On Non-Separating Contact Hypersurfaces in Symplectic 4-Manifolds
aa r X i v : . [ m a t h . S G ] J u l ON NON-SEPARATING CONTACT HYPERSURFACES IN SYMPLECTIC –MANIFOLDS PETER ALBERS, BARNEY BRAMHAM, AND CHRIS WENDL
Abstract.
We show that certain classes of contact 3–manifolds do not admit non-separatingcontact type embeddings into any closed symplectic 4–manifolds, e.g. this is the case for allcontact manifolds that are (partially) planar or have Giroux torsion. The latter impliesthat manifolds with Giroux torsion do not admit contact type embeddings into any closedsymplectic 4–manifolds. Similarly, there are symplectic 4–manifolds that can admit smoothlyembedded non-separating hypersurfaces, but not of contact type: we observe that this is thecase for all symplectic ruled surfaces. Introduction
Main results.
Let (
W, ω ) denote a closed symplectic manifold of dimension four. Aclosed hypersurface M ⊂ W is of contact type if it is transverse to a Liouville vector field,i.e. a smooth vector field Y defined near M such that L Y ω = ω . Then ι Y ω is a contact formon M , and we will denote the resulting contact structure by ξ = ker ι Y ω ; it is independentof Y up to isotopy. If M separates W into two components, then it is said to form a convexboundary on the component where Y points outward, and a concave boundary on the othercomponent. By constructions due to Etnyre-Honda [EH02] and Eliashberg [Eli04], everycontact 3–manifold can occur as the concave boundary of some compact symplectic manifold.This is not true for convex boundaries: for instance, Gromov [Gro85] and Eliashberg [Eli90]showed that overtwisted contact manifolds can never occur as convex boundaries, and a finerobstruction comes from Giroux torsion [Gay06].In this paper, we address the question of whether a given contact 3–manifold ( M, ξ ) can oc-cur as a non-separating contact hypersurface in any closed symplectic manifold, and similarly,whether a given symplectic 4–manifold (
W, ω ) admits non-separating contact hypersurfaces.Observe that separating contact hypersurfaces always exist in abundance, e.g. the boundariesof balls in Darboux neighborhoods. We will see in Example 1.2 that non-separating contacthypersurfaces sometimes exist, but there are restrictions, as the following Theorem shows.
Theorem 1.
Suppose ( M, ξ ) is a closed contact –manifold which has any one of the followingproperties: (1) ( M, ξ ) has Giroux torsion (2) ( M, ξ ) is planar or partially planar (see Definition 1.6 below) (3) ( M, ξ ) admits a symplectic cap containing a symplectically embedded sphere of non-negative self-intersection numberThen every contact type embedding of ( M, ξ ) into any closed symplectic –manifold is sepa-rating. Mathematics Subject Classification.
Primary 32Q65; Secondary 57R17.
Key words and phrases. symplectic manifolds, contact manifolds, pseudoholomorphic curves, separatinghypersurfaces.
Remark 1.1.
Theorem 1 admits an easy generalization as follows. We will say that (
M, ξ )has any given property after contact surgery if the property holds for some contact manifold( M ′ , ξ ′ ) obtained from ( M, ξ ) by a (possibly trivial) sequence of contact connected sum op-erations and contact ( − M, ξ ) to ( M ′ , ξ ′ ): recallthat a symplectic cobordism from ( M − , ξ − ) to ( M + , ξ + ) is in general a compact symplecticmanifold ( W, ω ) with ∂W = ( − M − ) ⊔ M + , such that there is a Liouville vector field near ∂W defining ( M − , ξ − ) and ( M + , ξ + ) as concave and convex boundary components respectively.The special case where M − = ∅ is a convex filling of ( M + , ξ + ). If M + = ∅ we instead get aconcave filling of ( M − , ξ − ), also known as a symplectic cap.It will follow from the more general Theorem 7 below that Theorem 1 also holds wheneverproperties (1) or (2) hold after contact surgery. (For property (3) this statement is trivial.)The following example shows that non-separating contact type hypersurfaces do exist ingeneral. Example 1.2 ( Etnyre ) . Suppose ( W , ω ) is a compact symplectic manifold with a convexboundary that has two connected components. In this case we say that ( W , ω ) is a con-vex semifilling of each of its boundary components; the existence of such objects was firstestablished by McDuff [McD91]. Produce a new symplectic manifold ( W , ω ) with convexboundary by attaching a symplectic 1–handle along a pair of 3–balls in different componentsof ∂W . Now cap W with a concave filling of ∂W as provided by [EH02]: this produces aclosed symplectic manifold ( W, ω ), which contains both of the components of ∂W as non-separating contact hypersurfaces (see Figure 1). Figure 1.
