t ) + e τ ds ( v )( − f e − τ )) + e τ = e τ . Hence the kernel (divided by e τ ) is ( I × Γ) ξ = S τ,ξ + dz ∧ ds ( S τ,ξ , v )( ∂ τ + f ∂ t − f e − τ R η )= S τ,ξ + dz ∧ ds ( S τ,ξ , v )( ∂ τ + f ∂ t ) − e − τ f dz ∧ ds ( S τ,ξ , v ) R η . (cid:3) ONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY 27
Note that Eq. (5.1.2) agrees with Eq. (5.1.1) at τ = 0 . Moreover ( I × Γ) ξ isnowhere-vanishing because the first two terms in Eq. (5.1.2) cannot simultaneouslyvanish. (At a singularity of S τ,ξ , we take a sequence of points limiting to thesingularity in order to interpret the right-hand side of Eq. (5.1.2). In that case, thesecond term on the right-hand side dominates the first.)We now describe the dynamics of ( Z ′ ∪ ( I × Γ)) ξ : Lemma 5.1.7.
There exists a C -small function σ : W cτ = W c ∪ ([0 , τ ] × Γ) → R ≥ which vanishes exactly on Sk( W ) such that: • σ < σ on W c \ Sk( W ) ; • for any x ∈ W cτ , each flow line passing through I σ ( x ) × { x } converges toa singularity of Z ′ ξ in forward time; and • for any x ∈ W cτ \ Sk( W ) , each flow line passing through ( I \ I σ ( x ) ) ×{ x } exits along S τ × ∂W cτ in finite time.Proof. We will describe the flow for the PL model; the lemma follows by smooth-ing in the usual way.In the PL model, any flow line passing through {− } s × (0 , t ) t × W cτ eventuallylimits to a negative singularity of ( Z ′ ∪ ( I × Γ)) ξ : Eq. (5.1.2) implies that anyflow line of ( I × Γ) ξ that intersects A := int( P τ ∪ P τ ∪ P τ ) × { τ } × Γ is tangentto A , i.e., has no τ -component. All flow lines passing through int( P τ ) × { τ } × Γ with the exception of those that go to ( P τ ∩ P τ ) × { τ } × Γ continue to B :=int( P τ ) × { τ } × Γ . Since dz ∧ ds ( S τ,ξ , v ) < on P τ , if a flow line enters B , thenit must be have negative ∂ τ + f ∂ t component. Moreover, since τ > is small,the flow line quickly flows into Z ′ = Z × W c and eventually limits to a negativesingularity.The situation for flow lines passing through { s + 1 } × (0 , t ) × W cτ is similar. (cid:3) Remark . Observe that the third term of Eq. (5.1.2) has a substantial contribu-tion in the R η -direction, since we are applying rapid damping and f is large. Thisis something we need to be careful about, but ultimately can be finessed away bystacking the C -folds in a particular way as in Section 7.S TEP From bump to flat R s,t . Consider the hypersurface I := ∪ τ ≤ τ ≤ τ ( S τ × { τ } ) ⊂ R z,s,t,τ , where S τ , τ ∈ [ τ , τ ] , is a 1-parameter family of surfaces such that S τ = R s,t and T I = R h T S τ , ∂ τ + f w i , where w = ∂ z − K∂ t , K > is a constant, and f ≤ is a τ -dependent functionon R z,s,t which vanishes when τ is close to { τ , τ } . Lemma 5.1.9.
The characteristic foliation ( I × Γ) ξ is given by: ( I × Γ) ξ = S τ,ξ + dz ∧ ds ( S τ,ξ , v )( ∂ τ + f w ) + f ( ds ( S τ,ξ ) v − ds ( v ) S τ,ξ )+ e − τ f ( − ds ∧ dt ( S τ,ξ , v ) + Kdz ∧ ds ( S τ,ξ , v )) R η . Proof.
The calculation is similar to the previous ones. We first compute T ( I × Γ) ∩ ξ = R h S τ,ξ , e τ + s v − R η , ∂ τ + f w + ( − e − τ − s + Ke − τ ) f R η , ker η i . As before we have e − s dα = ds ∧ dt + e τ ds ∧ η + e τ ( dτ ∧ η + dη ) . Setting X = S τ,ξ , Y = e τ + s v − R η , Z = ∂ τ + f w + ( − e − τ − s + Ke − τ ) f R η , h X, Y i = e τ + s ds ∧ dt ( S τ,ξ , v ) − e τ ds ( S τ,ξ ) = e τ ( dz ∧ ds ( S τ,ξ , v )) , h X, Z i = e τ ds ( S τ,ξ )( − e − τ − s f ) = − e − s f ds ( S τ,ξ ) , h Y, Z i = e τ + s ( e τ ds ( v )( − e − τ − s f )) + e τ = e τ (1 − f ds ( v )) . ( I × Γ) ξ = (1 − f ds ( v )) S τ,ξ + e − τ − s f ds ( S τ,ξ )( e τ + s v − R η )+ dz ∧ ds ( S τ,ξ , v )( ∂ τ + f w + ( − e − τ − s + Ke − τ ) f R η ) . A rearrangement of the terms gives the lemma. (cid:3)
The first two terms of ( I × Γ) ξ are analogous to those of ( I × Γ) ξ . The thirdterm f ( ds ( S τ,ξ ) v − ds ( v ) S τ,ξ ) lies in ker ds and, away from the corners, • vanishes on P ∪ P , • has negative ∂ t -component on P ∪ P , and • has positive ∂ t -component on P .See Figure 5.1.2. In other words, the third term, when we project out the s - and W -directions, is a flow in the clockwise direction around ∂P . As in Remark 5.1.8,the last term, i.e., the component in the R η -direction, is substantial, but will notmatter in the end. t sz F IGURE f ( ds ( S τ,ξ ) v − ds ( v ) S τ,ξ ) is de-picted in blue. ONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY 29
Definition 5.1.10.
The C -fold of Σ along H is defined to be the hypersurface (5.1.4) Z H := (cid:0) Σ \ ( (cid:3) × W cτ ) (cid:1) ∪ Z ′ ∪ ( I × Γ) ∪ ( I × Γ) , modulo corner rounding. The region (cid:3) × W cτ ⊂ Σ is called the base of Z H . By definition Z H = Σ away from the base.It remains to describe the dynamics of Z H,ξ . Suppose τ ′ ∈ ( τ , τ ) and σ : W cτ ′ → R ≥ , σ : W cτ ′ → R ≥ are functions satisfying the following: • σ vanishes exactly on Sk( W ) and σ < σ on W cτ \ Sk( W ) ; • σ vanishes on W cτ ; • on { τ ≤ τ ≤ τ ′ } , both σ = σ ( τ ) and σ = σ ( τ ) are strictly increasingand reach their maximum at τ = τ ′ ; • σ ( τ ′ ) + σ ( τ ′ ) = t .We then define the compact submanifold H in := { ( t, x ) | x ∈ W cτ ′ , κ + σ ( x ) ≤ t ≤ t + κ − σ ( x ) } which approximates H when all the parameters involved in the construction tendto . Similarly we define the compact submanifold H out approximating H , basedon outgoing flow lines. See Figure 5.1.3.Assuming all the parameters are sufficiently small, there exists a collar neighbor-hood [ − ǫ, ǫ ] ℓ × ∂H of the convex hypersurface ∂H (which we assume has roundedcorners) such that ∂H = { } × H , and ∂H in , ∂H out ⊂ [ − ǫ, ǫ ] × ∂H are graphicalover ∂H .We use the notation X ◦ (and also int( X ) ) to denote the interior of a space X . Proposition 5.1.11.
There exist H in and H out that approximate H and such that:(Z1) Any flow line of Z H,ξ that passes through H ◦ in ⊂ { s = − } converges toa negative singularity of Z H,ξ in forward time. Similarly, any flow line of Z H,ξ that passes through H ◦ out ⊂ { s = s + 1 } converges to a positivesingularity of Z H,ξ in backward time.(Z2) Any flow line of Z H,ξ that does not pass through H ∪ ([ − ǫ, ǫ ] × ∂H ) ⊂{ s = − } has trivial holonomy.(Z3) There exists a Morse function F on ∂H such that ∂H ξ is gradient-likefor F (and hence flows “from R + ( ∂H ) to R − ( ∂H ) ”) and such that anyflow line of Z H,ξ that passes through ( ℓ, x ) ∈ [ − ǫ, ǫ ] × ∂H ⊂ { s = − } and does not converge to a singularity of Z H,ξ passes through ( ℓ ′ , y ) ∈ [ − ǫ, ǫ ] × ∂H ⊂ { s = s + 1 } with F ( y ) ≥ F ( x ) . Construction of partial C -folds. Throughout this subsection ( W c , λ ) willbe a Weinstein cobordism from Γ − := ∂ − W c to Γ + := ∂ + W c . Let W := ([0 , ∞ ) × Γ + ) ∪ W c ∪ (( −∞ , × Γ − ) be the Weinstein completion of W c . Abusing notation, we also denote the Liouvilleform on W by λ . Then η ± := λ | Γ ± are contact forms on Γ ± , respectively. H in H out tκ t + κ τ τ Sk( W ) F IGURE H in and H out , respec-tively. The area of the complements of H in and H out in the rect-angles tend to 0 as all the parameters involved in the constructiontend to 0.Consider the contactization R t × W of ( W, λ ) with contact form β = dt + λ .Then we define the generalized contact handlebody H := [0 , t ] × W c ⊂ R × W with thickness t > . As in § M = R z,s,t × W with contact form α = dz + e s β . The hypersurface on which the partial C -foldwill be constructed is Σ = { z = 0 } .We define the partial C -fold modeled on H to be(5.2.1) Z PH := (cid:16) Σ \ ( (cid:3) × W cτ + ,τ − ) (cid:17) ∪ ( I + × Γ + ) ∪ Z ′ ∪ ( I − × Γ − ) . Let us explain the notation in (5.2.1). First (cid:3) = [0 , s ] × [0 , t ] ⊂ R s,t as usual; W cτ + ,τ − := ([0 , τ + ] × Γ + ) ∪ W c ∪ ([ − τ − , × Γ − ) is a slight extension of the compact Weinstein domain W c for τ ± > small; Z ′ = Z × W c is as before; the term I + × Γ defines a rapid damping of thepositive end, in particular τ + = τ , I + = I ∪ I in comparison to (5.1.4).The goal of this subsection is to construct the slow damping I − × Γ − on thenegative end [ − τ − , × Γ − .5.2.1. Slow damping.
Since we are given a finite size for the negative end, thefollowing assumption is crucial for our construction.
Assumption 5.2.1.
Assume for the rest of this subsection that the thickness t ≪ τ − . Recall from (TZ3) in Proposition 3.2.1 that for -dimensional C -folds Z , wehave t ≈ (1 − e − s ) z ≈ s z up to first order. It follows that we can also assume z , s ≪ τ − . This means that Z can be taken to be C -small compared to τ − .Hence there exists a τ -dependent vector field ν τ on R z,s,t satisfying the follow-ing properties:(F1) ν = ν − τ − = 0 .(F2) The time- τ flow of ν τ takes Z to the flat R s,t .(F3) With respect to the Euclidean metric on R z,s,t,τ , | ν | C ≪ | ∂ τ | .Consider the hypersurface I − := ∪ − τ − ≤ τ ≤ ( S τ × { τ } ) , ONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY 31 where S τ ⊂ R z,s,t , τ ∈ [0 , τ − ] is the image of Z under the time- τ flow of ν τ . Inpractice one can think of the 1-parameter family of surfaces S τ as being the sameas those considered in the rapid damping procedure in § R s,t (i.e., Condition (F3)). Lemma 5.2.2.
Under the assumption that | ν | is small, the characteristic foliation ( I − × Γ − ) ξ satisfies: (5.2.2) ( I − × Γ − ) ξ . = S τ,ξ + dz ∧ ds ( S τ,ξ , v ) ∂ τ , where . = means equality up to an error term which tends to as the size of Z tends to .Proof. First we have T ( I − × Γ − ) ∩ ξ = R h S τ,ξ , w − e s v + o ( ν ) , ker λ + o ( ν ) i , where o ( ν ) denotes possibly different vector fields on I − × Γ − which are on theorder of ν , and S τ,ξ denotes the characteristic foliation on S τ ⊂ ( R z,s,t , dz + e s dt ) .Then the same calculation as in Lemma 5.1.6 and Lemma 5.1.9 gives ( I − × Γ − ) ξ = S τ,ξ + dz ∧ ds ( S τ,ξ , v ) ∂ τ + | S τ,ξ | C o ( ν ) . Observe that dz ∧ ds ( S τ,ξ , v ) = 0 near the singularities of S τ,ξ . Hence the sumof the first two terms of ( I − × Γ − ) ξ is uniformly bounded away from . Thisimplies the lemma. (cid:3) Note that Eq. (5.2.2) agrees with Eq. (5.1.1) near Γ − .5.2.2. Dynamics of Z PH,ξ . Let W c be a Weinstein cobordism. If φ t : W ∼ → W isthe time- t flow of the Liouville vector field X λ , then the support Supp( W c ) andthe skeleton Sk( W c ) are defined as follows: Supp( W c ) = { x ∈ W c | lim t →−∞ φ t ( x ) ∈ W c } , Sk( W c ) = { x ∈ W c | lim t →±∞ φ t ( x ) ∈ W c } . For example, if W c is a Weinstein domain, i.e., ∂ − W c = ∅ , then Supp( W c ) = W c and Sk( W c ) is the usual isotropic skeleton. On the other hand, if W c has nocritical points, then Supp( W c ) = Sk( W c ) = ∅ .Since the dynamics of Z PH,ξ on Z ′ and I + × Γ + were already analyzed in § I − × Γ − .By Eq. (5.2.2), it suffices to describe the dynamics of S τ,ξ for each τ ∈ [ − τ − , .We assume that, after reparametrizing τ , the surfaces S τ ⊂ R agree with thesurfaces used in the rapid damping setting and that S τ,ξ is of generalized Morsetype for all τ . Fix an identification of S τ with R for all τ ∈ [ − τ − , . We canrewrite Eq. (5.2.2) as follows:(5.2.3) ( I − × Γ − ) ξ . = S τ,ξ + Ω τ ∂ τ , where Ω τ is a function on R which is positive (resp. negative) in a bounded regioncontaining the positive (resp. negative) singularities of S τ,ξ . The τ = 0 case is de-scribed by Proposition 3.2.1. Note that Ω τ may not vanish even if the correspond-ing S τ,ξ has no corresponding singularities. See Figure 5.2.1 for an illustration ofthe vector fields S τ,ξ , τ ∈ [ − τ − , , decorated by (the sign of) Ω τ .F IGURE S τ,ξ as τ goes from 0 to − τ − . The red (resp. blue) region indicates where Ω τ > (resp. Ω τ < ).We slightly change the definition of H so that H = [0 , t ] × W cτ + ,τ − and define ∂ + H := ∂H − ([0 , t ] × {− τ − } × Γ − . We consider the collar neighborhood [ − ǫ, ǫ ] ℓ × ∂ + H of the convex hypersurface ∂ + H , which we assume has rounded corners, such that ∂H = { } × H . If H in and H out are compact submanifolds whose boundaries approximate ∂H andagree with H near τ = − τ − , then we similarly define ∂ + H in and ∂ + H out . Also ∂ + H in , ∂ + H out ⊂ [ − ǫ, ǫ ] × ∂ + H are graphical over ∂ + H .The following is analogous to Proposition 5.1.11: Proposition 5.2.3.
There exist H in and H out that approximate H and such that:(Z1) Any flow line of Z PH,ξ that passes through H ◦ in ⊂ { s = − } either con-verges to a negative singularity of Z PH,ξ or passes through I − × Γ − ⊂ ONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY 33 { s = s + 1 } in forward time. Similarly, any flow line of Z PH,ξ that passesthrough H ◦ out ⊂ { s = s + 1 } either converges to a positive singularity of Z PH,ξ or passes through I − × Γ − ⊂ { s = − } in backward time.(Z2) Any flow line of Z PH,ξ that does not pass through H ∪ ([ − ǫ, ǫ ] × ∂ + H ) ⊂{ s = − } has trivial holonomy.(Z3) There exists a Morse function F on ∂ + H such that ∂ + H ξ is gradient-likefor F and such that any flow line of Z PH,ξ that passes through ( ℓ, x ) ∈ [ − ǫ, ǫ ] × ∂ + H ⊂ { s = − } and does not converge to a singularity of Z PH,ξ or pass through I − × Γ − ⊂ { s = s + 1 } passes through ( ℓ ′ , y ) ∈ [ − ǫ, ǫ ] × ∂ + H ⊂ { s = s + 1 } with F ( y ) ≥ F ( x ) .
6. Q
UANTITATIVE STABILIZATION OF OPEN BOOK DECOMPOSITIONS
Let M be a closed manifold of dimension n + 1 . An open book decomposition (OBD) of M is a pair ( B, π ) , where B ⊂ M is a closed codimension submanifoldand π : M \ B → S ⊂ C is a fibration such that π − ( e iθ ) is the interior of a compact codimension sub-manifold S θ ⊂ M with ∂S θ = B for any e iθ ∈ S . We call S θ , e iθ ∈ S , the pages of the OBD, and call B the binding .If ( M, ξ ) is a closed contact manifold, then an OBD ( B, π ) of M is ξ -compatible ,or just compatible , if there exists a contact form ξ = ker α (called an adapted con-tact form) such that the Reeb vector field R α of α is transverse to all the pages inthe interior and is tangent to B , and λ := α | B is a contact form on B . Following Assumption 1.2.2, we are assuming that all the pages of an OBD areWeinstein.
Recall that the α -action of a curve γ ⊂ M is A ( γ ) = R γ α . Definition 6.0.1.
Let ( B, π ) be a ξ -compatible OBD with an adapted contact form α . The action A ( B, π, α ) is (6.0.1) A ( B, π, α ) := sup γ ∈R ( α ) A ( γ ) , where R ( α ) is the set of Reeb chords γ in M \ S whose closures have endpointson S . The problem for us is that A ( B, π, α ) may be infinite if the actions of the Reebchords become unbounded near B . In § stronglyadapted α , which prohibits this from happening.One of the goals of this section is prove the following: Proposition 6.0.2 (Quantitative stabilization) . Let ( B, π ) be a compatible OBDof ( M, α ) and α be strongly adapted. Fix δ > . Suppose the binding ( B, α | B ) itself admits a compatible OBD ( B , π ) such that α | B is strongly adapted and theaction A ( B , π , α | B ) < δ . Then there exists an OBD ( B ′ , π ′ ) of ( M, α ′ ) with astrongly adapted contact form α ′ C -close to α such that A ( B ′ , π ′ , α ′ ) < δ . The notion of C -closeness will be made precise in § -dimensional contact topology, there exists a no-tion of stabilization of an OBD which involves changing the topology of the pageby a -handle attachment and composing the monodromy with a suitable Dehntwist; see [Etn06] for more details. This is not the kind of stabilization we areinterested in here. Instead, the stabilization which we will study in detail in § s : M → C and the stabilization, roughly speaking, amounts to consideringthe OBD determined by s k : M → C for some k ∈ Z > . In more geometric terms,a “large” part of the new page after stabilization consists of the union of k copies ofthe old page which are uniformly distributed with respect to the angular coordinateof the original OBD. It is therefore expected that, outside a small neighborhood N ǫ ( B ) of B , the action of the new OBD is roughly A ( B, π, α ) /k , which will besmall if k ≫ . Finally, the extension of the new OBD inside N ǫ ( B ) causes a greatdeal of complications, which corresponds to the fact that is never a regular valueof s k for k > .One can also view Proposition 6.0.2 as an inductive step where the inductionis on the dimension of the contact manifold. For example, the base case is when dim M = 1 ; in this case Proposition 6.0.2 holds trivially. The next case with dim M = 3 is also somewhat special since B = ∅ necessarily.The proof of Proposition 6.0.2 is given in § partial open book decompo-sitions (POBD’s) — is crucial to the proofs of our main theorems and is proved in § § Strongly adapted contact forms.