The construction from Example 1.2 of a symplectic manifold withnon-separating contact hypersurfaces.The example demonstrates that (
M, ξ ) can occur as a non-separating hypersurface in someclosed symplectic manifold whenever it arises from a convex filling with disconnected bound-ary. There are, however, contact manifolds that never arise in this way: McDuff [McD91]showed that this is the case for the tight 3–sphere, and the result was generalized by Etnyre[Etn04] to all planar contact manifolds, i.e. those which are supported by planar open books.The latter suggests that planar open books may provide an obstruction to non-separatingcontact embeddings, and this is indeed true due to Theorem 1. As we’ll see shortly, thereare also non-planar contact manifolds (e.g. the standard contact 3–torus) which satisfy theassumptions of Theorem 1, and thus also the following corollary:
Corollary 2.
Given the assumptions of Theorem 1 (see also Remark 1.1), every convexsemifilling of ( M, ξ ) has connected boundary. Actually one can use the same methods to give a slightly simpler proof of Corollary 2 whichis independent of the theorem; we’ll do this in § N NON-SEPARATING CONTACT HYPERSURFACES IN SYMPLECTIC 4–MANIFOLDS 3
In the case of Giroux torsion, a result of Gay [Gay06] shows that (
M, ξ ) does not admitany convex fillings, thus Theorem 1 has the following stronger consequence: Corollary 3. If ( M, ξ ) has Giroux torsion (possibly after contact surgery), then it does notadmit a contact embedding into any closed symplectic –manifold. Theorem 1 will follow from some more technical results stated in § M ⊂ W exists, one can use it to construct a special noncompact symplecticmanifold ( V , ω ) with convex boundary M . We do this by first cutting W open along M toproduce a symplectic cobordism ( V , ω ) from a concave copy of M to a convex copy of M , andthen removing the concave boundary by attaching an infinite chain of copies of ( V , ω ) alongmatching concave and convex boundaries; a picture of this construction appears as Figure 4in §
5, where it is explained in detail. Now our assumptions on (
W, ω ) or (
M, ξ ) guarantee theexistence of an embedded holomorphic curve in ( V , ω ) with certain properties: in particular,we’ll show in § V , ω ). But this would imply that ( V , ω ) is compact, and thus yields acontradiction. Remark 1.3.
A contact manifold (
M, ξ ) is said to be weakly fillable if it occurs as theboundary of a compact symplectic manifold (
W, ω ) such that ω | ξ > ∂W . A fundamentalresult of Eliashberg [Eli90] and Gromov [Gro85] shows that overtwisted contact manifoldsare never weakly fillable: the original proof is based on the existence of a so-called Bishopfamily of pseudoholomorphic disks with boundary on an overtwisted disk in ∂W , and derivesa contradiction using Gromov compactness (a complete exposition may be found in [Zeh03]).In the setting described above, one can adapt the Eliashberg-Gromov argument to show thatovertwisted contact manifolds do not occur as hypersurfaces of weak contact type in any closedsymplectic manifold. If we remove the word “weak”, then this is also implied by Corollary 3since overtwisted contact manifolds have infinite Giroux torsion.The third condition in Theorem 1 is satisfied by any contact 3–manifold that has a con-tact embedding into the standard symplectic R : indeed, the latter can be identified with C P \ C P , and C P is a symplectically embedded sphere with self-intersection 1. As YashaEliashberg has pointed out to us, Theorem 1 in this case also morally follows, via the infinitechain construction sketched above, from Gromov’s classification [Gro85] of symplectic mani-folds that are Euclidean at infinity—one just has to be a little more careful in the noncompactsetting (cf. Prop. 5.3). Natural examples are the unit cotangent bundles of all closed surfacesthat admit Lagrangian embeddings into R , i.e. the torus, and the connected sums of theKlein bottle with a positive number of oriented surfaces of positive, even genus. Furtherexamples of symplectic caps containing nonnegative symplectic spheres have appeared in thework of Ohta-Ono et al [OO05, BO] on contact manifolds obtained from algebraic surfacesingularities.We now explain the notion of a partially planar contact manifold, which is due to the thirdauthor (see [Wenf]). Recall that an open book decomposition for M consists of the data( B, π ) where B ⊂ M is an oriented link, and π : M \ B → S is a fibration for which eachfiber π − (point) is an embedded surface whose closure in M has oriented boundary B . These An alternative proof closely related to the arguments in this paper appears in [Wenc].