Let ( B, π ) be a ξ -compatible OBD. Wemay assume that there exists an adapted contact form α and a decomposition M = N ǫ ( B ) ∪ T φ , where N ǫ ( B ) and T φ are glued along their boundary and:(SA1) N ǫ ( B ) = D ( ǫ ) × B , where ǫ > is small, D ( ǫ ) = { r < ǫ } , and ( r, θ ) are polar coordinates on D ( ǫ ) , is a tubular neighborhood of the binding { } × B , with contact form(6.1.1) α | N ǫ ( B ) = (1 − c r ) λ + c r dθ where c , c > are constants.(SA2) On N ǫ ( B ) , S θ = { θ = θ } for all e iθ ∈ S .(SA3) T φ is the mapping torus of ( S ⋆ , φ ) , where S ⋆ = S ⋆ is a truncated page S ∩ ( M − N ǫ ( B )) and φ | ∂S ⋆ is a positive-time flow of the Reeb vectorfield R λ . Here S = S θ =0 . ONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY 35 On N ǫ ( B ) the Reeb vector field is given by(6.1.2) R α = R λ + ( c /c ) ∂ θ . Hence if γ is a Reeb chord in N ǫ ( B ) \ S , then(6.1.3) A ( γ ) = 2 πc /c < ∞ . Definition 6.1.1.
A contact form α adapted to ( B, π ) is strongly adapted to ( B, π ) if there exists ǫ > such that (SA1)–(SA3) hold. Given a strongly adapted contact form which takes the form of item (6.1.1) nearthe binding, one can easily rescale the r -coordinate such that c = 1 , which weassume to be the case from now on.If α is strongly adapted, then A ( B, π, α ) < ∞ . However, note that the con-dition (SA1), which is the most restrictive condition for strongly adapted contactforms, implies much more than the finiteness of A ( B, π, α ) . Indeed, it followsfrom Eq. (6.1.3) that all the Reeb chords in N ǫ ( B ) \ S have the same length. Onthe one hand, the introduction of strongly adapted contact form facilitates some ofthe computations in § § Definition 6.1.2.
Given a Riemannian manifold M , two contact forms α , α are C -close if both k α − α k C and k dα − dα k C , measured with respect to theRiemannian metric, are sufficiently small. Clearly our notion of C -closeness given in Definition 6.1.2 is strictly weakerthan the usual notion of C -closeness since we take the exterior derivatives ratherthan all the partial derivatives. This weaker notion gives us a bit more flexibility,while capturing the most important features of nearby contact forms as we explainin the following lemmas. Lemma 6.1.3.
Suppose ( B, π ) is a compatible OBD of ( M, ξ ) with an adaptedcontact form α . Then there exists a contact form α ′ which is C -close to α suchthat α ′ is strongly adapted to ( B, π ) .Proof. By assumption B ⊂ ( M, ξ ) is a contact submanifold and there exists atubular neighborhood N ǫ ( B ) = D ( ǫ ) × B such that π | N ǫ ( B ) is the projection ontothe D ( ǫ ) -factor. It follows that B p := { p } × B is contact for any p ∈ D ( ǫ ) if ǫ is sufficiently small.Note that the characteristic foliation on S θ ∩ N ǫ ( B ) has a nonzero ∂ r -componentfor any e iθ ∈ S . It follows that through any point x ∈ B = B there exists a -disk D x such that D x ∩ S θ is tangent to the characteristic foliation for all e iθ ∈ S , and moreover the family of disks D x , x ∈ B , varies smoothly in B . Usingthe disks D x we can reparametrize N ǫ ( B ) such that the characteristic foliation on S θ ∩ N ǫ ( B ) is parallel to ∂ r . Hence along each S θ ∩ N ǫ ( B ) we can write α | S θ ∩ N ǫ ( B ) as a symplectization of λ = α | B . This implies that(6.1.4) α | N ǫ ( B ) = F λ + Gdθ, where
F, G : N ǫ ( B ) → R ≥ satisfy:(i) F | B = 1 , G | B = 0 , and(ii) lim r → ∂ r Gr ≡ .Here (ii) is guaranteed by a suitable rescaling of D x for every x ∈ B such that thearea form ∂ r Gdr ∧ dθ ≡ rdr ∧ dθ along T B N ǫ ( B ) .We compute dα | N ǫ ( B ) = ( ∂ r F dr + ∂ θ F dθ + d B F ) ∧ λ + F dλ + d B G ∧ dθ + ∂ r Gdr ∧ dθ,R α | N ǫ ( B ) k ∂ r Gr R λ − ∂ r Fr ∂ θ + X, where X ∈ span { ∂ r , ker λ } and k means equality up to a positive scalar function.Since α is adapted by assumption, R α must have a positive ∂ θ -component for r > . It follows that ∂ r F < for r > .Finally, to obtain the strongly adapted α ′ , it suffices to pick < ǫ ′ ≪ ǫ andwrite α ′ | N ǫ ( B ) = F ′ λ + G ′ dθ such that • F ′ = (1 + c ) − c r and G ′ = r for r < ǫ ′ and some c , c > ; • F ′ = F and G ′ = G for r close to ǫ ; • ∂ r F ′ < and ∂ r G ′ > for all < r < ǫ .It is straightforward to check that α ′ is C -close to α and is strongly adapted. (cid:3) The following lemma characterizes another key feature of C -close contactforms which will be used repeatedly in this paper. Lemma 6.1.4. If α, α ′ are two C -close contact forms on M , then there exists adiffeomorphism φ : M ∼ → M isotopic to the identity and a function f ∈ C ∞ ( M ) which is C -close to the constant function such that φ ∗ ( α ′ ) = f α . The proof is a standard application of the proof of Gray’s theorem and is omitted.To wrap up this subsection, we note that the action A ( B, π, α ) is unstable under C -small perturbations of α . More precisely, for ǫ > sufficiently small, considerthe contact form α ′ := (1 + c ′ r ) λ + c ′ r dθ on N ǫ ( B ) . Then α is C -close to α ′ on N ǫ ( B ) if | c − c ′ | is small. This means that, in view of Eq. (6.1.3), A ( B, π, α ′ ) can be made arbitrarily large by taking c ′ > to be arbitrarily small.On the other hand, note that if α is C -close to α ′ , then R α is C -close to R α ′ .It follows that the action is stable under C -small perturbation of α when restrictedto M \ N ǫ ( B ) .6.2. Quantitative stabilization of OBD.
The goal of this subsection is to proveProposition 6.0.2.Let ( B, π ) be a compatible OBD of ( M, ξ ) and α be a strongly adapted contactform. Let N ǫ ( B ) ∼ = D ( ǫ ) × B be the ǫ -neighborhood of B such that α | N ǫ ( B ) sat-isfies (SA1)–(SA3). Here ǫ > is a small constant subject to conditions specifiedlater in the proof.We construct a map s : M → C as follows: First define s | N ǫ ( B ) : N ǫ ( B ) → D ( ǫ ) ⊂ C ONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY 37 as the projection to the first factor D ( ǫ ) ⊂ C , and then uniquely extend s to all of M by requiring that s restricts to constant maps on each S θ \ N ǫ ( B ) , ǫe iθ ∈ ∂D ( ǫ ) .Hence π = s/ | s | on M \ B . Strictly speaking, s is only piecewise smooth, but inwhat follows we will pretend that s is smooth since a smoothing can easily beconstructed. By definition B = s − (0) and is transversely cut out.We then consider the map s k : M → C for k ∈ Z > . Since is not a regularvalue of s k for any k > , we need to add a small perturbation term coming from B to s k . By assumption B also comes with a compatible OBD ( B , π ) such that λ := α | B is strongly adapted. Let s : B → C be the associated map, defined inthe same way as in the above paragraph . Pick a C ∞ -small nonincreasing bumpfunction ρ : [0 , ǫ ] → R ≥ with ρ (0) > supported on [0 , ǫ/ . Consider the map(6.2.1) s ( k ) := s k − ρ ( r ) s : M → C where s is first extended to N ǫ ( B ) by precomposing it with the projection ontothe second factor B , and then the cut-off function ρ ( r ) guarantees that ρ ( r ) s isglobally defined on M .We analyze the OBD ( B ( k ) , π ( k ) ) given by s ( k ) and the corresponding Reebdynamics in steps. Steps 1 and 2 give topological descriptions of the binding B ( k ) and the page S ( k ) and the remaining steps describe the compatibility with a suitably C -small perturbed α . We use the convention that the subscript (e.g., B , s , r )refers to subsets etc. of B that are analogous to those of M (e.g., B , s , r ).Note that besides the trivial case of dim M = 1 , the case dim M = 3 isslightly different from and substantially easier than the higher-dimensional casessince B = ∅ . We will point out such differences in the proof when applicable.S TEP The binding B ( k ) = s − k ) (0) . We can write s ( x ) = ( r ( x ) e iθ ( x ) , if x ∈ N ǫ ( B ) ,e iθ ( x ) , if x ∈ M \ N ǫ ( B ) , where e iθ ( x ) = π ( x ) for x / ∈ B . More concisely, we write s = re iθ with theunderstanding that r ( x ) = ǫ on M \ N ǫ ( B ) . Similarly we write s = r e iθ on B ,where r ( y ) ≡ ǫ for y ∈ B \ N ǫ ( B ) . Then(6.2.2) s ( k ) = r k e ikθ − ρ ( r ) r e iθ We claim that(6.2.3) B ( k ) = { r k e ikθ − ρ ( r ) r e iθ = 0 } is a k -fold branched cover of B with branch locus B . Clearly B ( k ) ∩ B = B and B ( k ) ∩ { ρ ( r ) = 0 } = ∅ . Therefore B ( k ) = { r = r k /ρ ( r ) , e iθ = e ikθ } . Thisimplies that, for each point in s − ( r , θ ) with r > , there exist k distinct valuesof ( r, θ ) for which s ( k ) = 0 . The choice to use the same ǫ for both M and B is superficial and does not affect the argument. Note that when dim M = 3 , we have dim B = 1 and r ≡ ǫ . Eq. (6.2.2) impliesthat the new binding B ( k ) is the closure of a k -strand braid around B .S TEP The page S ( k ) . It is an easy verification that the map π ( k ) = s ( k ) / | s ( k ) | : M \ B ( k ) → S , (6.2.4) π ( k ) = r k e ikθ − ρ ( r ) r e iθ | r k e ikθ − ρ ( r ) r e iθ | , is a submersion, and hence induces a smooth fibration.We analyze the page S ( k ) = π − k ) (1) , i.e., examine the solution set to:(6.2.5) r k e ikθ − ρ ( r ) r e iθ ∈ R ≥ . Observe that, since r k /ρ ( r ) is strictly increasing, there exists a unique a ∈ (0 , ǫ/ such that a k = ρ ( a ) ǫ ; moreover B ( k ) ⊂ { r ≤ a } .First consider P := S ( k ) ∩ { r > a } . In this case we always have r k > ρ ( r ) r since r ≤ ǫ . We claim that P ∼ = ∪ ≤ j One technical difficulty is that the Reeb vector field given by Eq. (6.1.2) is al-most never tangent to B ( k ) unless the Reeb flow of R λ on B \ B takes pages topages; see § dim M = 3 since dim B = 1 and B = ∅ in this case.) The maintask of this step therefore is to “synchronize” the Reeb flows on M \ B and B \ B .In particular, we apply a C -small perturbation to α | N ǫ ( B ) given by item (6.1.1)into the form given by Eq. (6.2.9).Fix δ > small as in the assumption of the proposition. Recall λ = α | B . Since λ is strongly adapted to ( B , π ) , we can write:(6.2.7) R λ = f ( r ) R λ + h∂ θ , where f ( r ) is supported in { r < ǫ } , f ( r ) ≡ for r ≤ ǫ/ , and h : B → R > is a function which restricts to a constant function on { r < ǫ } . The two terms inEq. (6.2.7) require some explanation: The issue is that the definition of the vectorfield ∂ θ on B requires a choice of a splitting of the exact sequence → T S ◦ → T ( B \ B ) → T S ◦ ( B \ B ) → , where S ◦ denotes the interior of the page. For r ≤ ǫ/ , we choose the splittinginduced by the product structure N ǫ/ ( B ) = D ( ǫ/ × B . On B \ N ǫ ( B ) ,we use the splitting T ( B \ N ǫ ( B )) = R h R λ i ⊕ T S ⋆ induced by λ . Here S ⋆ = S \ N ǫ ( B ) denotes the truncated page as before. Finally we interpolate betweenthe two splittings on { ǫ/ ≤ r ≤ ǫ } , e.g., one can use the linear interpolation with suitable smoothing at the endpoints. The function f depends on the choice of theinterpolation.The above definition of ∂ θ has the advantage that the time- t flow of ∂ θ takesany page S ,θ = c to S ,θ = c + t . Note also that although min( h ) > does not needto be large, the “average” (or rather the integral) of h along each Reeb chord in B \ S must be large because A ( B , π , λ ) < δ with δ > small by assumption.Since α is strongly adapted to ( B, π ) , there exists a small tubular neighborhood N ǫ ( B ) = B × D ( ǫ ) of B such that α | N ǫ ( B ) = (1 − c r ) λ + r dθ . Fix < ǫ ′ ≪ ǫ and perturb α such that(6.2.8) α | N ǫ ( B ) = F ( x, r ) λ + r dθ, where F : N ǫ ( B ) → R > depends on x ∈ B and r and satisfies:(i) F ( x, r ) = 1 − c r near r = ǫ ;(ii) F ( x, r ) = 1 − gr for r ≤ ǫ ′ , where g : B → R > will be specified later;(iii) F ( x, r ) is strictly decreasing with respect to r for any x ∈ B .Here (iii) guarantees that α | N ǫ ( B ) is contact and is adapted to ( B, π ) .For simplicity of notation, we will identify ǫ ′ with ǫ from now on. Then(6.2.9) α | N ǫ ( B ) = (1 − gr ) λ + r dθ, Using Eq. (6.2.7) we compute: dα | N ǫ ( B ) = (1 − gr ) dλ − r dg ∧ λ − grdr ∧ λ + 2 rdr ∧ dθ,R α | N ǫ ( B ) = R λ + g∂ θ + v = f ( r ) R λ + h∂ θ + g∂ θ + v, where v is the unique vector field tangent to η = ker λ solving the equation(6.2.10) (1 − gr ) i v dλ + r d η g = 0 . Here d η g = dg − dg ( R λ ) λ . Since both g and d η g are bounded, | v | is small as longas r is sufficiently small. Using the splitting T ( B \ B ) = R h ∂ θ i ⊕ T S , we canwrite v = ˜ v + µ∂ θ such that ˜ v is tangent to S . It follows that R α | N ǫ ( B ) = f ( r ) R λ + ˜ v + ( h + µ ) ∂ θ + g∂ θ . Now choose k ∈ Z > such that A ( B, π, α ) /k < δ . The choice of k dependsonly on A ( B, π, α ) and, in particular, not on ǫ . We then choose g = h/k . Writing ˜ h = h + µ , we have:(6.2.11) R α | N ǫ ( B ) = f ( r ) R λ + ˜ v + ˜ h∂ θ + g∂ θ . The C -norm k ˜ h − kg k = k µ k → as r → . In fact, we have k µ k = O ( r ) for small r by Eq. (6.2.10). Moreover, note that ˜ h = kg on { r < ǫ } since h is constant there, and ˜ v ( r ) ≡ . It follows from Eq. (6.2.3) that R α | N ǫ ( B ) istangent to B ( k ) ∩ { r < ǫ } . Unfortunately, since ˜ h = kg in general, R α | N ǫ ( B ) is not everywhere tangent to B ( k ) . In fact, the first-order PDE h = kg − µ , whosesolution would have solved the problem, has no solutions in g for general h . Wewill deal with this technical issue in the next few steps. In § k will also depend on A ( B , π , α | B ) in order to make the resulting OBDdamped. ONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY 41 As a motivation for the above construction, we state the following, which isproved in Step 5: Claim 6.2.1. There exists a small tubular neighborhood of the stabilized binding B ( k ) , away from which we have (6.2.12) A ( B ( k ) , π ( k ) , α ) ≤ max( A ( B , π , λ ) , A ( B, π, α ) /k ) < δ. S TEP − . At this point we clarify the order in which choose the constants. Given δ > , we choose k > such that A ( B, π, α ) /k < δ . Then we choose ǫ > .Finally we choose a small ρ ( r ) so that a > satisfying a k = ρ ( a ) ǫ is much smallerthan ǫ . For convenience we will assume that ρ is constant on a small neighborhoodof r = a .S TEP The stabilized binding B ( k ) is contact. The goal of this step is to show B ( k ) ⊂ ( M, ξ ) , as constructed in Step 1, is acontact submanifold. It follows immediately that a small tubular neighborhood of B ( k ) is foliated by contact submanifolds since the contact condition is open. Wewill prove Claim 6.2.2 which estimates the size of such a neighborhood.In the following we calculate modulo error terms of order O ( a ) . Starting from α satisfying Eq. (6.2.9), we obtain: α = λ + O ( a ) , dα = dλ − grdrλ + 2 rdrdθ + O ( a ) , (6.2.13) α ∧ dα n − = λ ∧ ( dλ n − + ( n − rdrdθdλ n − ) + O ( a ) . (6.2.14)First consider B ( k ) ∩ { r = a } , which is topologically a k -fold cover of B \ N ǫ ( B ) . Since the contact condition is local, it suffices to examine a neighborhood O p ( S ⋆ ) ⊂ B \ N ǫ ( B ) of a (truncated) page S ⋆ . Since r is constant, Eq. (6.2.14)becomes α ∧ dα n − | O p ( S ⋆ ) = λ ∧ dλ n − + O ( a ) , (6.2.15)which implies that B ( k ) ∩ { r = a } is contact.Next consider B ( k ) ∩ { r < a } , which is topologically B × D . Workinglocally, fix a point ( r, θ ) ∈ D ( a ) (the disk of radius a centered at the origin). Ituniquely determines a pair ( r , θ ) such that the equation r k e ikθ − ρ ( r ) r e iθ = 0 holds, i.e., θ = kθ, r = r k /ρ ( r ) . Hence on a neighborhood O p ( B ) ⊂ B ( k ) of B , identified with the set of points in B ( k ) ∩ { r < a } with fixed ( r, θ, r , θ ) ,Eq. (6.2.14) and its analog for λ yield α ∧ dα n − | O p ( B ) = 2( n − λ ( r dr dθ dλ n − + rdrdθdλ n − ) + O ( a ) . (6.2.16)Writing r dr dθ in terms of drdθ , it is immediate that B ( k ) ∩ { r < a } is contact. Claim 6.2.2. There exists a small constant c > such that for ρ (and hence a > )sufficiently small, the tubular neighborhood N ca ( B ( k ) ) of B ( k ) of size ca in N ǫ ( B ) is foliated by contact submanifolds. Here c depends only on the original contact form α and ca measures the distancein the D ( ǫ ) -direction for r = ǫ . One easily computes that s ( k ) maps N ca ( B ( k ) ) ∩{ r = ǫ } (more or less) to D ( kca k ) ⊂ C , so we take the definition of N ca ( B ( k ) ) to be s − k ) ( D ( kca k )) . Proof. Consider { s ( k ) = w }∩{ r = ǫ } for | w | ≤ kca k . Observe that the variationsin the D ( ǫ ) -direction go to zero as a → . Hence { s ( k ) = w } ∩ { r = ǫ } , | w | ≤ kca k , are contact for a > small, since the second term in Eq. (6.2.14) goesto zero.Next we consider N ca ( B ( k ) ) ∩ { r < ǫ } . If | w | ≤ c , then we can write theparallel copies { s ( k ) = w } ∩ { r < ǫ } as a graph: r e iθ = (1 /ρ ( r ))( r k e ikθ − w ) and by writing r dr dθ in terms of drdθ in Eq. (6.2.16) we see that { s ( k ) = w } ∩ { r < ǫ } is contact. (cid:3) S TEP Transversality away from B ( k ) . As observed in Step 4, R α is almost never completely tangent to the binding B ( k ) and transverse to the interior of the pages S ◦ ( k ) , unless h is constant. By Step3, when r < ǫ , h is constant by assumption and the desired tangency/transversalityconditions on R α hold. Hence we assume that r = ǫ throughout this step.We fix S ( k ) to be the page at angle . Then R α is transverse to S ◦ ( k ) when ds ( k ) ( R α ) = i ( kgr k e ikθ − ˜ hρ ( r ) r e iθ ) = i ˜ hs ( k ) − iµr k e ikθ (6.2.17)has positive i R -component. All the other pages can be treated in the same way. Case r < a . In this case r k < ρ ( r ) r since r = ǫ . Suppose that kθ ∈ ( − π/ , π/ . Observe that, in Figure 6.2.2 (a), kθ and θ rotate in “opposite”directions; namely, as sin kθ increases, sin θ decreases. This implies that R α istransverse to S ◦ ( k ) in view of Eq. (6.2.17).If kθ ∈ [ π/ , π/ , then kθ and θ rotate in the same direction, as illustratedin Figure 6.2.2 (b). Here the transversality may fail, but there exists a small collarneighborhood of B ( k ) ⊂ S ( k ) such that R α is transverse to S ( k ) away from thecollar, at least when r < a .More precisely, transversality holds outside of N ca ( B ( k ) ) from Claim 6.2.2:This follows from the estimate of k ˜ h − kg k = k µ k = O ( a ) from Step 3 andthe comparison of the terms k ˜ hs ( k ) k = O ( a k ) and k µr k e ikθ k = O ( a k +2 ) on theright-hand side of Eq. (6.2.17). Case r = a . A slice of S ( k ) in this case, in terms of angular coordinates, is illus-trated in Figure 6.2.2 (c). Then R α is transverse to S ◦ ( k ) ∩ { r = a } for the samereason as in the case of Figure 6.2.2 (a). ONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY 43 (a) (b)(c) (d)F IGURE r k < ρ ( r ) r with θ ∈ ( − π/ , π/ .