PETER ALBERS, BARNEY BRAMHAM, AND CHRIS WENDL fibers are called the pages of the open book (
B, π ), and B is called the binding. We recall thefollowing important concept introduced by Giroux [Gir]. Definition 1.4.
A contact structure ξ on M is said to be supported by an open bookdecomposition ( B, π ) if it admits a contact form λ such that the associated Reeb vector fieldis positively transverse to the pages and is positively tangent to the link B .In particular, the component circles of B are closed Reeb orbits for such a contact form λ .These are referred to as the binding orbits. Definition 1.5.
A contact manifold (
M, ξ ) is said to be planar if it admits a supporting openbook decomposition for which each page has genus zero.Giroux established that every contact structure on a closed 3-manifold is supported by someopen book decomposition. Entyre showed in [Etn04] that all overtwisted contact structuresare planar, though not all contact structures are.The notion of a planar contact manifold can be generalized using the contact fiber sum;the following is a special case of a construction originally due to Gromov [Gro86] and Geiges[Gei97] (see also [Gei08]). For i = 1 ,
2, suppose ( M i , ξ i ) are contact manifolds with supportingopen book decompositions π i : M i \ B i → S , and γ i ⊂ B i are connected components of thebindings. Each γ i is a transverse knot, thus one can identify neighborhoods N ( γ i ) with solidtori via an orientation preserving mapΦ : N ( γ ) ∪ N ( γ ) → S × D , thus defining coordinates ( θ, ρ, φ ), where θ ∈ S and ( ρ, φ ) are polar coordinates on D (forsimplicity we shall take φ ∈ S = R / Z , thus the actual angle is this times 2 π ). We will assumewithout loss of generality (and perhaps after a small isotopy of the open books) that thesecoordinates have the following properties:(1) The contact structure ξ i is the kernel of λ i = f ( ρ ) dθ + g ( ρ ) dφ for some pair offunctions f and g with f (0) > g (0) = 0.(2) The pages of π i have the form { φ = const } near γ i .Note that the contact condition requires f ( ρ ) g ′ ( ρ ) − f ′ ( ρ ) g ( ρ ) > ρ > g ′′ (0) > M , ξ ) Φ ( M , ξ )can be defined in two steps:(i) Modify ( M i , ξ i ) by “blowing up” γ i to produce a contact manifold ( c M i , ˆ ξ i ) with pre-Lagrangian torus boundary: we do this by removing a solid torus neighborhood { ρ ≤ ǫ } and replacing it with S × [0 , ǫ ] × S by the natural identification of the coordinates( θ, ρ, φ ) ∈ S × [0 , ǫ ] × S . We also modify λ i for ρ ∈ [0 , ǫ ) to define a smoothcontact form near ∂ c M i by making C –small changes to f and g so that they becomerestrictions of even and odd functions respectively, with g ′ (0) >
0. In terms of theReeb vector field defined by λ i , the result of this change is to replace the single Reeborbit originally at { ρ = 0 } by a torus S × S foliated by Reeb orbits of the form S × { pt } .(ii) Attach ( c M , ˆ ξ ) to ( c M , ˆ ξ ) along their boundaries as follows: first, define new coordi-nates (ˆ θ, ˆ ρ, ˆ φ ) ∈ S × R × S near ∂ c M i so that they are the same as the old coordinates N NON-SEPARATING CONTACT HYPERSURFACES IN SYMPLECTIC 4–MANIFOLDS 5 on c M , but on c M we set (ˆ θ, ˆ ρ, ˆ φ ) := ( θ, − ρ, − φ ) , so ˆ ρ ≤ ∂ c M . We now attach c M to c M via a diffeomorphism such that(ˆ θ, ˆ ρ, ˆ φ ) ∈ S × [ − ǫ, ǫ ] × S become well defined coordinates after attaching. Ourassumptions on the modified functions f and g imply also that f (ˆ ρ ) d ˆ θ + g (ˆ ρ ) d ˆ φ givesa smooth contact form on M Φ M which matches the original outside the region { ˆ ρ ∈ ( − ǫ, ǫ ) } .In a straightforward way, one can generalize this definition to a sum of two or more openbooks on contact manifolds ( M , ξ ) , . . . , ( M N , ξ N ) along multiple binding components: theneach of these components becomes a boundary component in its respective “blown up” man-ifold c M i , and it becomes a special pre-Lagrangian torus in the sum Φ ( M i , ξ i ) . Definition 1.6.