(b) The case r k < ρ ( r ) r with θ ∈ [ π/ , π/ . (c) The case r k = ρ ( r ) r , where the north and south poles correspond to apoint in the binding. (d) The case r k > ρ ( r ) r . The dashed lineis horizontal and pointing to the right since S ( k ) is identified withthe page at angle . Case r > a . This case is illustrated in Figure 6.2.2 (d). The situation is similarto that of Figure 6.2.2 (b), i.e., there exists a small collar neighborhood of B ( k ) ⊂ S ( k ) , away from which R α is transverse to S ( k ) .Summarizing, R α is transverse to S ( k ) ∩ ( M − N ca ( B ( k ) )) for all the pages S ( k ) .Claim 6.2.1 now readily follows from the observation that the maximal actionof Reeb chords of ( B ( k ) , π ( k ) ) in N ǫ ( B ) \ N ca ( B ( k ) ) is approximately equal to A ( B , π , λ ) .S TEP Transversality near the binding B ( k ) . Since it is not easy to write down an explicit contact form whose Reeb vectorfield is transverse to the pages and tangent to the binding, we proceed through adifferent route. Again, since the transversality is already achieved for r < ǫ , weassume r = ǫ throughout this step.Let B ⋆ ( k ) := B ( k ) ∩ { r = a } be the k -fold cyclic cover of B \ N ǫ ( B ) and let S ⋆ := S \ N ǫ ( B ) be a truncated page of ( B , π ) . We write θ ( k )1 ∈ R / kπ Z forthe angular coordinate in B ⋆ ( k ) . Consider the restriction of N ca ( B ( k ) ) to N ca ( B ⋆ ( k ) ) , viewed as a disk bundle p : N ca ( B ⋆ ( k ) ) → B ⋆ ( k ) whose fibers are disks ⊂ C r,θ ofradius ca centered at ae iθ ( k )1 /k for ≤ θ ( k )1 ≤ kπ .Our goal is to adjust the contact form on N ca ( B ⋆ ( k ) ) such that the Reeb vectorfield becomes compatible with the OBD ( B ( k ) , π ( k ) ) . It’s helpful to keep in mindthe dim M = 3 case, i.e., when S ⋆ is a point. If we trivialize N ca ( B ⋆ ( k ) ) ≃ ( R /C Z ) z × D x,y × S ⋆ , using the framing consistent with ∂N ǫ ( B ) , then R α = ∂ z + O ( a ) on ∂N ca ( B ⋆ ( k ) ) ,in view of Eq. (6.2.11). With respect to this trivialization, the pages S ( k ) intersect ∂N ca ( B ⋆ ( k ) ) along S ⋆ times a curve that winds − k times around the meridian and +1 times in the R /C Z -direction. Then Claim 6.2.2 implies that α | N ca ( B ⋆ ( k ) ) can bewritten as α | N ca ( B ⋆ ( k ) ) = F ( x, y, z )( dz + σ ) + xdy − ydx, where F ( x, y, z ) depends on S ⋆ and R α | N ca ( B ⋆ ( k ) ) k ∂ z + ( ∂ y F ) ∂ x − ( ∂ x F ) ∂ y , where ∂ x F and ∂ y F are small. Now we can replace F by another function G suchthat (a) F = G near ( R /C Z ) × ∂D × S ⋆ ;(b) k F − G k C is small;(c) G is almost constant on ( R /C Z ) × D × S ⋆ and constant near x = y = 0 .In view of the above description of the pages S ( k ) ∩ N ca ( B ⋆ ( k ) ) , the new contactform C -approximates α | N ca ( B ⋆ ( k ) ) and is strongly adapted.S TEP Weinstein structure on the page S ( k ) . In this step we describe the Weinstein structure on S ( k ) , which we fix to be thepage at angle . The other pages can be treated similarly. We will decompose S ( k ) = T ∪ T ∪ T into three pieces and study the characteristic foliation on eachpiece separately. Note that our decomposition of S ( k ) here will be different from,but based on, the one from Step 2. In this step, we distinguish between ǫ and ǫ ′ , as described right below Eq. (6.2.8). First let T := S ( k ) ∩{ r ≥ ǫ ′ } = ∪ ≤ j In Section 5, we constructed C -folds based on contacthandlebodies. One rich source of contact handlebodies is given by the sectors ofa compatible OBD, i.e., the region bounded between two pages. This observationwill be explored extensively in Section 7. However, one drawback of using thesesectors is that all the sectors meet at the binding and become a little crowded. In thissubsection we explain how to slightly shift the binding by C -small perturbationsof the contact form.Since the issue is completely local near the binding, we restrict to ( D ( ǫ ) × B, α = (1 − x − y ) λ + xdy − ydx ) , where λ is a contact form on B and ( x, y ) are Euclidean coordinates on D ( ǫ ) .Pick an interior point ( x , y ) ∈ D ( ǫ ) . Let f = f ( x, y ) be a function on D ( ǫ ) satisfying the following conditions: • f = 1 − ( x − x ) − ( y − y ) on a small neighborhood of ( x , y ) ; • f is strictly decreasing along each ray emanating from ( x , y ) ; • f = 1 − x − y near ∂D ( ǫ ) .Consider α ′ = f λ + xdy − ydx . Since dα ′ = ( ∂ x f dx + ∂ y f dy ) ∧ λ + f dλ + 2 dx ∧ dy, the Reeb vector field is given by: R α ′ = R λ + (( ∂ y f ) ∂ x − ( ∂ x f ) ∂ y ) f − x∂ x f − y∂ y f . It follows that α ′ is adapted to the new OBD ( B ′ , π ′ ) with binding B ′ = { ( x , y ) }× B and whose pages restrict to γ × B ′ , where γ ⊂ D ( ǫ ) are smoothings of rays em-anating from ( x , y ) so they are tangent to ∂ r on a small neighborhood of ∂D ( ǫ ) . ONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY 47 Quantitative stabilization of a POBD. The definition of a POBD in this pa-per is slightly different from the original -dimensional definition from [HKM09]and the higher-dimensional analog from [HHa]. Our model here is closer in spiritto those in [CGHH11]. Definition 6.4.1. Let ( M, ξ ) be a contact manifold with boundary and concavecorners and let Σ = ∂M . Then Σ is a sutured concave boundary with suture [0 , × Γ if it admits the decomposition Σ = R + ∪ ([0 , t × Γ) ∪ R − such thatthe following hold:(SCH1) M has concave corners along { , } × Γ ;(SCH2) R ± are compact manifolds with boundary, ∂R + = { } × Γ and ∂R − = { } × Γ , and the orientation of Σ = ∂M agrees with that of R + and isopposite that of R − . Moreover the t -coordinate can be extended to collarneighborhoods of R ± such that R + = { t = 0 } and R − = { t = 1 } ;(SCH3) On a neighborhood of Σ , a contact form for ξ can be written as α = dt + β ,up to a positive constant scalar, so that R ± is a Weinstein domain withrespect to the Liouville form β ± := β | R ± and β | Γ is a contact form on Γ . We are now ready to define a POBD for contact manifolds with sutured concaveboundary. Definition 6.4.2. Let ( M, ξ ) be a compact contact manifold with sutured concaveboundary. A generalized compatible partial open book decomposition of ( M, ξ ) isa pair ( B, π ) (see Figure 6.4.1), where(1) The binding B ⊂ int M is a closed codimension contact submanifold;(2) π : M \ B → S is a submersion whose fibers are Weinstein cobordismsthat are completed at the positive end but not at the negative end;(3) Each of R + and R − is contained in some (possibly the same) page S θ := π − ( e iθ ) , e iθ ∈ S . Moreover, if the negative end ∂ − S θ of S θ is nonempty,then it is contained in the suture.A generalized compatible partial open book decomposition is a compatible partialopen book decomposition if, in addition, ∂ − S θ = {∗} × Γ , {∗} ∈ (0 , , wheneverit is nonempty. If we replace the phrase “Weinstein cobordisms” by “Weinstein manifolds” inDefinition 6.4.2, then we recover the usual compatible OBD of a closed contactmanifold. Remark . Unlike in the closed case, the topology of S θ may change withdifferent choices of θ . Remark . In [HKM09] and [HHa], the binding partially lies on ∂M as partof the dividing set Γ ∂M and the pages are all Weinstein manifolds with a par-tially defined monodromy. While the setup of [HKM09] and [HHa] facilitates thecorrespondence between bypass attachment and modification of open books (cf.Section 9), it is nearly impossible to stabilize such partial open books in the styleof § R + R − F IGURE single page; the red part repre-sents the suture; and the black dot at the center represents a neigh-borhood of the binding.of M and noting that all the constructions in § § Definition 6.4.5. Suppose ( B, π ) is a generalized compatible POBD for ( M, ξ ) .A contact form α is strongly adapted to ( B, π ) if it satisfies the conditions ofDefinition 6.1.1 near B and ∂M is sutured concave with respect to α . In particular, if α is strongly adapted, then R α is outward-pointing along R + ( ∂M ) ,inward-pointing along R − ( ∂M ) , and tangent to ∂M along the suture. Definition 6.4.6. Given compatible POBD ( B, π ) with a strongly adapted contactform α , the action A ( B, π, α ) is defined by Eq. (6.0.1), where R ( α ) is the set ofReeb orbits of M which start from one page, end on another, and wind around thebinding at most once. Finally we state the relative version of Proposition 6.0.2, which is almost iden-tical to Proposition 6.0.2. In fact, the proof of Proposition 6.0.2 also carries oversince the only tricky part is contained in a small neighborhood of the binding,which makes no difference whether ∂M is empty or not. Proposition 6.4.7. Let ( B, π ) be a generalized compatible POBD of ( M, α ) and α be strongly adapted. Suppose the binding ( B, α | B ) itself admits a compatibleOBD ( B , π ) such that α | B is strongly adapted and the action A ( B , π , α | B ) < ONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY 49 δ . Then there exists a generalized compatible POBD ( B ′ , π ′ ) of ( M, α ′ ) with astrongly adapted contact form α ′ C -close to α such that A ( B ′ , π ′ , α ′ ) < δ .Remark . Even if we start with a compatible POBD ( B, π ) in Proposition 6.4.7,which will be the case in our applications, the resulting ( B ′ , π ′ ) will necessarily bea generalized compatible POBD given δ sufficiently small. Indeed, in this case, all the pages will have nonempty negative ends.It is worthwhile noting that the main assumption of Proposition 6.4.7, i.e., theestimate A ( B , π , α | B ) < δ , can only be achieved by (inductively) applyingProposition 6.0.2 to B , which is by itself a closed contact manifold.6.5. Damped OBD and POBD. As we have seen in § C -small in the sense of Definition 6.1.2,since we need to drastically change the lengths of Reeb chords, especially awayfrom the binding. In a sense the construction in this subsection is orthogonal to theone in § C -folds constructed there are embedded. The cases of OBD and POBDwill be considered separately.6.5.1. The OBD case. Consider a closed contact manifold ( M, ξ ) with a compati-ble OBD ( B, π ) and a strongly adapted contact form α . Pick any point x ∈ M \ B and consider the flow segment γ x : ( − ǫ, ǫ ) t → M \ B of R α such that γ x (0) = x and ˙ γ x ( t ) = R α . Since α is adapted, π ◦ γ x : ( − ǫ, ǫ ) → S = R / π Z is a smoothembedding, at least for ǫ sufficiently small. Define ρ : M \ B → R > by ρ ( x ) = ddt (cid:12)(cid:12) t =0 ( π ◦ γ x ) . Roughly speaking, ρ ( x ) measures infinitesimally how fast the Reeb flow through x traverses the pages. Definition 6.5.1. The infinitesimal variation on the interior of each page S ◦ θ is V θ := max x ∈ S ◦ θ ρ ( x ) / min x ∈ S ◦ θ ρ ( x ) ∈ [1 , ∞ ) , and the total infinitesimal variation by V := max θ ∈ [0 , π ] V θ . Note that V θ and V are well-defined if α is strongly adapted since ρ is constantnear B . Definition 6.5.2. A strongly adapted contact form is damped if V < / . Given a compatible OBD ( B, π ) with a damped contact form α , one can as-sume, up to a reparametrization of the angular coordinate θ , that ρ is almost con-stant. In particular, it implies that the actions of Reeb chords from one page S ◦ θ toanother S ◦ θ +2 π/k are bounded between k A ( B, π, α ) and k A ( B, π, α ) . Such areparametrization will be implicitly assumed in what follows. To construct a damped contact form adapted to an open book, it is more con-venient to work with abstract open books , whose construction we recall now: Let ( S, η ) be a Weinstein domain and let R t × S be its contactization with contactform α = dt + η . Consider an exact symplectomorphism φ : S ∼ → S such that φ = id near ∂S and φ ∗ ( η ) = η + dF for some F ∈ C ∞ ( S ) which vanishes near ∂S . Choose a constant C > such that F + C > . One then constructs anabstract open book from ( S, φ ) by assembling the pieces N ( B ) = D × B (here D = D (1) ) and the mapping torus T φ,C together in the obvious way, where T φ,C := { ( t, x ) ∈ R × S | ≤ t ≤ F ( x ) + C } / (0 , φ ( x )) ∼ ( F ( x ) + C, x ) . Then ∂T φ,C = R /C Z × ∂S . We extend α to N ( B ) by α | N ( B ) = f ( r ) λ + g ( r ) dθ, where ( r, θ ) ∈ D are the polar coordinates and f, g satisfy the following condi-tions:(1) There exists ǫ > small such that α | N ǫ ( B ) satisfies (SA1)–(SA3);(2) The contact condition ( f ′ , g ′ ) · ( − g, f ) > for r > ;(3) f ( r ) λ = η and g ( r ) ≡ C/ (2 π ) for r close to . In particular θ = 2 πt/C along { r = 1 } .The resulting (closed) contact manifold is denoted by M ( S,φ ) . Note that M ( S,φ ) depends on some choices including the choice of C . For simplicity, in what followswe will suppress C from the notation.Let T • φ = { ( t, x ) ∈ R × S | ≤ t ≤ F ( x ) + C } ⊂ R × S be the fundamental domain in the infinite cyclic cover. We foliate T • φ by pages S t := graph ( h t ) , t ∈ [0 , C ] , where h t : S → R are functions varying smoothly with respect to t and satisfy thefollowing conditions:(1) h = 0 , h C = F + C ;(2) h t ( x ) < h t ( x ) for any t < t and x ∈ S ;(3) h t ≡ t near ∂S .It follows that S t , t ∈ [0 , C ] , are the truncated pages of a compatible OBD of M ( S,φ ) . See Figure 6.5.1. Moreover each S t is naturally a Liouville domain withLiouville form λ + dh t . We remark that in general there is no guarantee that the S t are all Weinstein even if S and S C are Weinstein. The above discussion motivates the following assumption on the abstract openbook formulation, which is implicitly contained in Assumption 1.2.2. Assumption 6.5.3. For any abstract open book M ( S,φ ) as above, we assume theLiouville form λ + dh t defines a Weinstein structure on S t for all t ∈ [0 , C ] . Inparticular, the Weinstein structures defined by λ and φ ∗ λ are Weinstein homotopic. The following lemma states that any strongly adapted contact form can be iso-toped to a damped one. ONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY 51 tC F IGURE T • φ by the pages S t . Lemma 6.5.4. Suppose ( B, π ) is a compatible OBD of ( M, ξ ) and α is a stronglyadapted contact form. Then there exists another strongly adapted contact form α ′ ,whose kernel is isotopic to ξ , such that ( B, π ) is damped with respect to α ′ .Proof. Let ( S, φ ) be an abstract open book representing ( B, π ) . We first discussthe mapping torus part of the open book. Starting with the fundamental domain T • φ = { ( t, x ) ∈ R × S | ≤ t ≤ F ( x ) + C } ⊂ R × S foliated by S t = graph ( h t ) , t ∈ [0 , C ] , the trick is to take k ≫ , and consider anew angular variable τ ∈ [0 , kC ] . Now the graph of the functions H τ := h τ/k + k − k τ, τ ∈ [0 , kC ] defines a foliation on the mapping torus T kφ := { ( t, x ) ∈ R × S | ≤ t ≤ F + kC } / (0 , φ ( x )) ∼ ( F ( x ) + kC, x ) The Liouville form on each S kτ := graph ( H τ ) coincides with that on S τ/k , whichimplies that all the pages S kτ are Weinstein. Now observe that ˙ H τ = k ˙ h τ/k + k − k → uniformly as k → ∞ . Hence we can choose a large k such that T kφ is damped withrespect to the contact form α k | T kφ := dτ + λ .Now we extend α k to N ( B ) . By construction we can write α k = f k ( r ) λ + g k ( r ) dθ, near ∂D × B , where f k ( r ) = f ( r ) and g k ( r ) = kg ( r ) for r close to . Define g k := kg for r ∈ [0 , . Assume without loss of generality that g is constant for r ∈ [ , and is strictly increasing for r ∈ [0 , ] . We extend f k to r ∈ [0 , as a strictly decreasing function in three steps as follows. Fix ǫ > small. Firstextend f k to r ∈ [ , arbitrarily; then to r ∈ [ − ǫ, ] such that f ′ k ≤ − cg ′ k , withequality near r = − ǫ , where c > is a constant such that f k /c ≪ g k (1) for all r ∈ [ − ǫ, ] ; and finally to r ∈ [0 , − ǫ ] such that f ′ k = − cg ′ k holds. The reader might find it helpful to note that the curve { ( f k ( r ) , g k ( r )) | r ∈ [0 , } is close tothe line segment connecting ( cg k (1) , and (0 , g k (1)) .We claim that the total infinitesimal variation on N ( B ) with respect to α k | N ( B ) is less than / for ǫ sufficiently small. Indeed, the Reeb vector field is given by R α k | N ( B ) = g ′ k R λ − f ′ k ∂ θ f k g ′ k − f ′ k g k . The coefficient of ∂ θ is equal to g k (1) on r ≥ and is equal to cg ′ k f k g ′ k + cg ′ k g k = cf k + cg k ≈ g k (1) on r ≤ − ǫ . On − ǫ ≤ r ≤ , we can estimate | f k g ′ k /f ′ k | ≤| f k /c | ≪ g k (1) , which implies that the coefficient − f ′ k f k g ′ k − f ′ k g k of ∂ θ is close to g k (1) .The claim follows.The contact structure ker α ′ := ker α k is isotopic to ξ by varying the parameter k . Moreover α ′ is clearly strongly adapted by construction. (cid:3) Since the isotopy given by Lemma 6.5.4 is not necessarily C -small, for ourlater applications, we will need the fact that dampedness is preserved under quanti-tative stabilizations in the sense of § Lemma 6.5.5. In the hypothesis of Proposition 6.0.2, assume in addition that both ( B, π ) and ( B , π ) are damped and δ ≪ / . Then we can arrange so that thestabilized OBD ( B ′ , π ′ ) is also damped with respect to a C -small perturbation of α ′ .Proof. We continue to use the notation from § ( B ′ , π ′ ) =( B ( k ) , π ( k ) ) for k ≫ .In this proof we distinguish between ǫ and ǫ ′ , as described right below Eq. (6.2.8).On the region M − N ǫ ( B ) , the page S ( k ) restricts to ∪ ≤ j The POBD case is, in principle, the same as the OBDcase. The only difference is that we need to pay extra attention to the boundary forour later applications. ONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY 53 Let ( M, ξ ) be a compact contact manifold with sutured concave boundary Σ .As always, assume Σ ξ is Morse + , or equivalently, R ± (Σ) are Weinstein. As in theclosed case, it will be convenient to work with abstract (partial) open books. Description of abstract partial open book. The initial data, usually denoted by ( S, W, φ ) , consists of the following:(AP1) a Weinstein domain ( S, η ) ;(AP2) a subordinated Weinstein cobordism ( W, β ) ⊂ ( S, η ) in the sense that ∂ + W = ∂S and β = η | W ;(AP3) a partial monodromy map φ : W → S which restricts to the identity mapon ∂ + W ;(AP4) an auxiliary function F : S → R which vanishes near ∂S , such that φ ∗ ( η + dF ) = β , and a constant κ > .Choose a constant C such that F − C < . We first construct the partial mappingtorus T φ := { ( t, x ) ∈ (( −∞ , × S ) ∪ ([0 , ∞ ) × W ) | F ( x ) − C ≤ t ≤ κ } / ∼ , ( F ( x ) − C, φ ( x )) ∼ ( κ, x ) ∀ x ∈ W. The contact form α on T φ is given by dt + η for t ≤ and dt + β for t ≥ . Nowthe abstract partial open book is obtained from T φ by filling in N ( B ) = D × B as in the closed case, where B ∼ = ∂S . Here ∂D is identified with R / ( C + κ ) Z via the identification of variables θ = 2 πt/ ( C + κ ) . Finally the extension of α to N ( B ) and the foliation of T φ by graphical Weinstein pages are identical to theclosed case. In particular, an obvious variation of Assumption 6.5.3 applies to theabove constructed abstract partial open books. Lemma 6.5.6. Any contact manifold which admits a POBD is contactomorphic toan abstract partial open book.Proof. Note that the above construction of an abstract partial open book requires [0 , κ ] × W ⊂ T φ to be foliated by Weinstein pages {∗} × W , ∗ ∈ [0 , κ ] , withidentical Liouville -forms.On the other hand, by the definition of a compatible POBD (Definition 6.4.2),there exist a function G : W → R which is locally constant near ∂W and satisfies φ ∗ ( η + dF ) = β + dG , a constant C ′ > such that G + C ′ > , and a mappingtorus T ′ φ := { ( t, x ) ∈ (( −∞ , × S ) ∪ ([0 , ∞ ) × W ) | F ( x ) − C ≤ t ≤ G ( x ) + C ′ } / ∼ , ( F ( x ) − C, φ ( x )) ∼ ( G ( x ) + C ′ , x ) ∀ x ∈ W, such that ( M, ξ ) is obtained from T ′ φ by filling N ( B ) .To get from T ′ φ to T φ we increase C ′ so that G + C ′ ≫ κ , move { ( t, x ) ∈ [0 , ∞ ) × W | κ < t ≤ G ( x ) + C ′ } to the ( −∞ , × S side, and extend each page (which is a graph over φ ( W ) ) byattaching the appropriate {∗} × ( S − φ ( W )) . In other words, we can “absorb G into F ”. One can verify Assumption 1.2.2 holds for T φ . (cid:3) Under the above identification between a POBD of ( M, ξ ) and an abstract partialopen book determined by ( S, W, φ ) , we have R + = S \ W, R − = S \ φ ( W ) , and [0 , × Γ = [0 , × ∂ − W using the notation of Definition 6.4.1. Remark . Lemma 6.5.6 provides us with a key property of abstract partialopen books: the portion [0 , × W is already damped and moreover the infinitesi-mal variation on any page in [0 , × W is exactly .Note that Definition 6.5.1 and Definition 6.5.2 carry over to the case of compat-ible POBDs since the infinitesimal variation is locally constant on the suture.We are now ready to state the relative analog of Lemma 6.5.4, which will play akey role in the construction of plugs in Section 8. Lemma 6.5.8. Suppose ( M, ξ ) is a compact contact manifold equipped with acompatible POBD ( B, π ) and α is a strongly adapted contact form. Then thereexists another strongly adapted contact form α ′ such that • ker α ′ is isotopic to ξ ; • ( B, π ) is damped with respect to α ′ ; • Σ ξ = Σ ker α ′ , where Σ = ∂M .Proof. Fix an abstract partial open book ( S, W, φ ) for ( B, π ) . By construction, α | { F − C ≤ t ≤ } = dt + η and α | { ≤ t ≤ G + C } = dt + β . Let α k = α = dt + β on { ≤ t ≤ G + C } and let α k = dt + η on { F − kC ≤ t ≤ } . It follows fromthe proof of Lemma 6.5.4 that we can extend α k to N ( B ) so that α k is damped.Finally we map α k to α ′ via a diffeomorphism which fixes ∂M . (cid:3) We conclude this subsection with the analog of Lemma 6.5.5 with the sameproof. Lemma 6.5.9. In the hypothesis of Proposition 6.4.7, assume in addition that both ( B, π ) and ( B , π ) are damped and δ ≪ / . Then we can arrange so that theresulting generalized compatible POBD is also damped with respect to α ′ . 