We say that (
M, ξ ) is partially planar if it can be constructed in the abovemanner as a contact fiber sum along binding orbits of open book decompositions, at least oneof which is planar.Obviously, every planar contact manifold is also partially planar. Since there exist contact3–manifolds that admit semifillings with disconnected boundary, a consequence of Corollary 2is now the following:
Corollary 4.
Not every contact manifold is partially planar.
Example 1.7.
McDuff showed in [McD91] that for any closed oriented surface Σ of genus atleast two, if ST ∗ Σ denotes the unit cotangent bundle, then there is a symplectic structure on[0 , × ST ∗ Σ which is convex on the boundary and induces the canonical contact structureat { } × ST ∗ Σ. More generally, Geiges [Gei95] constructed a class of closed 3–manifolds M which admit pairs of contact forms λ ± such that λ + ∧ dλ + = − λ − ∧ dλ − > λ + ∧ dλ − = λ − ∧ dλ + = 0 . In this situation, [0 , × M admits a symplectic structure such that both boundary componentsare convex, giving a convex filling of ( M, ker λ + ) ⊔ ( − M, ker λ − ). It follows from Corollary 2that none of these contact manifolds are partially planar. Moreover by Example 1.2, each ofthem admits a non-separating contact type embedding into some closed symplectic manifold.The next example shows that there are also partially planar contact manifolds that are notplanar. Example 1.8.
The standard contact S × S is planar: it admits a supporting open bookdecomposition with two binding orbits connected by cylindrical pages. If we take two copiesof this, pair up both of their respective binding components and construct the fiber sum,we obtain the standard contact T , which is not planar due to a result of Etnyre [Etn04].In fact, each of the tight contact tori ( T , ξ n ), where ξ n = ker [cos(2 πnθ ) dx + sin(2 πnθ ) dy ]in coordinates ( x, y, θ ) ∈ S × S × S , can be obtained as a fiber sum of 2 n copies of thestandard S × S ; see Figure 2. By a result of Kanda [Kan97], this includes every tight contactstructure on T .By the above example, every contact structure on T is partially planar. In fact, otherthan the standard torus ( T , ξ ), all contact 3–tori also have Giroux torsion, thus ξ is the PETER ALBERS, BARNEY BRAMHAM, AND CHRIS WENDL
Figure 2.
At left, we see four copies of the tight S × S , represented by openbooks with two binding components and cylindrical pages. For each dottedoval surrounding two binding components, we construct the contact fiber sumto produce the manifold at right, containing four special pre-Lagrangian tori(the black line segments) that separate regions foliated by cylinders. The resultis the tight 3–torus ( T , ξ ). In general, one can construct ( T , ξ n ) from 2 n copies of the tight S × S .only convex fillable contact structure on T . Theorem 1 therefore implies that every contacttype embedding of T into a closed symplectic 4–manifold separates (and the induced contactstructure must be ξ ). This result is not true for embeddings of weak contact type: in factall of the tight tori ( T , ξ n ) admit weak symplectic semifillings with disconnected boundary[Etn], and thus by the construction in Example 1.2, they also admit non-separating weaklycontact type embeddings.Recall however that if ( W, ω ) is a weak filling of (
M, ξ ) and M is a homology 3–sphere,then ω can always be deformed in a collar neighborhood of ∂W to produce a convex filling of( M, ξ ); see for instance [Gei08, Lemma 6.5.5]. Thus our results have corresponding versionsfor weakly contact hypersurfaces that are homology 3–spheres. For example, since the onlytight contact structure on S is planar, every weakly contact type embedding of S into aclosed symplectic 4–manifold must separate.Here is a more general example that also implies the observation made above about the3–torus. Let Σ = Σ + ∪ Γ Σ − denote any closed oriented surface obtained as the union of two nonempty surfaces withboundary Σ ± along a multicurve Γ ⊂ Σ. By results of Giroux [Gir01] and Honda [Hon00],the manifold M Γ := S × Σ admits a unique (up to isotopy) S –invariant contact structure ξ Γ which makes Γ the dividing set on { const } × Σ. We claim that ( M Γ , ξ Γ ) is partially planarwhenever there exists a connected component of Σ \ Γ having genus zero. Indeed, for anyconnected component Σ ⊂ Σ \ Γ, the closure of S × Σ may be viewed as an open book withpage Σ and trivial monodromy, blown up at all its binding circles; the entirety of ( M Γ , ξ Γ )can thus be obtained by attaching these blown up open books. (The tight 3–tori arise fromthe case where Σ ∼ = T and Γ is a union of parallel curves that are primitive in H ( T ).) N NON-SEPARATING CONTACT HYPERSURFACES IN SYMPLECTIC 4–MANIFOLDS 7
Moreover, using Etnyre’s obstruction [Etn04] it is easy to construct many examples ( M Γ , ξ Γ )which are partially planar (as just explained) but not planar. Theorem 1 now implies: Corollary 5. If Σ \ Γ has a connected component of genus zero, then the S –invariant contactmanifold ( S × Σ , ξ Γ ) does not admit any non-separating contact type embeddings into closedsymplectic –manifolds. Finally, the following demonstrates that in some settings where non-separating hypersur-faces can be embedded smoothly , they can never be contact type. In contrast to Theorem 1,here the assumptions are on the ambient symplectic 4-manifold and not the contact manifold.