7. A TOY EXAMPLE Consider the contact manifold M = R r,θ × Y with contact form α = η + r dθ ,where η is a contact form on Y . For each a > , consider the hypersurface Σ a := S a × Y , where S a = { r = a } ⊂ R . In this section, we show, as a warm-up to thefollowing section, how to make Σ a convex by an explicit C -small perturbation.In principle, the constructions in this section, together with the analysis of by-pass attachments, should give a way to study the relationship between contact struc-tures on Y and M . An outstanding question in this direction is the following: Question 7.0.1. Is ( M, α ) tight whenever ( Y, η ) is tight? See [CMP19, HCMMP] for closely related works. ONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY 55 The -dimensional case. In this case Y is a transverse knot and up to rescal-ing we can write α = dz + r dθ , where z ∈ R / Z ∼ = Y .It is well-known that Σ a can be C ∞ -approximated by a convex torus e Σ a suchthat R ± ( e Σ a ) ∼ = T ∗ S as Weinstein manifolds. Moreover the slope dz/dθ of the -section S ⊂ T ∗ S ⊂ e Σ a tends to as a → . Such S is called a Legendrianapproximation of the transverse knot Y . It is a good exercise to C -perturb Σ to aconvex torus using the techniques developed in Section 3.7.2. A Peter-Paul contactomorphism. Let ( Y, η ) be a contact manifold with afixed choice of contact form η . Let S be a hypersurface of Y transverse to theReeb vector field R η . Then S has a neighborhood S × [ − ǫ, ǫ ] τ ⊂ Y on which R η = ∂ τ .The following is well-known: Lemma 7.2.1. If R η = ∂ τ on S × [ a, b ] τ ⊂ Y , a < b , then η = dτ + β , where β is the pullback of a -form on S . Moreover, dβ is symplectic on S . In other words, η is the contactization of ( S, β ) . In particular, if ( S, β ) is Wein-stein then S × [ a, b ] is a contact handlebody. Proof. We first write η = f dτ + β , where f ∈ Ω ( S ) and β ( τ ) ∈ Ω ( S ) . Since η ( R η ) = 1 , we have f = 1 . Also, since L R η η = 0 , β ( τ ) must be τ -independent.Finally, dβ is symplectic on S due to the contact condition on Y . (cid:3) Now consider the contactization of the symplectization of ( Y, η ) given by ( M = R z,s × Y, α = dz + e s η ) . Let φ t : Y ∼ → Y be the time- t flow of R η . Lemma 7.2.2. The diffeomorphism Ψ : R × Y ∼ −→ R × Y, (7.2.1) ( z, s, y ) (cid:16) e ( − /C ) s · Cz, s/C, φ (1 − C ) e − s z ( y ) (cid:17) , where C > , is a contactomorphism.Proof. We compute Ψ ∗ ( α ) = d ( e ( − /C ) s · Cz ) + e s/C ( η + d ((1 − C ) e − s z ))= e ( − /C ) s (1 − C ) zds + e ( − /C ) s Cdz + e s/C η − e s/C (1 − C ) e − s zds + e s/C (1 − C ) e − s dz = e ( − /C ) s ( dz + e s η ) = e ( − /C ) s α. We explain the first line: By Lemma 7.2.1, η can locally be written as dτ + β ,where β is a -form on a hypersurface S ⊂ Y transverse to R η = ∂ τ . Then φ (1 − C ) e − s z ( τ, x ) = ( τ + (1 − C ) e − s z, x ) , where x is the coordinate on S , and dφ ∗ (1 − C ) e − s z η = η + d ((1 − C ) e − s z ) . (cid:3) As an immediate corollary, by taking C ≫ , we have: Lemma 7.2.3. Let M ( z,s ) = [0 , z ] × [0 , s ] × Y be a contact submanifold in ( M, α ) .Then for any < s ′ ≤ s , there exists < z ≤ z ′ , such that M ( z ,s ) contactlyembeds into M ( z ′ ,s ′ ) . We call the contactomorphism Ψ given by (7.2.1) a Peter-Paul contactomor-phism for the following reason: In Lemma 7.2.3, Σ = { } × [0 , s ] × Y ⊂ M ( z,s ) is the hypersurface on which we want to create C -folds. The length of the interval [0 , z ] can be regarded as the given size of a neighborhood of Σ . Then the terminol-ogy comes from the desire to rob the (already small) size of the neighborhood of Σ to pay for a large size in the s -direction.To explain why a large size in the s -direction is desirable, note that not everycontact form η , even up to a C ∞ -small perturbation, is compatible with an openbook decomposition. However, according to Corollary 1.3.1, there exists anothercontact form η ′ = e f η on Y which is compatible, where f is a smooth functionon Y . Now if we let Y ′ = { s = f } ⊂ Σ be the graph of f , then clearly α | Y ′ is a contact form on Y ′ which is compatible with some open book decomposition.Then the machinery that we developed in Section 6 can be applied to most of theargument that will be carried out in the rest of this section as well as in Section 8.Observe that the Peter-Paul contactomorphism is not needed in Section 4 tomake any -dimensional surface convex. We do not know whether ingredients suchas the Peter-Paul contactomorphism and the plug to be constructed in Section 8 arereally necessary or just reflect our ignorance regarding the nature of convexity.7.3. The higher-dimensional case. Recall the setup: Y n − is a closed contactmanifold with contact form η and ( M = R r,θ × Y, ξ = ker α ) , α = η + r dθ , is acontact manifold of dimension n + 1 . Consider the hypersurface Σ a = S a × Y for a > . Then Σ a,ξ = ∂ θ − a R η , where R η is the Reeb vector field of η on Y .The goal of this subsection is to fold Σ a in such a way that Σ a,ξ becomes Morse + .Slightly abusing notation, let Y ∼ = { pt } × Y ⊂ Σ a . We emphasize the spe-cial feature of this example which is that Y ⊂ Σ a is a global transversal, i.e., anyflow line of Σ a,ξ passes through Y . Hence it suffices to fold Σ on a neighbor-hood N ( Y ) ⊃ Y such that no (broken) flow line of the new characteristic foliationpasses through N ( Y ) . For a general hypersurface in a contact manifold, one can-not hope to find such a global transversal and we resort to a more local blockercalled the Y -shaped plug (where Y is compact contact manifold with boundary) inSection 8.To carry out the above plan, note that, by Lemma 2.0.3, Y admits a neighbor-hood in ( M, ξ ) which is contactomorphic to ( U := [0 , z ] × [ − ǫ, s + ǫ ] × Y, ker( dz + e s η )) , for some z , s > and ǫ > , such that N ( Y ) ∼ = { } × [ − ǫ, s + ǫ ] × Y underthis identification. We assume that the values of z , s , ǫ are fixed. We denote ∂ − N ( Y ) := {− ǫ } × Y and ∂ + N ( Y ) := { s + ǫ } × Y .Assume the following inductive step holds: Corollary 1.3.1 holds for any dimension ≤ n − . ONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY 57 By Lemma 7.2.3 and the discussion immediately after it, we can assume that η is strongly adapted to a compatible OBD ( B, π ) of ( Y, ker η ) . Next, by the dis-cussion in § ( B, π ) is damped with respect to R η . ByProposition 6.0.2, we can assume, up to a C -small perturbation of η , that aftera suitable stabilization the action A := A ( B, π, η ) is small with respect to thefixed z and s . Finally, the dampedness condition is preserved by the stabilizationaccording to Lemma 6.5.5.For the rest of this subsection, the small constants ǫ , ǫ , . . . that we use satisfy min( z , s ) ≫ ǫ ≫ ǫ ≫ · · · > . The idea to make Σ a convex by folding N ( Y ) is as follows: First cover Y by afinite number — say, six as in the -dimensional case in § sectors of thecompatible OBD, where a sector refers to a region bounded between two pages.Since each sector can be identified with a contact handlebody in the sense of § C -folds based on the sectors as constructed in § C -folds in the s -direction (the order is veryimportant!), then the flow lines of the perturbed hypersurface e Σ a are completelyblocked by the folded N ( Y ) , and therefore e Σ a,ξ is Morse.The argument consists of three steps.S TEP Dividing the OBD into sectors. Let S θ , θ ∈ S ∼ = R / π Z , be the pages of the open book ( B, π ) for Y and letus identify a tubular neighborhood of B with D ( ǫ ) × B . By § C -small perturbation η ′ of η which shifts the binding to ( − ǫ / , × B . Let b S = S ∪ ([ − ǫ / , × { } × B ) be the truncated page at angle for the shifted OBD, which contains the originalbinding. Let N ǫ ( b S ) be the collar neighborhood of S of width ǫ with respectto the Reeb flow of η ′ , i.e., any point in N ǫ ( b S ) is connected to b S along an arctangent to R η ′ of length ≤ ǫ ; see Figure 7.3.1. In particular N ǫ ( b S ) is a contacthandlebody in the sense of § H .Returning to the original OBD ( B, π ) with the contact form η , by constructionthere exists a smaller neighborhood D ( ǫ ) × B of B which is contained in H .Define the truncated pages S ∨ θ := S θ − ( D ( ǫ ) × B ) ; we also write S ∨ = S ∨ .Let π ′ : Y − ( D ( ǫ ) × B ) → S be the restriction of π such that ( π ′ ) − ( θ ) = S ∨ θ . Consider five truncated sectors σ j , ≤ j ≤ , defined by σ j := ( π ′ ) − [ ( j − π , ( j +1) π ] . Let ∂ ± σ j be the maximal subsets of ∂σ j transverse to R η such that R η flows from ∂ − σ j to ∂ + σ j .Each sector σ j contains some contact handlebody H j of thickness (cf. § A/ by the dampedness assumption and Definition 6.5.2. We now explainhow to pick specific H j . Let H ± j be the contact handlebody of maximal thick-ness in σ j such that ∂ − H − j = ∂ − σ j and ∂ + H + j = ∂ + σ j , respectively. Here we F IGURE b S . The shadedregion represents H = N ǫ ( b S ) .use the usual convention that R η is inward-pointing along ∂ − H ± j and outward-pointing along ∂ + H ± j . Intuitively, ∂ + H + j is “straight” in the sense that it is tan-gent to a page, while ∂ − H + j is “wiggly” since it is not tangent to a page in general.Similarly, ∂ − H − j is straight and ∂ + H − j is wiggly. We then set H = H +1 and H j = H − j for ≤ j ≤ . Clearly Y = ∪ ≤ j ≤ H j .We explain the reason for our choices of H j . Observe that the overlap betweenadjacent H j , ≤ j ≤ , measured in terms of the θ -coordinate, is approximately π/ . The overlap between H (or H ) and H however can be very complicateddepending on whether we represent H by H +1 or H − , since the thickness of H is much smaller than A . This motivates the choices H = H +1 and H = H − ; thesigns for H j , ≤ j ≤ can be chosen arbitrarily.S TEP Folding the sectors. By assumption and Proposition 6.0.2, A ≪ min( z , s ) . Using § C -folds Z − j , ≤ j ≤ , with bases B − j := [ j , j +111 ] s × H − j , and rapid damping from § e Σ a be the resulting hypersurface. Remark . The rapid damping is necessary since there is not enough room forthe slow damping near the binding.Note that Z can obviously be constructed since the thickness of H is muchsmaller than A . Next consider Z . Referring to Figure 3.1.1, where the t -coordinatecorresponds to the θ -coordinate here, the θ -width of the base B is the θ -thicknessof H , which is bounded from above by π/ . It follows from Assumption 3.1.5and the dampedness condition that the θ -width of Z (i.e., the θ -width of the top ONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY 59 face of Figure 3.1.1) is at most π/ < π . Hence Z is embedded. The sameargument applies to Z j , ≤ j ≤ . s θ F IGURE H , . . . , H , from top to bottom.S TEP Verification that the characteristic foliation e Σ a,ξ is Morse. Claim 7.3.2. There exists no broken flow line from ∂ − N ( Y ) to ∂ + N ( Y ) .Proof of Claim 7.3.2. This is a direct consequence of Proposition 5.1.11 and ourchoice of the ordering of the C -folds in the s -direction.We first give names to regions of ∂ − N ( Y ) : Viewing H j as a subset of ∂ − N ( Y ) ,let e H j be the closure of the union of H j and the set of points such that the holonomyfrom s = − j − ǫ to − j + ǫ (for ǫ > small) is not trivial or does not exist;note that e H j is contained in a small neighborhood of H j . We denote the portion of e H j that closely approximates it and acts as a sink by H j, in and e H j − H j, in by H j,∂ .The dynamics of e Σ a,ξ is described as follows:(a) If x ∈ H j, in , j = 0 , . . . , , and is not in H i, in or H i,∂ for i > j , then the flowline ℓ x of e Σ a,ξ passing through x converges to a singularity in Z j .(b) If x ∈ H j,∂ , j = 1 , . . . , , and is not in H i, in or H i,∂ for i > j , then the flowline ℓ x exits Z j at y near ∂H j and one of the following will happen:(1) ℓ x follows ∂ s until { s = 10 / } , and converges to a singularity in Z ;(2) ℓ x follows ∂ s until H j − , in ⊂ { s = − j } , and then converges to a singu-larity in Z j − ;(3) ℓ x follows ∂ s until H j − ,∂ ⊂ { s = − j } . If (3) holds, then we inductively apply (b) with j − instead of j . Also, if j = 1 ,then only (1) holds. (cid:3) A similar analysis as above can be applied to any point x ∈ N ( Y ) to yield a listof all the possibilities for the behavior of the flow line ℓ x passing through x in bothforward and backward time. If e Σ a,ξ ( x ) = 0 , then the forward flow of ℓ x eitherconverges to a singularity of e Σ a,ξ contained in { s ≥ s ( x ) } or reaches ∂ + N ( Y ) infinite time. Similarly, the backward flow of ℓ x either converges to a singularity of e Σ a,ξ contained in { s ≤ s ( x ) } or reaches ∂ − N ( Y ) in finite time.In light of Proposition 2.1.3 and Lemma 2.2.2, we conclude that e Σ a,ξ is Morse,and hence can be made Morse + and convex after a C ∞ -small perturbation. Remark . In fact e Σ a,ξ can be explicitly described on N ( Y ) using the tech-niques of Section 12: There exist two disjoint copies Y i , i = 1 , , of Y containedin the interior of N ( Y ) , which can be identified with { s = i s } × Y , respectively,such that e Σ a,ξ is tangent to Y i , i = 1 , . Moreover, the vector field e Σ a,ξ | Y on Y isMorse and the zeros of e Σ a,ξ | Y consist of the following:(1) negative singularities on Z , . . . , Z ;(2) half of the positive singularities on Z corresponding to the positive saddlein Proposition 3.2.1; and(3) half of the negative singularities on Z corresponding to the negative sinkin Proposition 3.2.1.All the other zeros of e Σ a,ξ are contained in Y . Finally, in the ( -dimensional)normal directions to Y i , i = 1 , , Y is attracting, Y is repelling, and e Σ a,ξ flowsfrom Y to Y in the region bounded between Y and Y .8. C ONSTRUCTION OF THE PLUG The goal of this section is to generalize the -dimensional plug constructed in § -dimensional case.Let us rephrase the -dimensional case considered in § ( R , ker( dz + e s dt )) and the surface Σ = { z = 0 } ⊂ R . The plug isobtained by “wiggling” Σ in a box U = [0 , s ] × [0 , t ] , where we are viewing U as the truncated symplectization of the -dimensional compact contact manifold ∂ − U = { } × [0 , t ] with contact form dt .In higher dimensions, let ( Y, ker β ) be a compact contact manifold of dimension n − with (not necessarily convex) boundary. Let ( N ǫ ( Y ) := Y ∪ ([0 , ǫ ] × ∂Y ) , ker β ) be a small extension of ( Y, ker β ) . Now we consider ( M n +1 := R z,s × N ǫ ( Y ) , ξ = ker( dz + e s β )) ONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY 61 and the hypersurface Σ := { z = 0 } . Let U := [0 , s ] × N ǫ ( Y ) and let ∂ − U := {− } × N ǫ ( Y ) and ∂ + U := { s + 1 } × N ǫ ( Y ) . From now on, we fix a Riemannian metric on M , which induces a metric on anysubmanifold. Definition 8.0.1. A Y -shaped plug is a C -small perturbation e U of U supportedin the interior U ◦ of U such that:(1) all the flow lines of e U ξ that pass through {− } × Y ◦ flow to a negativesingularity,(2) all the flow lines of e U ξ that pass through { s + 1 } × Y ◦ flow from a positivesingularity,(3) for all flow lines of e U ξ that go from ∂ − U to ∂ + U , the holonomy map is ǫ -close to the identity when defined. We now give an outline of the construction of a Y -shaped plug: First, we slightlyenlarge Y and apply the inductive assumption to C -perturb ∂Y such that ( ∂Y ) ξ is Morse + and, moreover, any smooth trajectory of ( ∂Y ) ξ is short with respect tothe metric. We then generalize the construction from § Y so that Definition 8.0.1(1) and (2) hold. The purpose of the first step is toguarantee Definition 8.0.1(3).8.1. ǫ -convex hypersurfaces. In this subsection we strengthen Theorem 1.2.3 ina quantitative way. Namely, in addition to the requirement that Σ ξ be Morse + ,we also require all the smooth flow lines of Σ ξ to be short. Here is the formaldefinition. Definition 8.1.1. A closed hypersurface Σ ⊂ ( M, ξ ) is ǫ -convex if Σ ξ is Morse + and the length of any smooth trajectory of Σ ξ is shorter than ǫ with respect to theinduced metric on Σ . Of course any closed convex hypersurface Σ is ǫ -convex for sufficiently large ǫ which depends on Σ . For our purposes we take ǫ > to be a small numberwhich is independent of the choice of convex hypersurface. Theorem 1.2.3 can bestrengthened as follows: Theorem 8.1.2. Given ǫ > , any closed hypersurface in a contact manifold canbe C -approximated by an ǫ -convex one. Theorem 8.1.2 holds in dimension by § Construction of the Y -shaped plug. We make the following inductive as-sumption: Theorem 8.1.2 and Theorem 11.0.3 hold for any dimension ≤ n − . Let Y be a compact contact manifold with boundary as before. We assume that ∂Y is ǫ -convex after a C -small perturbation. The goal of this subsection is toprove the following: Theorem 8.2.1. A Y -shaped plug exists.Proof. The hypersurface under consideration is U = [0 , s ] × N ǫ ( Y ) with Liouvilleform e s β . Choose z > sufficiently small.We first apply the Peter-Paul contactomorphism to [0 , z ] × U so that ∂Y be-comes sutured concave in the sense of Definition 6.4.1 with respect to the rescaledcontact form, which we still denote by β , on N ǫ ( Y ) . Strictly speaking, since ∂Y = ∅ , one must be careful when applying (7.2.1) as Ψ can potentially mapa point in Y outside N ǫ ( Y ) . This disaster, however, can be avoided by assumingthat z is sufficiently small with respect to ǫ and the initial contact form β .Theorem 11.0.3, together with another application of the Peter-Paul contacto-morphism, allows us to assume that β is compatible with a POBD ( B, π ) of Y such that ∂Y remains ǫ -convex. Here B ⊂ Y ◦ is the binding and π : Y \ B → S is a fibration whose fibers are naturally Weinstein cobordisms. Finally we applyLemma 6.5.8 to assume that ( B, π ) is damped while keeping ∂Y ǫ -convex.We remark that this is the end of our freedom to rescale the contact form usingthe Peter-Paul contactomorphism. For the rest of the proof the values of s , z willbe fixed and the contact form can only be C -small perturbed using Lemma 6.1.4.Note also that at this stage, the action A ( B, π, β ) can be very large compared to min( s , z ) .We continue to use the same convention for small constants as § ǫ ≫ ǫ ≫ ǫ ≫ · · · > . Following the conventions from Definition 6.4.1, there existsa decomposition ∂Y = R + ∪ ([0 , ǫ ] × Γ) ∪ R − , where R ± are Weinstein domainsand Γ = ∂R ± is the contact boundary.Next we define b R − := ([ − ǫ , ǫ ] × Γ) ∪ R − , modulo corner rounding and a slight extension. Note that by definition the Reebvector field R β is transverse to R − and tangent to [ − ǫ , ǫ ] × Γ . We choose a C ∞ -small perturbation of β such that R β becomes transverse to b R − . Still writing β forthis perturbation, b R − admits a collar neighborhood H := N ǫ ( b R − ) = [ − ǫ , ǫ ] τ × b R − on which R β = ∂ τ , i.e, it is a contact handlebody. Construction of U ∨ . The Y -shaped plug U ∨ consists of one C -fold Z and sixpartial C -folds Z , . . . , Z , which are analogous to the six C -folds constructed in § I − j := [ js , (2 j +1) s ] ⊂ [0 , s ] , ≤ j ≤ . The C -fold Z is constructed with base I × H ; Z has rapid damping at thepositive end.To construct the partial C -folds Z j , ≤ j ≤ , with base I j × H j , we ap-ply Proposition 6.4.7 and Lemma 6.5.9 to the triple ( B, π, β ) . After a C -smallperturbation of β , the stabilized POBD, still denoted by ( B, π, β ) , is damped and A ( B, π, β ) = ǫ . The construction of Z j , ≤ j ≤ , is essentially identical tothe construction in § H is the contact handlebody which ONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY 63 contains the binding and H j , ≤ j ≤ , are ordered in a clockwise manner aroundthe binding. The only difference is that we require slow damping (cf. § ǫ ≪ ǫ . Verification of the dynamics. It remains to verify that U ∨ indeed blocks flow linesfrom passing through a region approximating Y and only affects the holonomy onthe unblocked part by a small fluctuation. Let us define e H j ⊃ H j , j = 1 , . . . , , asin Step 3 of § Z j , j = 1 , . . . , , we define H j, in as the set ofpoints that limit to a singularity in Z j or to I − × Γ − corresponding to H j , whenwe flow from s = (12 − j ) s to s = (13 − j ) s , and let H j,∂ = e H j − H j, in .By the argument in Step 3 of § x ∈{− } × N ǫ ( Y ) , the flow line ℓ x of U ∨ ξ passing through x (i) reaches { (12 − j ) s } × H j, in and limits to a singularity of Z j ,(ii) stays near [0 , s ] × R + , or(iii) reaches { s = 12 s / } at some point x ′ ∈ H , in .If (iii) holds, then ℓ x continues from x ′ to a singularity in Z . In view of (ii), theonly flow lines that do not limit to a singularity of U ∨ travel from R + to R − closeto ∂Y . Finally, the ǫ -convexity of ∂Y implies that the flow of U ∨ ξ does not move x ∈ {− } × N ǫ ( Y ) more than ǫ in the N ǫ ( Y ) -direction. (cid:3) 9. T HREE DEFINITIONS OF THE BYPASS ATTACHMENT The goal of this section is to relate certain codimension degenerations ofMorse + hypersurfaces to bypass attachments introduced in [HHa]. Such a corre-spondence is fundamental in bridging the more dynamical approach [Gir91, Gir00]and the more combinatorial approach [Hon00] of convex surface theory in dimen-sion . Unfortunately the details never existed in the literature.To facilitate the exposition, we slightly repackage the Morse theory on Morsehypersurfaces from Section 2 in terms of folded Weinstein hypersurfaces .9.1. Definitions and examples. In this subsection we define folded Weinstein hy-persurfaces and look at a few examples. Definition 9.1.1. An oriented hypersurface Σ ⊂ ( M, ξ ) is a folded Weinsteinhypersurface if the characteristic foliation Σ ξ satisfies the following properties:(FW1) There exist pairwise disjoint closed codimension submanifolds K i ⊂ Σ , i = 1 , . . . , m − , which cut Σ into m pieces, i.e., Σ = W ∪ K · · · ∪ K m − W m , where W i are compact with boundary. We also set K , K m = ∅ . We call K i the folding loci of Σ .(FW2) The singular points of Σ ξ in each W i have the same sign, and the signchanges when crossing K i . We assume the singular points in W are posi-tive. (FW3) There exists a Morse function f i on each W i such that K i − and K i areregular level sets and ( W i ) ξ is gradient-like with respect to f i . In particu-lar, Σ ξ is transverse to all the K i . W W W . . .W m − W m Σ ξ + − ++ − F IGURE Σ . The top arrows indicate the direction of the Liouvillevector fields on each W i and the bottom arrow indicates the direc-tion of the characteristic foliation.Observe that if Σ = W ∪ · · · ∪ W m ⊂ ( M, ξ ) is a folded Weinstein hyper-surface, then there exists a contact form α for ξ whose restriction to the interior ofeach W i defines a Weinstein cobordism. Moreover, the orientation on W i given bythe Weinstein structure agrees with (resp. is opposite to) the orientation inheritedfrom Σ if the singular points of ( W i ) ξ are positive (resp. negative). We say a fold-ing locus K i is maximal (resp. minimal ) if the Liouville vector fields on W i and W i +1 are pointing towards (resp. away from) K i .Note that by definition any folded Weinstein hypersurface is Morse, and anyMorse hypersurface can be equipped with the structure of a folded Weinstein hy-persurface.We end this subsection with examples of folded Weinstein hypersurfaces andexplain why they are called “folded”. Example . If Σ is a convex hypersurface such that R ± (Σ) are Weinstein man-ifolds, then Σ is a folded Weinstein hypersurface. The folding locus coincideswith the dividing set Γ Σ and is maximal. A folded Weinstein hypersurface is notalways convex because there may exist flow trajectories of Σ ξ from a negative sin-gularity to a positive one. Nevertheless, since a C ∞ -small perturbation of a Morsehypersurface is Morse + by Lemma 2.2.2, any folded Weinstein hypersurface is C ∞ -generically convex by Proposition 2.2.3. Example . Consider ( R n +1 , ξ std ) with contact form α = dz + P ni =1 r i dθ i .The unit sphere S n is convex with respect to the contact vector field z∂ z + P ni =1 r i ∂ r i .We will slightly generalize this example as follows, which motivates our defini-tion of a “folded” Weinstein hypersurface: We refer the reader to Definition A.3.1for the definition of a v -folded hypersurface and a seam , where v is a vector field;for example, the graph of y = x is a ∂ x -folded hypersurface in R . Taking v to be ONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY 65 R α = ∂ z , we consider closed R α -folded hypersurfaces Σ ⊂ R n +1 with seam C and decomposition Σ \ C = Σ + ∪ Σ − such that R α is positively (resp. negatively)transverse to Σ ± . It follows that Σ ± are naturally exact symplectic manifolds withsymplectic forms dα | Σ ± .We now make the following nontrivial assumption:(W) Each component of Σ ± is a (completed) Weinstein cobordism.For example (W) holds if each component of C is contained in { z = const } ∼ = R n and is transverse to the radial vector field. Moreover, Σ ± are graphical over R n ,and in particular, subdomains of ( R n , ω std ) up to Weinstein homotopy.Any R α -folded hypersurface satisfying (W) is clearly folded Weinstein with thefolding locus equal to C , and hence can be made convex by a C ∞ -small perturba-tion. Note, however, that if Σ happens to be convex, R ± (Σ) = Σ ± in general.This explains our terminology but at the same time raises a hard problem: Question . Characterize or classify convex hypersurfaces (e.g., spheres) in aDarboux chart. Any answer to this question will be of fundamental importance in understandingcontact manifolds. See [Eli92] for a complete answer to this question in the case S ⊂ ( R , ξ std ) .9.2. Normalization of contact structure near a folded Weinstein hypersurface. Recall that if Σ ⊂ ( M, ξ ) is a convex hypersurface, then there exists a collarneighborhood U (Σ) ≃ R t × Σ of Σ such that ξ | U (Σ) = ker( f dt + β ) , where f ∈ C ∞ (Σ) and β ∈ Ω (Σ) . The goal of this subsection is to generalize this tofolded Weinstein hypersurfaces.Let Σ ⊂ ( M, ξ ) be a folded Weinstein hypersurface. Following Definition 9.1.1,we write Σ = W ∪ K ∪ · · ·∪ K m − W m . Choose a collar neighborhood U ( K i ) foreach K i and identify it with [ − , τ × K i such that Σ ξ is directed by ∂ τ on U ( K i ) .(In particular, this means that {− } × K i ⊂ W i and { } × K i ⊂ W i +1 .) Identify acollar neighborhood of Σ ⊂ M with a neighborhood of Σ = Σ ⊂ R t × Σ , where Σ t := { t } × Σ .Let ξ = ker α . In the following three steps we construct a preferred contact form α Σ on a collar neighborhood U (Σ) of Σ such that α Σ | Σ = α | Σ , up to rescaling bya positive function.S TEP Construct the contact form on R × (Σ \ ∪ m − i =1 U ( K i )) . Let W ◦ i := W i \ ( U ( K i − ) ∪ U ( K i )) . After possibly rescaling α by a positivefunction as in Proposition 2.2.3, we may assume that β i := α | W ◦ i is Liouville for all i . Moreover, we can arrange so that the Liouville vector field X β i equals ∂ τ / (2 τ ) near ∂U ( K i ) if i is even, and equals − ∂ τ / (2 τ ) if i is odd. This is a purely technicalarrangement which makes the gluing of contact forms below easier. We define(9.2.1) α Σ := ( − i +1 dt + β i on R × (Σ \ ∪ m − i =1 U ( K i )) . S TEP Construct the contact form on R × U ( K i ) for i even. In this case K i is minimal. Assume without loss of generality that α | U ( K i ) = e τ λ , where λ is a contact form on K i . We will choose α Σ of the form(9.2.2) α Σ = − f ( τ ) dt − tg ( τ ) dτ + e τ λ on R × U ( K i ) . Clearly α Σ | U ( K i ) = α | U ( K i ) . A straightforward computation showsthat α Σ is contact if and only if(9.2.3) f ′ − τ f − g > . We choose f to be a decreasing odd function which equals ± when τ is close to ∓ , and then choose g to be a nonpositive even function which equals when τ isclose to ± , subject to (9.2.3); see Figure 9.2.1. − − − τ τf ( τ ) g ( τ ) F IGURE α Σ restricts to the Liouville form β i,t = − tg ( τ ) dτ + e τ λ on { t } × ( U ( K i ) \ K i ) for any t ∈ R . We compute the Liouville vector fields X β i,t = 1 / (2 τ )( ∂ τ + te − τ g ( τ ) R λ ) , where R λ denotes the Reeb vector field on ( K i , λ ) .It follows that(9.2.4) U ( K i ) t,ξ := ( { t } × U ( K i )) ξ = ∂ τ + te − τ g ( τ ) R λ . S TEP Construct the contact form on R × U ( K i ) for i odd. In this case K i is maximal. This step is analogous to the construction of thecontact form on Γ × [ − , in the proof of Proposition 2.2.3. Assume without lossof generality that α | U ( K i ) = e − τ λ . We define the contact form(9.2.5) α Σ = f ( τ ) dt + e − τ λ on R × U ( K i ) , where f ( τ ) is as above.We compute the Liouville vector fields X β i,t = − / (2 τ ) ∂ τ on { t } × ( U ( K i ) \ K i ) , and note that it is independent of t . In fact U ( K i ) is convex with respect tothe contact vector field ∂ t . ONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY 67 Combining Eq. (9.2.1), Eq. (9.2.2), and Eq. (9.2.5), we obtain a contact form α Σ on R × Σ such that α Σ | Σ = α | Σ , up to rescaling by a positive function. Therefore byLemma 2.0.3, we can assume, up to an isotopy, that ξ = ker α Σ in a neighborhoodof Σ . Remark . A crucial difference between the normal forms of contact structuresnear a convex hypersurface and a folded Weinstein hypersurface is the following:For convex hypersurfaces, since ∂ t is a transverse contact vector field, any smallneighborhood of Σ is in fact contactomorphic to the entire R t × Σ . On the otherhand, the above constructed α Σ is not t -invariant, and hence only specified by thedatum on Σ for | t | sufficiently small.9.3. Bypass attachment as a bifurcation. By the previous subsection we can as-sociate to any folded Weinstein hypersurface Σ ⊂ ( M n +1 , ξ ) a preferred contactform α Σ on R × Σ such that (( − ǫ, ǫ ) × Σ , ker α Σ ) is contactomorphic to a collarneighborhood of Σ in ( M, ξ ) , where ǫ > is sufficiently small.Now suppose Σ = Σ is convex. Then by Proposition 2.2.3, Σ t is convex for all ≤ t < ǫ small. Although a generic Σ t is Morse for all t ≥ , the Morse + con-dition fails at isolated instances. In particular, there exists a first instance t > such that Σ t is convex for any t = t sufficiently close to t but there exists a“retrogradient” trajectory of (Σ t ) ξ from a negative index n singularity to a posi-tive one. Such a phenomenon is called a bifurcation of the characteristic foliationin [Gir00]. As we will see, crossing such t corresponds precisely to a bypassattachment as introduced in [HHa].To set up the “bypass–bifurcation correspondence”, it is convenient to reformu-late the bypass attachment in the language of folded Weinstein hypersurfaces.9.3.1. Bypass attachments. We briefly review bypass attachments from [HHa],leaving the details of contact handle attachments and Legendrian (boundary) sumsto [HHa].Let Σ be a convex hypersurface with the usual decomposition Σ \ Γ = R + ∪ R − .The bypass attachment data (Λ ± ; D ± ) is given as follows: Let D ± ⊂ R ± beLagrangian disks with cylindrical ends which are regular in the sense of [EGL18],i.e., the complement in R ± of a standard neighborhood of D ± is still Weinstein.Let Λ ± = ∂D ± be Legendrian spheres in Γ equipped with the contact form α | Γ ,which we assume have a unique ξ | Γ -transversal intersection point.Next we discuss Reeb pushoffs. If Λ is Legendrian submanifold in Γ , then let Λ ǫ be the Reeb pushoff of Λ in the Reeb direction by ǫ . Clearly Λ ǫ is embedded for | ǫ | sufficiently small. Moreover, if Λ bounds a Lagrangian disk D in some Weinsteinfilling, then there exists a corresponding Lagrangian D ǫ in the same filling with ∂D ǫ = Λ ǫ .We now explain how to attach a bypass to Σ using the bypass attachment data (Λ ± ; D ± ) to obtain a contact structure on [0 , × Σ . The bypass attachment is a Since Morse-Smale vector fields are considered in [Gir00], there exists a different kind of bi-furcation where a pair of periodic orbits appear or disappear. This phenomenon does not occur heresince we are dealing with Morse gradient vector fields. topologically canceling pair of contact handle attachments in the middle dimen-sions. The first is a contact n -handle attachment to Σ along the Legendrian sphere Λ − ⊎ Λ + ⊂ Γ obtained by Legendrian sum. This step produces a new convexhypersurface S . It turns out the pushoffs Λ ∓ ǫ ± of Λ ± become Legendrian isotopicwhen viewed on Γ S . Hence we can attach a contact ( n + 1) -handle to S along theLegendrian sphere that we denote by D − ǫ + ∪ D + ǫ − and is obtained by gluing D − ǫ + and D + ǫ − via the Legendrian isotopy. Remark . It is not necessary to assume that D ± are regular in the definitionof a bypass attachment. It is an outstanding, and of course hard, problem to evenfind an irregular Lagrangian disk in any Weinstein domain. One consequence ofour work in this paper is that, as far as convex hypersurface theory and open bookdecompositions are concerned, one can completely stay in the world of Morsetheory, e.g., avoid using any irregular Lagrangian disks, regardless of their veryexistence, without losing any generality.Let (Σ × [0 , , ξ ) be the contact manifold resulting from the bypass attachment.Write Σ t := Σ × { t } , where Σ = Σ . We have the usual decomposition Σ i \ Γ i = R i + ∪ R i − , i = 0 , . Then by [HHa, Theorem 5.1.3]: • R is obtained from R by removing a standard neighborhood of D − ǫ + andattaching a Weinstein handle along Λ − ⊎ Λ + . • R − is obtained from R − by removing a standard neighborhood of D + ǫ − andattaching a Weinstein handle along Λ − ⊎ Λ + . • Γ , viewed as the boundary of R , is obtained from Γ by a contact (+1) -surgery along Λ − ǫ + and a contact ( − -surgery along Λ − ⊎ Λ + . Γ , viewedas the boundary of R − , is obtained from Γ by a contact (+1) -surgeryalong Λ + ǫ − and a contact ( − -surgery along Λ − ⊎ Λ + . These two presen-tations of Γ are canonically identified by a handleslide.9.3.2. Folded Weinstein description. We will now describe the convex hypersur-face Σ as a folded Weinstein hypersurface.Let W ⊂ R + be the Weinstein subdomain obtained by digging out a standardneighborhood W ′ of D − ǫ + . Then D − ǫ + is the unstable manifold of q + (with respectto the Liouville flow of R + ), and W ′ is a Weinstein cobordism with a unique index n critical point q + . Similarly, let W ⊂ R − be the Weinstein subdomain such that R − is the concatenation of W and a Weinstein cobordism W ′ with a unique index n critical point q − , whose unstable manifold is D ǫ − (with respect to the Liouvilleflow on R − ). Since Λ ∓ ǫ ± = ∂D ∓ ǫ ± are disjoint, we can shuffle the critical values of q ± to obtain the following decomposition Σ = W ∪ K W ∪ K W ∪ K W , where W , W are Weinstein cobordisms (slight variants of W ′ , W ′ ) associatedwith the critical points q − , q + , respectively. See Figure 9.3.1.In particular we have:(FBP1) As contact manifolds, K , oriented as ∂W (and also as ∂W ), is obtainedfrom Γ by a contact (+1) -surgery along Λ − ǫ + ; K , oriented as ∂W (and ONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY 69 also as ∂W ), is obtained from Γ by a contact (+1) -surgery along Λ ǫ − ; and K , oriented as − ∂W (and also as − ∂W ), is obtained from Γ by contact (+1) -surgeries along Λ − ǫ + and Λ ǫ − .