Theorem 6.
If the closed and connected symplectic 4-manifold ( W, ω ) contains a symplec-tically embedded sphere S ⊂ W with self-intersection number S • S ≥ , then every closedcontact type hypersurface in W is separating. The reason for this is closely related to McDuff’s results [McD90], which imply that (
W, ω )in this situation is always rational or ruled (up to symplectic blowup). In fact, the case where S • S > W is then a blowup of either S × S or C P and thus simply connected, so it does not admit non-separating hypersurfacesat all (contact or otherwise). The case S • S = 0 is more interesting: the key fact here isthat one can choose a compatible almost complex structure J for which any given contacthypersurface M ⊂ W is J –convex, and W is foliated by a family of embedded J –holomorphicspheres (possibly including some isolated nodal spheres unless ( W, ω ) is minimal). If M doesnot separate, then there exists a connected infinite cover ( f W , ˜ J ) of ( W, J ), constructed bygluing together infinitely many copies of W \ M in a sequence. Now the J –holomorphicspheres in W lift to f W and form a foliation, which must include a J –holomorphic sphere thattouches a lift of M tangentially from below, violating J –convexity. That’s a quick sketch ofthe proof—we’ll give an alternative proof in § ℓ × S ⊂ Σ × S , where Σ is any closedoriented surface of positive genus and ℓ ⊂ Σ is a non-separating closed curve. It follows thata hypersurface isotopic to this one is never contact type.1.2.
Open questions.
Let Ξ(3) denote the collection of closed 3–manifolds with positive,cooriented contact structures, and consider the inclusionsΞ nonsep (3) ( Ξ embed (3) ( Ξ(3) , where Ξ embed (3) denotes all ( M, ξ ) ∈ Ξ(3) that admit a contact type embedding into someclosed symplectic manifold, and Ξ nonsep (3) denotes those that admit a non-separating embed-ding. The results stated in § M, ξ ) is convex fillable then it is also in Ξ embed (3), since a filling canalways be capped to produce a closed symplectic manifold. Conversely, if (
M, ξ ) admits a separating contact type embedding, then it is fillable. While the same is not strictly truefor a non-separating embedding, the construction depicted in Figure 4 of § geometrically bounded , which makes it a good setting for J –holomorphic curves. In this context, any filling obstruction that involves J –holomorphiccurves can also serve as an obstruction to non-separating contact embeddings (cf. Corollary 3),thus implying that ( M, ξ ) Ξ embed (3). This motivates the conjecture that, in fact, Ξ embed (3)is the same as the set of convex fillable contact 3–manifolds. PETER ALBERS, BARNEY BRAMHAM, AND CHRIS WENDL
Conjecture 1.
If (
M, ξ ) is not convex fillable, then it admits no contact type embeddingsinto any closed symplectic manifold.Equivalently, this would mean there is no contact 3–manifold that admits only non-separating contact type embeddings.A more ambitious conjecture would arise from Example 1.2, which is the only method weare yet aware of for constructing non-separating contact embeddings: (
M, ξ ) ∈ Ξ nonsep (3)whenever it admits a convex semifilling with disconnected boundary. The latter class ofcontact manifolds is evidently somewhat special, and one wonders whether it might be equalto Ξ nonsep (3). Question 1.