(FBP2) Let D †± be the stable manifolds of q ± in W and W with respect to theLiouville flows. Then D † + ∩ K = Λ − ǫ + and D †− ∩ K = Λ ǫ − , where Λ − ǫ + and Λ ǫ − are the core Legendrians of the contact (+1) -surgeries.(FBP3) The ξ | Γ -transverse intersection point between Λ + and Λ − turns into a short(i.e., length ǫ ) Reeb chord γ ⊂ K from Λ − ǫ + to Λ ǫ − . q + q − W W W W F IGURE α Σ be the contact form defined in § R × Σ , where Σ is identifiedwith { } × Σ . Then by (9.2.4) there exists a unique t = a γ > near t = 0 such that the corresponding stable manifolds of q ± on { a γ } × Σ (with respectto the Liouville flows) intersect at a unique ξ | K -transversal point in K . Thiscorresponds to collapsing the short Reeb chord γ to a point.9.3.3. Bypass-bifurcation correspondence. Proposition 9.3.2 (Bypass–bifurcation correspondence) . Let the convex hypersur-face (or equivalently the folded Weinstein hypersurface) Σ , the quadruple (Λ ± ; D ± ) ,and the contact form α Σ be as above. Then for any small δ > , the contact man-ifold ([ a γ − δ, a γ + δ ] × Σ , ker α Σ ) is contactomorphic, relative boundary, to thebypass attachment to Σ along (Λ ± ; D ± ) . We will implicitly use the fact the Σ t is convex for all t ∈ [0 , a γ − δ ] , and hence Σ = Σ may be canonically identified with Σ a γ − δ .Since the proof of Proposition 9.3.2 is somewhat complicated, we start by ex-plaining the key ideas involved and also highlight the difference between the usual -dimensional strategy and the higher-dimensional approach.First note that the bypass attachment is a local operation, i.e., the hypersurface isonly affected in a neighborhood of D + ∪ D − . Let B ⊂ Σ be a small neighborhoodof D + ∪ D − , which is diffeomorphic to a ball. The question is then reduced tounderstanding the contact structure on I × B given by the bypass attachment, where I = [ a γ − δ, a γ + δ ] . At this point, two “miracles” happen in dimension (i.e., dim Σ = 2 ) whichmake the -dimensional proof easy. The first is that one can take ∂B to be Legen-drian using the Legendrian realization principle (cf. [Hon00, Theorem 3.7]). Thisgives us good control over the contact structure near I × ∂B . The second, andmore significant, miracle is Eliashberg’s theorem (cf. [Eli92, Theorem 2.1.3]) onthe uniqueness of tight contact structures on the -ball. Using these two facts, onecan prove Proposition 9.3.2 in dimension by arguing that both the bifurcation andthe bypass attachment produce tight contact structures on the -ball I × B up toedge-rounding, and hence must coincide.Unfortunately, both of the above-mentioned miracles fail in dimension > :the first one fails for dimensional reasons and the second one fails by results of[Eli91, Ust99]. Nevertheless, the proof of Proposition 9.3.2 follows the same gen-eral outline as in dimension by replacing the Legendrian boundary condition on ∂B by a transverse boundary condition and Eliashberg’s theorem by a direct proofthat both the bifurcation and the (trivial) bypass attachment produce the standardball in a Darboux chart. Proof of Proposition 9.3.2. The proof follows the above outline and consists ofseveral steps.S TEP Localizing the problem to B . By (FBP1), the contact manifold K is obtained from Γ by a contact (+1) -surgeries along Λ − ǫ + and Λ ǫ − . Abusing notation, K sits in every Σ t , t ∈ I . Let Λ ± ⊂ K be the Legendrian spheres at level { t = a γ } which ξ | K -transverselyintersect at one point and let D † + and D †− be the corresponding Lagrangian disksin W and W that are given by (FBP2). In what follows the subscript t , e.g., Λ ± ,t and D †± ,t , will be used to denote their parallel copies at different t -levels and willbe omitted if it is understood.We now describe a small closed neighborhood B of D † + ∪ D †− in Σ a γ . We take B ∩ K to be a small contact handlebody neighborhood C = [ − κ, κ ] z × A , κ > small, of Λ + ∪ Λ − , where A is the plumbing of two copies of disk bundles D ∗ S n − with the canonical Liouville form and Λ ± are the -sections of the corresponding D ∗ S n − in { } × A . The restriction of B to a collar neighborhood [ − , τ × K with the -form e τ λ is [ − , × C . Then B is obtained from [ − , × C by attaching Weinstein handles along {− } × Λ − and { } × Λ + . The boundarydecomposes as ∂B = C ∪ C h ∪ C , where C (resp. C ) is the compact contactmanifold obtained from C by a contact ( − -surgery along {− } × Λ − (resp. { } × Λ + ) and C h = [ − , × ∂C . We are viewing B ⊂ W ∪ W , C ⊂ K ,and C ⊂ K .Note that the C i , i = 1 , , , are all contactomorphic since applying a contact ( − -surgery along { } × Λ + to [ − κ, κ ] × T ∗ Λ + still yields [ − κ, κ ] × T ∗ Λ + . Thecase for { } × Λ − is identical.The characteristic foliation B ξ is inward-pointing along C , outward-pointingalong C , and tangent to C h . By slightly tilting C h , we may assume that B ξ is ONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY 71 outward-pointing along C ∪ C h and inward-pointing along C . This results in afold-type tangency roughly along ∂C .Moreover, for δ > sufficiently small, we can construct parallel copies B t ∈ Σ t , t ∈ I , of B = B a γ such that the characteristic foliation is t -invariant near ∂B t .The copies B t are obtained from [ − , × A by attaching cores of the handlesalong {− } × Λ − ,t and { } × Λ + ,t , and δ > small ensures that we can attachhandles (i.e., the thickened cores) to {± } × C along the same locus independentof t .By using certain folding techniques similar to (and in fact simpler than) those inSection 5, one can reverse the direction of the characteristic foliation on C throughan isotopy of B in a suitably wiggled Σ such that Σ ξ is everywhere outward-pointing along ∂B . This will be achieved in Step 3. The folding technique is calledthe Creation Lemma which in dimension is the converse of the usual EliminationLemma (cf. [Gei08, § TEP The Creation Lemma. In this step, we describe the effect of applying a C -small perturbation calleda box-fold . This is the content of the Creation Lemma, which we do not stateformally.We closely follow the discussion of § z and t here, since we are already using t to parametrize the hypersurfaces Σ t .Consider R t,s,z × V equipped with the contact form α = dt + e s ( dz + λ ) , where ( V, λ ) is a complete Weinstein manifold. Let F := { t = 0 } be the hypersurface onwhich we will create singularities. Clearly F ξ = ∂ s where ξ = ker α .Fix t , s , z > . Let Π ⊂ R t,s,z be a surface obtained from the flat R s,z bygrowing a box with base (cid:3) := [0 , s ] × [0 , z ] and height t ; see Figure 9.3.2. Ofcourse, as in the construction of Z in Section 3, one needs to round the cornersof Π and Morsify the resulting characteristic foliation Π ξ . These operations aresuppressed from the notation. We say Π is obtained from R s,z by a -dimensional box-fold . z t s (cid:3) F IGURE Π .Comparing Figure 9.3.2 with Figure 3.1.1, we note that the key difference is that Π ξ admits only positive singularities: one source and one saddle. See Figure 9.3.3.This is the content of the Creation Lemma in dimension 3. e + h + F IGURE Π × V c , where V c ⊂ V is the compact domain, i.e., V \ V c ∼ = [0 , ∞ ) τ × ∂V c is symplectomorphic to ahalf-symplectization of ∂V c . Following the strategy from § box-fold Π to be the hypersurface obtained by slowly damping out the Π -factorin Π × V c as τ increases. Then Eq. (5.1.1) implies that Π ξ is Morse with a pairof canceling critical points for each one in V c . In particular, let D ⊂ Π bea disk containing the source e + such that Π ξ is transverse to ∂D . Then Π ξ iseverywhere outward-pointing along ∂ ( D × V c ) . Note that instead of creating apair of canceling critical points as in dimension (cf. [CE12, Proposition 12.21]for the higher-dimensional version), our Creation Lemma produces many pairs ofcanceling critical points at once, in fact as many as the number of critical points of V .S TEP Modification from B to b B . In Step 1 we constructed the family B t ⊂ Σ t , t ∈ I , such that Σ t,ξ is inward-pointing along C . The goal of this step is to modify B t (and Σ t,ξ ) to b B t so that Σ t,ξ is outward-pointing along ∂ b B t . Write C = [ − κ, κ ] z × A , where κ > issufficiently small (this we need for slow damping). Since Σ ξ points into B along C , we can choose t , s > such that there exists an embedding U ( C ) := I t × [0 , s ] × C ⊂ M, I t = [ a γ − t , a γ + t ] , such that t ≪ δ , U ( C ) ∩ B t = { ( t, s ) } × C , and { t } × [0 , s ] × C ⊂ W .Write the contact form as α | U ( C ) = dt + e s ( dz + λ ) , where λ is the standardLiouville form on A .We then apply the Creation Lemma with V = A to install/uninstall a box-foldalong [0 , s ] × C , which we are assuming is contained in W . This is the higher-dimensional analog of the procedure in § I × Σ by leaves Σ t to obtain a C -close foliation by leavesstill denoted by Σ t such that the following hold:(i) Σ t,ξ is unchanged for t ∈ ∂I and on W ∪ W ∪ W for t ∈ I . ONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY 73 (ii) The box-fold is installed along [0 , s ] × C for t ∈ [ a γ − δ, a γ − t ] and isuninstalled along [0 , s ] × C for t ∈ [ a γ + t , a γ + δ ] . The characteristicfoliation on W is t -invariant for t ∈ I t .(iii) For t ∈ I t , W contains a subdomain symplectomorphic to D × A , where D ⊂ R is a disk containing e + as in Step 2 and there exists an arc µ ⊂ ∂D such that µ × A is identified with { s } × C ; see Figure 9.3.4. e + B F IGURE B to b B so it en-compasses e + . The blue arc represents µ and the red arc represents ∂D \ µ .In order to achieve the transversal boundary condition on B t , t ∈ I t , it remainsto isotop µ through D to ∂D \ µ and use the fact from Step 2 that Σ t,ξ is everywheretransverse to ∂ ( D × A ) , to obtain the new b B t ⊂ Σ t such that Σ t,ξ is everywhereoutward-pointing along ∂ b B t . In particular ∂ b B t , t ∈ I t , are contact submanifoldsof M . Remark . Similar ideas will be exploited in greater generality in Section 12. Claim 9.3.4. Σ t,ξ is Morse for all t ∈ I and Morse + (hence Σ t is convex) for t = a γ .Proof of Claim 9.3.4. By the folded Weinstein structure for Σ t and the fact that in-stalling/uninstalling the box-fold induces a Weinstein homotopy on W , it followsthat Σ t,ξ is Morse for all t ∈ I . As for the Morse + property, it suffices to considerthe stable submanifold of the unique singular point in W . There is a stable tra-jectory that comes from a negative singularity precisely when t = a γ (this is thesame as the situation before Σ t was perturbed). The convexity of Σ t , t = a γ , thenfollows from Proposition 2.2.3. (cid:3) Hence we may restrict attention to the new Σ t , t ∈ I t .S TEP Triviality of the contact structure on I t × b B for t > small. Let S := ∂ ( I t × b B ) = b B a γ − t ∪ b B a γ + t ∪ ( I t × ∂ b B ) . Suppose t > issufficiently small. Claim 9.3.5. After corner rounding, S is convex and isomorphic to the unit spherein a Darboux chart.Proof of Claim 9.3.5. The key point is to describe R + ( S ) , i.e., the positive criticalpoints and the stable submanifolds of these critical points; the situation for R − ( S ) is similar. The critical points of R + ( S ) are as follows:(1) sitting over e + in b B a γ + t are one index critical point q and two index ( n − critical points q ± corresponding to Λ ± ⊂ A ; and(2) the critical points p + on b B a γ + t ∩ W and p − on b B a γ − t ∩ W have index n , where + indicates being on the “top sheet” b B a γ + t .We denote the analogous critical points of R − ( S ) by q ′ , q ′± , p ′± .We denote the stable manifold of a critical point p by W p . For definiteness, weassume that there exists ǫ ′ > small such that(i) W p + , W p ′ + intersect C ⊂ b B a γ + t along the pushoffs Λ ǫ ′ + , Λ − ǫ ′ − ;(ii) W p − , W p ′− intersect C ⊂ b B a γ − t along the pushoffs Λ ǫ ′ − , Λ − ǫ ′ + .By (i), W p + intersects K = ∂W along Λ ǫ ′ + and therefore limits to Λ + ⊂ A over e + ; moreover, there is a unique trajectory from p + to q + . Next, W p − intersects C along Legendrian which is isotopic to a positive pushoff of Λ − , continues inside b B a γ + t to a Legendrian isotopic to a positive pushoff of Λ − on K , and limits to Λ − ⊂ A over e + . Moreover, there is a unique trajectory from p − to q − . Thisimplies that S ξ is Morse + and convex, with Weinstein structures on R ± ( S ) justdescribed.The index ( n − and index n critical points cancel in pairs and R ± ( S ) areWeinstein homotopic to the standard ball with a unique critical point. (cid:3) Remark . The reader might find it instructive to consider the n = 1 (i.e., dim M = 3 ) case, where we have three index critical points “sitting over e + ”. Claim 9.3.7. The contact structure ξ on I t × b B is standard, i.e., contactomorphicto the unit ball in a Darboux chart.Proof of Claim 9.3.7. Consider the -parameter family of spheres S t := ∂ ([ a γ − t + t, a γ + t ] × b B ) , for ≤ t < t .We show that all the S t can be made convex after a small perturbation. We usethe argument of Claim 9.3.5, but this time there are two moments t ′ < t , where abifurcation occurs. There is still a unique trajectory from p + to q + for all t .Next we describe the trajectories of W p − for ≤ t < t . When t = t ′ , all thetrajectories of W p − reach C but one continues to the critical point of b B a γ + t ∩ W ;when t = t , a trajectory of W p − limits to the critical point of b B a γ ∩ W . We have:(1) For t < t ′ , W p − ∩ K is Legendrian isotopic to a positive pushoff of Λ − .(2) For t > t , W p − ∩ K is Legendrian isotopic to a negative pushoff of Λ − .(3) For t ′ < t < t , W p − ∩ K is Legendrian isotopic to Λ + ⊎ Λ − . ONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY 75 (4) For t = t ′ , t , W p − ∩ K is Legendrian isotopic to Λ + ∪ Λ − intersectingat a point and W p − ∩ K corresponds to Λ − − Λ + .Here K is understood to be on t = a γ + t . (3) is a consequence of corner-rounding along C , which has the effect of introducing a slight negative Reeb flowalong the corner as we go from the bottom sheet to the top. (4) is the limitingconfiguration of (1)–(3). In all the cases, there is a unique trajectory from p − to q − , although there may be trajectories from p − to q + for t ′ < t < t .Now by the usual Elimination Lemma (cf. [CE12, Proposition 12.22]) and atrick from [Hua13, Lemma 3.3], S t is convex for all ≤ t < t : By a C -smallperturbation one can simultaneously eliminate the pairs ( p + , q + ) and ( p − , q − ) on S t for all t ∈ [0 , t ] (since the trajectories from p + to q + and p − to q − varycontinuously with respect to t ), which in turn implies that all the S t are convex.Since the ball bounded by S t for t sufficiently close to t is standard, the claimfollows. (cid:3) Finally we observe that the bypass attachment to Σ along (Λ ± ; D ± ) restricts tothe trivial bypass attachment to S in the sense of [HHa, Definition 6.1.1]. It followsfrom [HHa, Proposition 8.3.2] that the contact structure on I t × b B given by a trivialbypass attachment is standard. By Claim 9.3.7, ξ on I t × b B is standard, hence isequivalent to a bypass attachment. This finishes the proof of the proposition. (cid:3) Bypass attachment as a partial open book. The goal of this subsection isto summarize the main constructions and results from [HHa, Section 8]. Noth-ing is new here and the reader is referred to the original paper for details. Theonly notable difference in our current exposition is that every Liouville manifold isassumed to be Weinstein.The subsection is organized as follows: First we adapt the constructions in § § From cornered Weinstein to subordinated Weinstein. Let S be the idealcompactification of a complete Weinstein manifold ( S ◦ , λ ) and S c ⊂ S ◦ be a sub-domain such that S ◦ \ S c can be identified with the positive half-symplectization (0 , ∞ ) × ∂S c of the contact boundary ∂S c = { } × ∂S c .A cornered Weinstein subdomain W c ⊂ S c satisfies the following properties:(CW1) There exists a decomposition ∂W c = ∂ in W c ∪ ∂ out W c such that(1) ∂ in W c and ∂ out W c are compact manifolds with smooth boundary,(2) ∂ ( ∂ in W c ) = ∂ ( ∂ out W c ) is the codimension corner of ∂W c , and(3) W c ∩ ∂S c = ∂ out W c .(CW2) The Liouville vector field X λ on S is inward-pointing along ∂ in W c andoutward-pointing near ∂ out W c . A particularly useful class of cornered Weinstein subdomains consists of regularneighborhoods of Lagrangian cocore disks in S c . Using the flow of X λ , one canextend W c ⊂ S c to W ⊂ S in the obvious way.In order to apply the construction of an abstract partial open book from § § S . This isobtained by taking the union of W and a collar neighborhood of ∂S . Slightlyabusing notation, we will denote by ( S, W, φ ) the abstract partial open book in thesense of § Remark . The notion of an abstract partial open book in § ∂ ( ∂ in W c ) = ∂ ( ∂ out W c ) = ∅ in (CW1).9.4.2. From bypass attachment to partial open book. We follow the recipe from[HHa, Section 8.3] to translate a bypass attachment into a certain modification of apartial open book.Continuing to use the notation from § ( M, ξ, Γ) be the compact contactmanifold associated to an abstract partial open book ( S, W, φ ) . Write Σ := ∂M for the convex boundary. Then in the usual decomposition Σ \ Γ = R + ∪ R − , R + = S \ W and R − = S \ φ ( W ) are Weinstein domains.Let (Λ ± ; D ± ) be bypass attachment data for Σ , where D ± ⊂ R ± are regular inthe sense of [EGL18]. Denote the resulting contact manifold by ( M ♭ , ξ ♭ , Γ ♭ ) . Wedescribe the partial open book ( S ♭ , W ♭ , φ ♭ ) corresponding to ( M ♭ , ξ ♭ , Γ ♭ ) : Abus-ing notation, we will not distinguish S , S c , and S ◦ for the rest of this subsection.Fix ǫ > small.