Is there a contact 3–manifold that admits a non-separating contact type em-bedding but not a convex semifilling with disconnected boundary?Finally, observe that while Theorem 6 rules out the existence of a non-separating contacthypersurface (
M, ξ ) ⊂ ( W, ω ) if (
W, ω ) is rational or ruled, it still allows the possibility that(
M, ξ ) ∈ Ξ nonsep (3) but admits a separating embedding into ( W, ω ). There is some reasonto suspect that this could still never happen. There are indeed cases where the existenceof a contact embedding of (
M, ξ ) into some particular symplectic manifold implies (
M, ξ ) Ξ nonsep (3), e.g. this is true if ( M, ξ ) ֒ → ( R , ω ). Moreover, the simplest known example of amanifold in Ξ nonsep (3), the unit cotangent bundle of a higher genus surface, has been shownby Welschinger [Wel07] to admit no contact type embeddings into rational or ruled symplectic4–manifolds. Question 2.
Is there a contact 3–manifold that admits a contact type embedding into somerational/ruled symplectic 4–manifold and also admits a non-separating contact type embed-ding into some other closed symplectic manifold?2.
Pseudoholomorphic curves in symplectizations
Technical background.
In this section we collect a number of important technicaldefinitions. A positive contact form on a 3–manifold M is a 1–form λ for which λ ∧ dλ > ξ := ker λ is then a contact structure. The equations ι X λ dλ = 0 and λ ( X λ ) = 1 uniquely determine a vector field X λ , called the Reeb vector field associated to λ .Since X λ is everywhere transverse to ξ , one obtains a splitting T M = R X λ ⊕ ξ . Moreover,( ξ, dλ | ξ ) is a symplectic vector bundle, and the flow of X λ preserves λ , hence also ( ξ, dλ | ξ ).A periodic Reeb orbit of period T > λ is a smooth map γ : R /T Z → M satisfying ˙ γ ( t ) = X λ ( γ ( t )). We identify all possible reparametrizations t γ ( t + const). AReeb orbit is called simply covered if it has degree 1 onto its image, i.e. it is an embedding.If γ covers a simply covered orbit with period τ >
0, we call τ the minimal period of γ .Since the Reeb flow preserves the symplectic vector bundle ( ξ, dλ | ξ ), linearizing about aperiodic orbit γ determines a symplectic linear map dφ T ( p ) : ξ p → ξ p for each p in the imageof γ . Then γ is said to be nondegenerate if 1 is not an eigenvalue of this map; this conditionis independent of the point p . More generally, an orbit γ of period T is Morse-Bott if it liesin a submanifold N ⊂ M foliated by T –periodic orbits, such that the 1–eigenspace of dφ T ( p )is precisely T p N . We then call N a Morse-Bott submanifold. A contact form λ is said to benondegenerate if all of its periodic Reeb orbits are nondegenerate, and Morse-Bott if everyperiodic orbit belongs to a Morse-Bott submanifold. N NON-SEPARATING CONTACT HYPERSURFACES IN SYMPLECTIC 4–MANIFOLDS 9
Given a symplectic trivialization Φ of ( ξ, dλ ) along a T –periodic orbit γ , the linearizedflow dφ t ( p ) for t ∈ [0 , T ] defines a continuous family of symplectic matrices, which has a welldefined Conley-Zehnder index if γ is nondegenerate: we denote this index by µ ΦCZ ( γ ) ∈ Z .It is convenient also to express this in terms of asymptotic operators: associated to any T –periodic Reeb orbit γ is a linear operator A γ : Γ( x ∗ ξ ) → Γ( x ∗ ξ ), where x : R / Z → M is the reparametrization x ( t ) := γ ( T t ). If ∇ is a symmetric connection on T M and J is acomplex structure on ξ → M compatible with the symplectic structure dλ | ξ , then A γ can bedefined on smooth sections by A γ η = − J ( ∇ t η − T ∇ η X λ ) . This expression is independent of the choice of connection. Choosing a unitary trivializationΦ of x ∗ ξ , A γ is identified with the operator C ∞ ( S , R ) → C ∞ ( S , R ) : η
7→ − J ddt η − S · η, (2.1)where S ( t ) is some smooth loop of symmetric 2–by–2 matrices. Thus the equation A γ η = 0defines a linear Hamiltonian flow, and one can show that the resulting family of symplecticmatrices matches the family obtained from dφ t ( p ). It follows that A γ has trivial kernel if andonly if γ is nondegenerate, and we can use the linear Hamiltonian flow determined by (2.1)to define an integer µ ΦCZ ( A γ ), which matches µ ΦCZ ( γ ). The advantage of this definition is thatit does not reference the orbit directly, but makes sense for any operator that takes the formof (2.1) in the trivialization: in particular we can define µ ΦCZ ( A γ − c ) ∈ Z whenever c ∈ R isnot an eigenvalue of A γ , even if γ is degenerate. For this we will use the shorthand notation µ ΦCZ ( γ − c ) := µ ΦCZ ( A γ − c ) . We now recall some of the important spectral properties of asymptotic operators. For moredetails and proofs we refer to [HWZ95]. A γ extends to an unbounded self-adjoint operator on the complexified Hilbert space L ( x ∗ ξ );its spectrum σ ( A γ ) consists of real eigenvalues of multiplicity at most 2 that accumulateonly at infinity. Generalizing the statement above about nondegeneracy, if γ belongs to aMorse-Bott submanifold of dimension n ∈ { , , } , then the 0–eigenspace of A γ is ( n − η of A γ has a well defined winding number wind Φ ( η ) ∈ Z relativeto the trivialization, which is independent of the choice of η in its eigenspace. Thus we mayspeak of the winding number wind Φ ( µ ) ∈ Z for each eigenvalue µ ∈ σ ( A γ ), and it turns outthat the map σ ( A γ ) → Z : µ wind Φ ( µ ) is non-decreasing and attains every value exactlytwice (counting multiplicity). The following integers α Φ − ( γ ) := max { wind Φ ( µ ) | µ < A γ } α Φ+ ( γ ) := min { wind Φ ( µ ) | µ > A γ } are therefore determined by the eigenfunctions with eigenvalues closest to 0 that are negativeand positive respectively. The number p ( γ ) := α Φ+ ( γ ) − α Φ − ( γ ) is called the parity of γ ; it isindependent of Φ and necessarily equals 0 or 1 if γ is nondegenerate. More generally, we canreplace A γ by A γ − c for some c ∈ R and similarly define α Φ ± ( γ − c ) and p ( γ − c ); then if c σ ( A γ ), a result in [HWZ95] implies the relation µ ΦCZ ( γ − c ) = 2 α Φ − ( γ − c ) + p ( γ − c ) = 2 α Φ+ ( γ − c ) − p ( γ − c ) . (2.2) Observe that every Morse-Bott submanifold of dimension 2 admits a nonzero vector fieldand is thus either a torus or a Klein bottle. The following characterization of Morse-Bott toriis a simple consequence of the spectral properties of A γ (cf. [Wenb, Prop. 4.1]). Proposition 2.1.
Suppose γ is a Morse-Bott periodic orbit of X λ belonging to a Morse-Bottsubmanifold N ⊂ M diffeomorphic to T . Then the Morse-Bott property is satisfied for allcovers of all orbits in N , and they all have the same minimal period. We will also need a relative version of the standard genericity result for nondegeneratecontact forms.
Lemma 2.2.
Suppose N ⊂ M is a union of 2–tori which are Morse-Bott submanifolds forsome contact form λ . Then for any T >
0, there exists an arbitrarily small perturbation λ of λ such that λ = λ on a neighborhood of N and every periodic orbit of X λ with periodless than T is Morse-Bott. Proof . Since all orbits in N are Morse-Bott (including all multiple covers, due to Prop. 2.1),for any T > U of N such that U \ N contains no periodicorbits with period less than T . By Theorem 13 in the appendix, one can then find a genericsmall perturbation of λ with support in M \ U so that all orbits passing through M \ U arenondegenerate. (cid:3) We now recall the basic notions of holomorphic curves in symplectizations and their asymp-totic properties. The symplectization of a contact manifold (
M, ξ = ker λ ) is the product space R × M equipped with the exact symplectic form d ( e a λ ), where a : R × M → R refers to the R coordinate. An almost complex structure J on the symplectization is said to be admissibleif it is R –invariant, restricts to the symplectic vector bundle ( ξ, dλ ) as a compatible complexstructure, and satisfies J ∂ a = X λ . Any admissible J tames the symplectic form d ( e a λ ), andmore generally tames every symplectic form d ( ϕλ ) where ϕ : R → (0 , ∞ ) is smooth with ϕ ′ > pseudoholomorphic (or J –holomorphic or simply holomorphic ) curve from a puncturedRiemann surface ( ˙Σ , j ), into an almost complex manifold ( W, J ) is a solution u : ˙Σ → W tothe nonlinear Cauchy-Riemann equation T u ◦ j = J ( u ) ◦ T u . Here we take ˙Σ := Σ \ Γ forsome finite set of points Γ ⊂ Σ, where (Σ , j ) is a closed connected Riemann surface.For the rest of this section, let us consider only the case where the target is the symplecti-zation of (