(1) S ♭ is obtained from S by attaching a Weinstein handle along Λ − ⊎ Λ + ;(2) W ♭ = W ⊔ N ǫ/ ( D − ǫ + ) ⊂ S ♭ where N ǫ/ ( D − ǫ + ) , as a cornered Weinsteinsubdomain, is a standard ǫ/ -neighborhood of the Lagrangian disk D − ǫ + ;and(3) φ ♭ : W ♭ → S ♭ is determined by specifying the Lagrangian φ ♭ ( D − ǫ + ) ⊂ S ♭ .For (3), note that we have the Lagrangian disk D ǫ − ⊂ S ♭ \ φ ♭ ( W ) with Legendrianboundary Λ ǫ − ⊂ ∂S ♭ . We can slide Λ ǫ − in the negative Reeb direction across theWeinstein handle along Λ − ⊎ Λ + so it precisely matches Λ − ǫ + = ∂D − ǫ + . The slidingis induced by a Weinstein isotopy τ s : S ♭ ∼ → S ♭ , s ∈ [0 , , with τ = id . Then wedefine φ ♭ ( D − ǫ + ) = τ ( D ǫ − ) . Lemma 9.4.2 ([HHa], Proposition 8.3.1) . Let ( M, ξ, Γ) be a compact contactmanifold with convex boundary supported by a partial open book ( S, W, φ ) . If ( M ♭ , ξ ♭ , Γ ♭ ) is the contact manifold obtained by attaching a bypass along (Λ ± ; D ± ) on ( M, ξ, Γ) , then ( M ♭ , ξ ♭ , Γ ♭ ) is supported by the partial open book ( S ♭ , W ♭ , φ ♭ ) described above. Note that Lemma 9.4.2 is a direct consequence of the interpretation of a by-pass attachment as a topologically canceling pair of contact handle attachments. If dim M = 2 n + 1 , then the handles have indices n and n + 1 . ONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY 77 Contact Morse functions and vector fields. It is helpful, although not tech-nically necessary in this paper, to interpret the contact handle attachments in termsof contact Morse functions . This is the contact-topological analog of the corre-spondence between handle decompositions and Morse function presentations of agiven smooth manifold. See Sackel [Sac] for a more thorough discussion of contactMorse functions.Recall that a vector field v on ( M, ξ ) is a contact Morse vector field if v isgradient-like for some Morse function f : M → R and the flow of v preserves ξ ,i.e., L v α = gα where ξ = ker α and g ∈ C ∞ ( M ) . The Morse function f is calleda contact Morse function . The zeros of v are precisely the critical points of f andhence it makes sense to refer to the (Morse) indices of the zeros of v .If (Σ × [0 , , ξ ) is the contact manifold corresponding to a bypass attachmentas above, then there exists a contact Morse vector field v on Σ × [0 , satisfyingthe following properties:(BM1) v is inward-pointing along Σ and outward-pointing along Σ .(BM2) v has exactly two zeros — p of index n and q of index n + 1 — which areconnected by a unique flow line of v .(BM3) For any x ∈ Σ × [0 , , the flow line of v passing through x either convergesto a zero of v or leaves Σ × [0 , in both forward and backward time.(BM4) The unstable manifold of p intersects Σ along the Legendrian Λ − ⊎ Λ + ⊂ Γ , and the stable manifold of q intersects Σ along the Legendrian Λ − ǫ + ⊂ Γ , viewed as the boundary of R .Note, however, that in general Σ , Σ are not regular level sets of f since contactvector fields are not stable under rescaling by positive functions.10. C - APPROXIMATION BY CONVEX HYPERSURFACES In this section we complete the proofs of Theorem 8.1.2 and Theorem 1.2.4. Themain technical ingredient is the higher-dimensional plug constructed in Section 8.In fact our proofs are basically the same as those for the -dimensional case dis-cussed in Section 4. Proof of Theorem 8.1.2. Fix a metric on M . Given a closed hypersurface Σ ⊂ ( M, ξ ) , we may assume that the singularities of Σ ξ are isolated and Morse after a C ∞ -small perturbation. Let N be the union of small open neighborhoods of thesingularities.The higher-dimensional analog of Lemma 4.3.1 holds with small disjoint foli-ated charts of the form B i = [0 , s i ] × Y i , i ∈ I , where Y i is a compact (2 n − -dimensional submanifold with boundary. Start with a finite cover of Σ − N bysmall foliated charts B ′ j = [0 , s ′ j ] s × D j , j ∈ J , such that Σ ξ | B ′ j = R h ∂ s i and D j is a (2 n − -dimensional disk for each j ∈ J . Let D mj := { s ′ j / } × D j . Wesplit the disks D mj if D mj ∩ D mj ′ = ∅ as in Lemma 4.3.1 to obtain a collection Y i , i ∈ I , such that Y i ⋔ Σ ξ , Y i ∩ Y j = ∅ , and any flow line of Σ ξ passes throughsome Y i . (The slight difference here is that we need to extend the intersection D mj ∩ D mj ′ ⊂ D mj so it reaches the boundary of D mj , provided j > j ′ .) The foliated charts B i are slight thickenings of Y i in the direction of Σ ξ . As before, B I called a barricade on Σ .It remains to replace each B i with a Y i -shaped plug constructed in Section 8.Let Σ ∨ be the resulting hypersurface. Then clearly Σ ∨ ξ satisfies Conditions (M1)–(M3) of Proposition 2.1.3. Hence a further C ∞ -small perturbation of Σ ∨ will makeit convex by Proposition 2.2.3. Finally, the ǫ -convexity is guaranteed if all the B i , i ∈ I , are sufficiently small. (cid:3) Proof of Theorem 1.2.4. Let (Σ × [0 , , ξ ) be a contact manifold such that the hy-persurfaces Σ i , i = 0 , , are convex. The proof from § -dimensional one constructed in § B = [0 , s ] × Y .Then the diameter of Y is not necessarily small. Hence even though the foldedhypersurface Σ ∨ can be made convex, the intermediate hypersurfaces appearing inthe procedure of installing and uninstalling the Y -shaped plug need not be Morse.To remedy this defect, we take a cover Y = ∪ ≤ i ≤ K U i by a finite number ofballs of small diameter for which there exists a partition φ : { , . . . , K } → { , . . . , n } such that U i ∩ U j = ∅ if φ ( i ) = φ ( j ) . Now we choose n pairwise distinct valuesin (0 , s ) and position U i along [0 , s ] such that all the U i with the same φ -valuehave the same s -value. Clearly all the U i are disjoint from each other. Finally,by installing and uninstalling U i -shaped plugs and proceeding as in § Σ × [0 , by hypersurfaces of the form Σ t which are all Morse. The onlyobstruction to convexity occurs when (Σ t ) ξ fails to be Morse + , which genericallyoccurs at isolated moments. The theorem then follows from Proposition 9.3.2. (cid:3) 11. T HE EXISTENCE OF ( PARTIAL ) OPEN BOOK DECOMPOSITIONS The goal of this section is to prove Corollary 1.3.1 and a stronger/more preciseversion of Corollary 1.3.2. The proofs are, again, essentially the same as the proofsin the -dimensional case; see [Gir02] for the absolute case and [HKM09] for therelative case. Proof of Corollary 1.3.1. Let ( M, ξ ) be a closed contact manifold of dimension n + 1 . Choose a generic self-indexing Morse function f : M → R . Thenthe regular level set Σ := f − ( n + ) is a smooth hypersurface which divides M into two connected components M \ Σ = Y ∪ Y . It follows that Y i , i =0 , , deformation retracts (along ±∇ f ) to the skeleton Sk( Y i ) , which is a finite n -dimensional CW-complex.Writing Y for either of Y or Y , we now construct N (Sk( Y )) as a compactcontact handlebody. There exists a neighborhood of the -cells of Y that can bewritten as a contact handlebody H = [ − , × W , where W is Weinstein.Arguing by induction, assume that the k -skeleton of Sk( Y ) can be realized asa contact handlebody H k = [ − , × W k , where W k is Weinstein and Γ k = ONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY 79 { } × ∂W k is the dividing set of ∂H k . We explain how to attach the ( k + 1) -handles to ∂H k , where k + 1 ≤ n . Write K for the core of a ( k + 1) -handle. Then dim ∂K = k and by dimension reasons ∂K ⊂ ∂H k , after possible perturbation,can be isotoped into Γ k using the Liouville flow on W k .We then isotop ∂K to an isotropic submanifold ∂K ′ in Γ k (it may be Legendrianif k + 1 = n ) and then isotop K to an isotropic submanifold K ′ ⊂ Y − int H k with boundary ∂K ′ , using Gromov’s h -principle [Gro86, p. 339] for isotropicsubmanifolds in a contact manifold. For this we need to show that: Claim 11.0.1. K is formally isotropic with respect to ( M, ξ ) , subject to ∂K beingformally isotropic with respect to Γ k .Proof of Claim 11.0.1. We will explain the Legendrian (i.e., k +1 = n ) case, whichis the hardest case. Since K is a disk, it is clearly formally Legendrian inside itsdisk neighborhood N ( K ) . The key point is to make ∂K formally isotropic as well.Let τ be a trivialization of ξ | N ( K ) . Projecting out the Reeb direction and usingthe trivialization τ , the embedding K ֒ → N ( K ) can be converted into the map φ : K → G ( n, n ) , where G ( n, n ) is the Grassmannian of n -planes in R n .Since K is a disk, φ is homotopic to φ : K → L n , where L n ⊂ G ( n, n ) is theLagrangian Grassmannian. Next, writing v for a nonvanishing vector field along ∂K that is transverse to ∂H k and tangent to ξ , we would like to further homotop φ to φ : K → L n such that v ( x ) ∈ φ ( x ) for all x ∈ ∂K . Since v is homotopicto the constant vector field, we can view φ | ∂K as a map ∂K → L n − ⊂ L n .We claim that π n − L n − → π n − L n (this corresponds to a standard inclusion R n − ֒ → R n ) is surjective, which then implies Claim 11.0.1. Using the fact that L n = U ( n ) /O ( n ) , we have: π n − U ( n ) −−−−→ π n − L n −−−−→ π n − O ( n ) −−−−→ π n − U ( n ) x a x b x c x d π n − U ( n − −−−−→ π n − L n − −−−−→ π n − O ( n − −−−−→ π n − U ( n − Using the homotopy exact sequences for U ( n ) /U ( n − 1) = S n − and O ( n ) /O ( n − 1) = S n − , it follows that a, c are surjective and d is injective. The claim then fol-lows from the five lemma. (cid:3) Hence H k +1 = [ − , × W k +1 is a contact handlebody and N (Sk( Y )) ∪ N (Sk( Y )) can be realized as a compact contact handlebody with sutured convexboundary and its complement in M has sutured concave boundary.Now identify M \ ( N (Sk( Y )) ∪ N (Sk( Y ))) with Σ × [0 , such that if we write Σ t := Σ × { t } , then Σ i = ∂N (Sk( Y i )) , i = 0 , , are convex with dividing setscorresponding to the sutures. By Theorem 1.2.4 and Proposition 9.3.2, ξ | Σ × [0 , isgiven by a finite sequence of bypass attachments, which can be further turned intoa sequence of modifications of the trivial POBD of N (Sk( Y )) = [ − , × W ,n , according to Lemma 9.4.2 (here W ,n is W n for Y ). In this way we obtain aPOBD of M \ N (Sk( Y )) viewed as a contact manifold with sutured concave boundary. (A slight technical point is that while partial open books naturally havesutured concave boundary, the bypass attachment is attached to a sutured convexboundary. The transition between sutured convex and concave boundaries can bedone as explained in [CGHH11, Section 4].) It remains to fill in N (Sk( Y )) in theobvious manner to get a compatible OBD. (cid:3) Remark . We can also complete the proof of Corollary 1.3.1 as in the -dimensional case: The contact Morse function f on Σ × [0 , given by a sequenceof bypass attachments only has critical points of indices n and n + 1 . We can thenshuffle the critical values of f so that it is self-indexing. Then the compatible OBDof ( M, ξ ) is obtained by gluing the two contact handlebodies { f ≤ n + } and { f ≥ n + } together along their common boundary.Next we turn to the relative case, i.e., to contact manifolds with boundary. Wenote that Corollary 1.3.2 is somewhat loosely stated since the boundary condi-tion is vague. In the following we state a more precise, and stronger, version ofCorollary 1.3.2 by taking into account the characteristic foliation on the boundary.It is this version of the existence of POBD which we use in the induction (cf. § Theorem 11.0.3. If ( M, ξ ) is a compact contact manifold with sutured concaveboundary in the sense of Definition 6.4.1, then there exists a compatible partialopen book decomposition preserving ( ∂M ) ξ on the boundary.Proof. Choose a generic self-indexing Morse function f : M → [ , ∞ ) such that f ≡ on ∂M . In other words, f has no index critical points. As in the absolutecase, consider the hypersurface Σ := f − ( n + ) which divides M into two com-ponents Y i , i = 0 , , such that Y contains all the critical points of index at most n and Y contains all the critical points of index at least n + 1 . By the handle at-tachment discussion in the proof of Corollary 1.3.1, we can turn the critical pointsin Y into isotropic handles attached to ∂M along the suture which we still denoteby N (Sk( Y )) although it is no longer a contact handlebody, and the critical pointsin Y into the handle decomposition of a contact handlebody N (Sk( Y )) with su-tured convex boundary, as in the closed case. The rest of the proof proceeds as inthe closed case. (cid:3) 12. A PPLICATIONS TO CONTACT SUBMANIFOLDS In this section, we apply the techniques developed in Section 7 and Section 8 toprove Corollary 1.3.5 and Corollary 1.3.6.Recall that Ibort, Mart´ınez-Torres, and Presas [IMTP00] constructed contactsubmanifolds Y of ( M, ξ ) as the zero loci of “approximately holomorphic” sec-tions of a complex line or vector bundle over M . Our strategy for constructingcontact submanifolds is rather different: the key observation is that if Σ ⊂ M is ahypersurface which contains a codimension submanifold Y such that the charac-teristic foliation Σ ξ is transverse to Y , then Y ⊂ ( M, ξ ) is a contact submanifold. This is precisely the contact analog of the so-called symplectic reduction. ONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY 81 Until the last paragraph of this section we assume that Y ⊂ M is a closed codi-mension submanifold with a trivial normal bundle. The proofs of Corollary 1.3.5and Corollary 1.3.6 in full generality, i.e., only assuming Y is almost contact, is aconsequence of Gromov’s h -principle for open contact (sub-)manifolds, and willbe given at the end of this section.12.1. Some Morse-theoretic lemmas. The goal of this subsection is to presentsome technical facts about Morse vector fields. One can intuitively think of theseMorse vector fields as characteristic foliations.Our analysis of Morse vector fields consists of two parts: the absolute caseand the relative case, which will be applied to Corollary 1.3.6 and Corollary 1.3.5,respectively. Absolute case. Let Y be a closed manifold and M = Y × [0 , t . Write Y t := Y × { t } . Suppose v is a C ∞ -generic vector field on M which is gradient-like forsome Morse function, and in addition satisfies the following:(B1) v is inward-pointing along Y and outward-pointing along Y .(B2) There exist no (broken) flow lines of v from Y to Y .(B3) For any z ∈ M such that v ( z ) = 0 , let W s ( z ) be the stable manifoldof z and S ( z ) ⊂ W s ( z ) be a small sphere centered at z . In particular dim S ( z ) = dim W s ( z ) − . Then one of the following three scenarioshappens:(B3-1) For any x ∈ S ( z ) , there is a (broken) flow line of v from Y to x .(B3-2) For any x ∈ S ( z ) , there are no (broken) flow lines of v from Y to x .(B3-3) There is a closed codimension disk K ⊂ S ( z ) such that there is a(broken) flow line of v from Y to x ∈ S ( z ) if and only if x ∈ K .Here a broken flow line is a map ℓ : [0 , → M such that there exists an increasingsequence a < a < · · · < a m = 1 such that ℓ ( a j ) , < j < m , are zerosof v and ℓ | ( a j ,a j +1 ) are smooth (i.e., unbroken) oriented flow lines of v , up to areparametrization. The C ∞ -genericity of v is necessary to ensure that stable andunstable manifolds intersect transversely and that trajectories can be glued. Let U ⊂ M be the open subset of points that can be connected to Y by asmooth flow line of v . Define its closure U to be the stump of the pair ( M, v ) .Clearly Y ⊂ U ⊂ M \ Y by (B1) and (B2). Let ∂ + U be the set of points in U − Y that are not interior points in U . See Figure 12.1.1.Let z := { z i } i ∈ I be the set of zeros of v contained in ∂ + U and let Cl( z ) be theclosure of the union of smooth flow lines of v between the critical points in z . Lemma 12.1.1. ∂ + U = Cl( z ) = ∅ and is the closure of the union of the unstablesubmanifolds of z .Proof. First observe that ∂ + U = ∅ since U ⊂ M \ Y . Also observe that:(*) if x ∈ ∂ + U such that v ( x ) = 0 , then y ∈ M that lies on the same smoothflow line of v as x is in ∂ + U .(i) ∂ + U contains at least one zero of v : Indeed, pick x ∈ ∂ + U such that v ( x ) =0 . By definition there exists a broken flow line ℓ x : [0 , → U such that ℓ x (0) ∈ Y Y Y F IGURE v on an an-nulus satisfying (B1) and (B2). The shaded pair-of-pants is thestump in this case.and ℓ x (1) = x . Let z = ℓ x ( a m − ) be the last zero of v on ℓ x . Then ℓ x (( a m − , is a smooth flow line contained in ∂ + U by (*). This implies that z ∈ ∂ + U .(ii) ∂ + U = Cl( z ) : Indeed, by (*), every point of ∂ + U is in z or on some flowline of Cl( z ) . On the other hand, every point y ∈ Cl( z ) − z is in ∂ + U : Thereexist z ∈ z , a flow line ℓ from z to y , and a possibly broken flow line ℓ from Y to z . Then ℓ ℓ is a broken flow line that can be approximated by a smooth flowline from Y to y ; this is where the C ∞ -genericity of v is crucial. This implies that y ∈ U . The fact that y int( U ) follows from (iii).(iii) No unstable trajectories of z can point into U : Arguing by contradiction, if y ∈ int( U ) and ℓ is a flow line from z to y , then there exists a local codimension slice V ⊂ M through y that is transverse to v and is contained in U . Flowing V backwards along ℓ to z , it follows that all the stable trajectories of v that limitto z are contained in U . Together with the C ∞ -genericity of v , we obtain that z ∈ int( U ) , a contradiction.(iv) No unstable trajectories of z can point out of U : If y U and ℓ is a flowline from z to y , then ℓ ℓ is a broken flow line that can be approximated by asmooth flow line from Y to y , a contradiction.By (iii) and (iv) all the unstable trajectories of z are contained in ∂ + U . (cid:3) Lemma 12.1.2. (1) U ⊂ M is a submanifold with boundary Y ∪ ∂ + U .(2) v | ∂ + U is gradient-like for some Morse function on ∂ + U .(3) There exists a small collar neighborhood N ǫ ( ∂ + U ) of ∂ + U such that v isinward-pointing along ∂N ǫ ( ∂ + U ) .Proof. Since ∂ + U = Cl( z ) must block the flow of v , it must have codimension in M . By Lemma 12.1.1, ∂ + U is the union of z and the unstable manifolds of z .This gives a handle description of ∂ + U but it does not imply, a priori, that ∂ + U is a manifold by itself. Now the key observation is that all the z i ∈ z belong to ONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY 83 type (B3-3). Indeed, zeros of v of type (B3-1) are contained the interior of U , whilezeros of type (B3-2) are contained in the complement of U . The assumption (B3-3)implies that the restriction of the stable manifold of z i to ∂ + U is itself a manifoldfor all i . This, in turn, implies that ∂ + U is a manifold such that v | ∂ + U is Morse.Finally, (3) follows from the fact that no unstable trajectories of z can point out of ∂ + U . (cid:3) Lemma 12.1.3. Let M = Y × [0 , be as above and v be a C ∞ -generic, gradient-like vector field satisfying (B1)–(B3). Then there exist a (possibly disconnected)hypersurface Y ′ ⊂ int( M ) and a codimension submanifold K ⊂ M such that ∂K = Y ⊔ Y ′ and v points into K along Y ′ .Proof. Since v satisfies (B1)–(B3), there exists a stump U ⊂ M which is a compactsubmanifold with boundary and satisfies Lemma 12.1.2 (3). Hence K := U ∪ N ǫ ( ∂ + U ) satisfies the conclusions of the lemma. (cid:3) Remark . In the proof of Lemma 12.1.3, if we further assume that the interiorof U contains no zeros of v , then Y ′ is isotopic to Y . See Figure 12.1.1 for a non-example. Relative case. Let Y be a compact manifold with boundary and let M = Y × [0 , .Suppose v is a C ∞ -generic vector field on M which is gradient-like for someMorse function, and in addition satisfies the following:(RB1) v is inward-pointing along Y , outward-pointing along Y , and tangent to ∂Y × [0 , ;(RB2) There exist no (broken) flow lines of v from int( Y ) to int( Y ) ;(RB3) The same as (B3).As in the absolute case, let U ⊂ M be the open subset of points that can beconnected to int( Y ) by a smooth flow line of v . Define its closure U to be the relative stump of the pair ( M, v ) .By essentially the same argument as in the absolute case, one can show that U ⊂ M is a smooth codimension submanifold with boundary and corners.We formulate the relative version of Lemma 12.1.3 as follows. Lemma 12.1.5. Let M = Y × [0 , be as above and v be a C ∞ -generic, gradient-like vector field satisfying (RB1)–(RB3). Let Σ ⊂ int( Y ) be a hypersurface ob-tained by slightly pushing ∂Y into the interior, and Y σ ⊂ Y be the domainbounded by Σ . Then there exist a properly embedded hypersurface Y ′ ⊂ M such that Y ′ ∩ ∂M = Σ and a codimension submanifold K ⊂ M such that ∂K = Y σ ∪ Σ Y ′ and v is inward-pointing along ∂K . The proof is similar to that of Lemma 12.1.3 and is omitted. The followingremark is similar to Remark 12.1.4. Remark . Suppose the relative stump involved in the proof of Lemma 12.1.5contains no zeros of v in the interior. Then Y ′ is isotopic to Y σ relative to theboundary. See Figure 12.1.2 for an example. Y Y F IGURE v on a solidcylinder satisfying (RB1)–(RB3). The shaded region is the relativestump in this case.12.2. Existence h -principle for contact submanifolds. We use the Morse-theoretictechniques developed in § Y ⊂ ( M, ξ ) has a trivial normal bundle for the moment. Proof of Corollary 1.3.6. By assumption Y ⊂ ( M, ξ ) is a contact submanifold.Consider a hypersurface Σ := Y × [ − , s ⊂ M such that Y is identified with Y , where Y s := Y × { s } . The trivial normal bundle condition is used to constructthe hypersurface Σ . Observe that Σ ξ is transverse to Y . Assume without loss ofgenerality that Σ ξ = ∂ s . By the argument in § C -perturb Σ on a smallneighborhood of Y / to obtain a new hypersurface Σ ∨ such that Σ ∨ ξ is C ∞ -genericand satisfies (B1)–(B3) from § U contains no zeros of Σ ∨ ξ . Refer tothe description of the zeros e x − , e x + , h x − , h x + of Z ′ ξ from § Z ξ outside of Z ′ ξ . Using the notation from § Z ξ (also refer to Figure 3.2.1), we see that e x − , h x − , h x + ∈ U and e x + U . Since there are trajectories from e + to e − , h − , and h + , it follows that e x − , h x − , h x + cannot be in int( U ) . This implies the claim.The corollary then follows from Lemma 12.1.3 and Remark 12.1.4. (cid:3) Proof of Corollary 1.3.5. We continue to use the notation from the above proof, butnow Σ ξ is not necessarily transverse to Y . By Proposition A.3.2, we can assumethat Y is Σ ξ -folded with folding locus C ⊂ Y . We then have the decomposition Y = Y +0 ∪ C Y − such that Σ ξ is positively transverse to int( Y +0 ) and negatively ONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY 85 transverse to int( Y − ) . We may also perturb C so that C is a convex hypersur-face with respect to the contact submanifolds Cusp( Y ± ) obtained from Y ± byconverting folds to cusps; for more details see Appendix A.Since we can deal with the connected components of Y − one at a time, assume Y − is connected.(i) Suppose for the moment that the characteristic foliation Σ ξ , viewed as anoriented line field, points out of ∂Y − everywhere or points into ∂Y − everywhere.In other words, writing C = C + ⊔ C − , where Σ ξ points into Y ± along C ± , wewant ∂Y − = C + or ∂Y − = C − . We refer to C + (resp. C − ) as a folding locus of positive type (resp. negative type ). Assuming ∂Y − = C + (the case ∂Y − = C − is similar), there exist an identification N ǫ ( Y − ) = Y − × [ − ǫ, ǫ ] s and a piecewisesmooth approximation Y ⋆ of Y such that: • Σ ξ | N ǫ ( Y − ) = − ∂ s ; and • Y ⋆ ∩ N ǫ ( Y − ) = ( Y − × { } ) ∪ ( C × [ − ǫ, ∪ ( N ǫ ( C ) × {− ǫ } ) , where N ǫ ( C ) ⊂ Y − is a collar neighborhood of the boundary.Informally, Y ⋆ is sutured with respect to Σ ξ , where the suture is C × [ − ǫ, .By the argument in § C -perturb Σ on a neighborhood of Y − × { ǫ } to obtain a new hypersurface Σ ∨ such that Σ ∨ ξ is C ∞ -generic and satisfies (RB1)–(RB3) from § Y − × [0 , ǫ ] . Moreover, the relative stump contains no zeros of Σ ∨ ξ in the interior by the same argument as before. We then apply Lemma 12.1.5and Remark 12.1.6 to Y ⋆ and replace Y ⋆ ∩ N ǫ ( Y − ) by a hypersurface which ispositively transverse to Σ ∨ ξ and has the same boundary.(ii) Next suppose that ∂Y − has components in C + and C − . In this case we cancreate extra folds near C − so that the new Y is Tr( Y − ) ∪ A + ∪ A − ∪ Tr( Y +0 ) , where(1) Tr( Y ± ) is the truncation Y ± − N ǫ ′ ( C − ) , where ǫ ′ is small;(2) A + = C − × [ − , is positively transverse to Σ ξ and A − = C − × [0 , is negatively transverse to Σ ξ ;(3) the boundary components of Tr( Y − ) and Tr( Y +0 ) corresponding to C − are glued to C − × {− } and C − × { } ;(4) C − × {− } is of positive type and C − × { , } is of negative type.Note that after the modification both Tr( Y − ) and A − satisfy the conditions of (i).Hence we can now apply (i) to conclude the proof. (cid:3) Finally we explain how to remove the requirement that Y ⊂ M have trivialnormal bundle. Let D ⊂ Y be a closed ball around a point. Then Y \ D is anopen manifold. If Y is an almost contact manifold, then so is Y \ D . By Gromov’s h -principles for open contact manifolds (cf. [EM02, 10.3.2]) and open isocontactembeddings (cf. [EM02, 12.3.1]), we may assume that Y \ D ⊂ ( M, ξ ) is an opencontact submanifold. Hence the problem reduces to an extension problem over the ball D , which clearly has a trivial normal bundle in M . Observe that the proof ofCorollary 1.3.5 is essentially a relative extension problem.A PPENDIX A. W RINKLED AND FOLDED EMBEDDINGS The technique of wrinkled maps and wrinkled embeddings, developed by Eliash-berg and Mishachev in the series of papers [EM97, EM98, EM00, EM09], is ex-tremely powerful in dealing with homotopy problems of smooth maps betweenmanifolds. The goal of this appendix is to give a brief overview of their theory andprove a technical result, Proposition A.3.2, which is only used in Section 12.This appendix is organized as follows. First we review several fundamentaldefinitions and results in the theory of wrinkled embeddings following Eliashbergand Mishachev. Then we use the wrinkling technique to put a generic hypersurfacein a “good” position with respect to a nonvanishing vector field.A.1. Wrinkled and cuspidal embeddings. In this subsection we review the mainresults of [EM09].Let f : Σ → M be a smooth map between smooth manifolds. In this subsectionwe assume that dim Σ = dim M − k , unless otherwise specified. All theresults in this subsection are also valid whenever dim Σ < dim M . If dim Σ ≤ dim M , then we say that the singular set of f : Σ → M is the set of points in Σ where df is not injective and is denoted by Sing( f ) . Definition A.1.1 (Wrinkled embedding) . A smooth map f : Σ → M is a wrinkledembedding if:(WE1) f is a topological embedding.(WE2) Sing( f ) is diffeomorphic to a disjoint union of spheres S i ∼ = S k − , eachof which bounds a k -disk in Σ . Each such S i is called a wrinkle of f .(WE3) The map f near each wrinkle is equivalent to a map O p R k ( S k − ) → R k +1 , ( y, z ) (cid:18) y, z + 3( | y | − z, Z z ( z + | y | − dz (cid:19) . Here ( y, z ) = ( y , . . . , y k − , z ) denotes the Cartesian coordinates on R k such that S k − = {| y | + z = 1 } is the unit sphere. Let f : Σ → M be a wrinkled embedding. Consider a wrinkle S ∼ = S k − of f given in the local model specified by (WE3). Let S ′ := { z = 0 } ⊂ S be theequator of S . By identifying the wrinkled map f with its image in M which wealso denote by Σ , we say that Σ has cusp singularities along S \ S ′ and unfurledswallowtail singularities along S ′ . See Figure A.1.1. Remark A.1.2 . Although a wrinkled embedding f : Σ → M is in general not asmooth embedding, it follows from (WE3) that the image f (Σ) has a well-defined k -dimensional tangent plane everywhere. We shall denote by Gdf : Σ → Gr k ( M ) the corresponding “Gauss map”, where π : Gr k ( M ) → M is the k -plane bundleon M . ONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY 87 F IGURE A.1.1. Left: cusp singularity; Right: unfurled swallow-tail singularity.According to [EM09], the significance of wrinkled embeddings is that they sat-isfy an h -principle with respect to tangential rotations . Definition A.1.3 (Tangential rotation) . Given a smooth embedding f : Σ → M , a tangential rotation is a smooth homotopy G t : Σ → Gr k ( M ) , t ∈ [0 , , such that G = df and f = π ◦ G t . The following theorem was proved by Eliashberg and Mishachev in [EM09,Theorem 2.2]. Although we are only interested in codimension submanifolds Σ ⊂ M , the theorem holds for embedded submanifolds of any codimension. Theorem A.1.4 (Wrinkled approximation of a tangential rotation) . Let G t : Σ → Gr k ( M ) be a tangential rotation of a smooth embedding f : Σ → M . Then thereexists a homotopy of wrinkled embeddings f t : Σ → M with f = f such that Gdf t : Σ → Gr k ( M ) is arbitrarily C -close to G t . If the rotation G t is fixed on aclosed set K ⊂ Σ , then the homotopy f t can also be chosen to be fixed on K . Here a homotopy of wrinkled embeddings allows birth-death type singularities.It turns out that the unfurled swallowtail singularities in a wrinkle can be elim-inated by a C -small operation called Whitney surgery . Whitney surgery involvesfirst choosing an embedded ( k − -disk D in the wrinkled Σ such that ∂D = S ′ for some wrinkle S ⊂ Σ and the interior of D is disjoint from the wrinkles. (Theexistence of such a disk D is immediate.) Then one removes the unfurled swallow-tail singularities along S ′ and adds a family of zigzags along D as in Figure A.1.2.The formal treatment of Whitney surgery can be found in [EM09, § Definition A.1.5 (Cuspidal embedding) . A smooth map f : Σ → M is a spheri-cally cuspidal embedding (or simply a cuspidal embedding ) if the following hold:(CE1) f is a topological embedding.(CE2) Sing( f ) is a finite disjoint union of smoothly embedded spheres S i ∼ = S k − , called cusp edges , in Σ . D F IGURE A.1.2. Left: before the Whitney surgery; Right: afterthe Whitney surgery. The vertical sides are identified in these pic-tures. (CE3) The map f restricted a collar neighborhood S i × ( − ǫ, ǫ ) of S i in Σ isequivalent to a map S k − × ( − ǫ, ǫ ) → S k − × R , ( y, z ) ( y, z , z ) . Remark A.1.6 . As in the case of wrinkled embeddings, the image of a cuspidalembedding f : Σ → M also has well-defined tangent planes everywhere. Wedenote by Gdf : Σ → Gr k ( M ) the corresponding Gauss map. Remark A.1.7 . Our cuspidal embeddings are called folded embeddings in [EM09],where the cusp edges are not necessarily diffeomorphic to the sphere. The reasonwe use the terminology “cuspidal embedding” is that a “folded embedding” meanssomething else in this paper. See Definition A.3.1.The following result follows immediately from Theorem A.1.4 and the Whitneysurgery on wrinkles discussed above. Theorem A.1.8 (Cuspidal approximation of tangential rotation) . Let G t : Σ → Gr k ( M ) be a tangential rotation of a smooth embedding f : Σ → M . Then thereexists a homotopy of cuspidal embeddings f t : Σ → M with f = f such that Gdf t : Σ → Gr k ( M ) is arbitrarily C -close to G t . If the rotation G t is fixed on aclosed set K ⊂ Σ , then the homotopy f t can be chosen to be fixed on K . We conclude this subsection with a smoothing operation which turns a cuspidalembedding into a smooth embedding. Suppose f : Σ → M is cuspidal embed-ding with cusp edges S i . Using the local model near cusps given by (CE3), thesmoothing operation amounts to replacing each fiber { ( y , z , z ) | z ∈ ( − ǫ, ǫ ) } at y ∈ S i by { ( y , z , z ν ( z ) ) | z ∈ ( − ǫ, ǫ ) } . Here ν : ( − ǫ, ǫ ) → [1 , is an increas-ing function which equals near and equals near ± ǫ . We denote the resultingsmooth embedding by Sm( f ) : Σ → M and the image by Sm(Σ) .A.2. Cuspidal embeddings of a disk. In the previous subsection, we saw thatany tangential rotation of a smooth embedding can be C -approximated by a ho-motopy of wrinkled or cuspidal embeddings. However, for our purposes, we alsoneed to change the homotopy class of the tangential distribution, and ask if it canbe approximated by cuspidal embeddings. This was done in great generality by ONVEX HYPERSURFACE THEORY IN CONTACT TOPOLOGY 89 Eliashberg and Mishachev in [EM00]. In this subsection we review their work in aspecial case.Let D k be the unit disk in R k . For simplicity we only consider maps f : D k → R k × R s such that f is positively transverse to ∂ s on a neighborhood of ∂D k .If f is a smooth embedding, then we identify D k with its image in R k +1 . In thiscase, we coorient D k by declaring that ∂ s is positively transverse to it near ∂D k .Using the Euclidean metric on R k +1 , let n be the positive unit normal vector fieldalong D k . Remark A.2.1 . Since f is codimension embedding, specifying a hyperplane dis-tribution along D k is equivalent to specifying a nonvanishing vector field along D k .We now define a nonvanishing vector field n ( C + , C − ) along D k . Let C ⊂ int( D k ) be an embedded codimension submanifold which divides D k into twoparts D k \ C = D + ⊔ D − such that ∂D k ⊂ D + and the sign switches when wecross C . Identify a small collar neighborhood N ( C ) ⊂ D k of C with C × [ − ǫ, ǫ ] .Choose a decomposition C = C + ∪ C − and define a vector field v on N ( C ) suchthat v points into D ± along C ± . Then n ( C + , C − ) is defined as follows: • n ( C + , C − ) = n along D + \ N ( C ) . • n ( C + , C − ) = − n along D − \ N ( C ) . • Along each fiber { y } × [ − ǫ, ǫ ] ⊂ C × [ − ǫ, ǫ ] = N ( C ) , n ( C + , C − ) rotatescounterclockwise from n to − n in the oriented -plane spanned by ( n , v ) .Roughly speaking, C + becomes a convex suture and C − becomes a concave suturewith respect to n ( C + , C − ) .We state the following result [EM00, Theorem 1.7], adapted to our special case;see also [Eli72]. Theorem A.2.2. Suppose the manifolds C + and C − are nonempty and the vectorfield n ( C + , C − ) is homotopic to ∂ s rel ∂D k . Then there exists a cuspidal embed-ding f ′ : D k → R k × R that is everywhere transverse to ∂ s , such that f ′ = f near ∂D k and Sm( f ′ ) is C -small isotopic to f rel ∂D k .Remark A.2.3 . In fact a stronger result is given in [EM00], i.e., one can furtherarrange so that the cusp edges of f ′ coincide with C . This fact, however, is notneeded in this paper.Given any smooth embedding f : D k → R k × R which is positively transverseto ∂ s on a neighborhood of ∂D k , one can always find C = C + ∪ C − such that n ( C + , C − ) is homotopic to ∂ s rel ∂D k , where C ± can be taken to be sphericalboundaries of small neighborhoods of points in D k . This implies the followingcorollary of Theorem A.2.2: Corollary A.2.4. Given any smooth embedding f : D k → R k × R which ispositively transverse to ∂ s on a neighborhood of ∂D k , there exists a cuspidal em-bedding f ′ : D k → R k × R that is everywhere transverse to ∂ s , such that f ′ = f near ∂D k and Sm( f ′ ) is C -small isotopic to f rel ∂D k . A.3. Folding hypersurfaces. Using the techniques reviewed in § A.1 and § A.2,we show in this subsection how to “fold” a generic hypersurface with respect to anonvanishing vector field.Let Σ ⊂ M be a closed cooriented hypersurface and v be a nonvanishing vectorfield defined on a neighborhood of Σ . In general it is not possible to find a C -smallisotopy φ t : M ∼ → M with φ = id M such that Σ is everywhere transverse to v ,where Σ t := φ t (Σ) . However, if we allow Σ t to have cusp singularities (here weare implicitly allowing birth-death type singularities), then there exists a cuspidalembedding Σ ⊂ M which is everywhere transverse to v , and whose smoothing Sm(Σ ) is C -small isotopic to Σ . We say Sm(Σ ) is a v -folded hypersurface inthe sense of the following definition. Definition A.3.1 ( v -folded hypersurface) . Let Σ ⊂ M be a closed, coorientedhypersurface. If v is a nonvanishing vector field defined on a collar neighborhoodof Σ , then Σ is v -folded if there exists a codimension submanifold C (Σ) ⊂ Σ such that:(1) Σ \ C (Σ) = Σ + ⊔ Σ − , where v is positively (resp. negatively) transverse to Σ + (resp. Σ − ) with respect to the coorientation of Σ and the sign switcheswhen we cross C (Σ) .(2) For each connected component C of C (Σ) , there exists an orientation-preserving diffeomorphism from C × R x ,x to a tubular neighborhood U of C in M such that Σ ∩ U is identified with C × { x = x } , C is identifiedwith C × { } , and v | U is identified with ∂ x .The submanifold C (Σ) is called the v -seam (or the seam if v is understood) of Σ .Then C (Σ) = C + (Σ) ∪ C − (Σ) , where a component C of C (Σ) belongs to C + (Σ) (resp. C − (Σ) ) if, in the local model described in (2) above, Σ + ∩ U is identified with C × { x = x , x > } (resp. Σ + ∩ U is identified with C × { x = x , x < } ). Proposition A.3.2. Given any closed cooriented hypersurface Σ ⊂ M and a non-vanishing vector field v defined on a collar neighborhood of Σ , there exists a C -small isotopy φ t : M ∼ → M with φ = id M such that Σ = φ (Σ) is v -folded.Proof. Fix a Riemannian metric on M such that v has unit length. Let n be thepositive unit normal vector field along Σ . For generic Σ , there exists a finite set ofpoints { x i } i ∈ I in Σ where v = − n . Let D i ⊂ Σ be a small disk neighborhood of x i and let S i = ∂D i . Choose nested collar neighborhoods S i ⊂ N ǫ ( S i ) ⊂ N ǫ ( S i ) of S i in Σ .It is not hard to see that there exists a homotopy n t , t ∈ [0 , , of nonvanishingvector fields along Σ with n = n such that:(1) n t = n on ∪ i ∈ I ( D i \ N ǫ ( S i )) ;(2) n = v on the complement of ∪ i ∈ I ( D i \ N ǫ ( S i )) .Now we apply Theorem A.1.8 to the tangential rotation induced by n t to obtaina C -approximation of Σ by a cuspidal hypersurface Σ ′′ , whose smoothing is v -folded on the complement of ∪ i ∈ I ( D i \ N ǫ ( S i )) and such that the v -seam is disjointfrom S i for all i ∈ I . 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