M, λ ), and J is an admissible almost complex structure on R × M . The simplestcase of a punctured J –holomorphic curve in this setting is the so-called trivial cylinder u : S \ { , ∞} ∼ = R × S → R × M : ( s, t ) ( T s, γ ( T t )) , where T > γ is any T –periodic Reeb orbit. Following [Hof93, BEH + J –holomorphic curve u : ˙Σ → R × M can be defined as follows. Fix any constant C > E ( u ) := sup ϕ ∈T Z ˙Σ u ∗ d ( ϕλ ) (2.3)where T is the set of smooth maps ϕ : R → (0 , C ) with ϕ ′ >
0. Since J is compatible with d ( ϕλ ) for all ϕ ∈ T , the integrand in (2.3) is always nonnegative, thus u is constant if andonly if its energy vanishes. Observe that the integrand of R ˙Σ u ∗ dλ is also nonnegative, andthis integral is finite if u has finite energy: it vanishes identically if and only if u is a branchedcover of a trivial cylinder. N NON-SEPARATING CONTACT HYPERSURFACES IN SYMPLECTIC 4–MANIFOLDS 11
Definition 2.3.
We will say that u : ˙Σ → R × M is a finite energy J –holomorphic curve ifit is proper and E ( u ) < ∞ .Note that properness only fails when there exist punctures having neighborhoods whichare mapped into a compact set, in which case these punctures can be removed by Gromov’sremovable singularity theorem. Since d ( ϕλ ) is exact, Stokes’ theorem implies that not allpunctures are removable unless u is constant.Let us recall now the behaviour of a finite energy J –holomorphic curve u : ˙Σ → R × M in theneighborhood of a puncture. Each puncture z ∈ Γ has a neighborhood on which the R –value of u tends to + ∞ or −∞ , and we say that z is a positive/negative puncture respectively. Denotethe resulting partition into positive and negative punctures by Γ = Γ + ∪ Γ − . Restricting toa neighborhood of a puncture, we obtain a curve whose domain is the punctured closed disc,which is biholomorphic to both Z + := [0 , ∞ ) × S and Z − := ( −∞ , × S with the standardcomplex structure. It is convenient to choose the domain of the restricted curve to be Z + or Z − for z ∈ Γ + or z ∈ Γ − respectively, and we will write u : Z ± → R × M . It was shown byHofer in [Hof93] that for any sequence | s k | → ∞ , there exists a subsequence such that u ( s k , · )converges in C ∞ ( S , M ) to γ ( T · ), where γ is a T –periodic Reeb orbit for some T >
0. Wesay in this case that u is asymptotic to γ , and γ is an asymptotic orbit of u .In the following statement, we choose any R –invariant connection on R × M to define theexponential map, and use the term asymptotically trivial coordinates to refer to a diffeomor-phism ( σ, τ ) : Z ± → Z ± such that σ ( s, t ) − s and τ ( s, t ) − t approach constants as | s | → ∞ and their derivatives of all orders decay to zero. Theorem ([HWZ96a, HWZ96b, Mor03]) . Suppose u : Z ± → R × M has finite energy and isasymptotic to a Morse-Bott Reeb orbit γ of period T > . Then there exist asymptoticallytrivial coordinates ( σ, τ ) such that for sufficiently large | σ | , either u ( σ, τ ) = ( T σ, γ ( T τ )) or u ( σ, τ ) = exp ( T σ,γ ( T τ )) [ e µσ ( e µ ( τ ) + r ( σ, τ ))] , (2.4) where e µ is an eigenfunction of A γ with eigenvalue µ ∈ σ ( A γ ) such that ± µ < , and the“remainder” term r ( σ, τ ) ∈ ξ γ ( T σ ) decays to zero uniformly with all derivatives as | σ | → ∞ . Definition 2.4.
When (2.4) holds, we call e µ the asymptotic eigenfunction of u at thepuncture, and say that u has transversal convergence rate | µ | . In the case where u ( σ, τ ) =( T σ, γ ( T τ )), we define the asymptotic eigenfunction to be 0 and the transversal convergencerate to be ∞ .Observe that the asymptotic eigenfunction e µ is determined uniquely once a parametriza-tion of γ is fixed. We know also from the monotonicity of winding numbers that wind Φ ( e µ ) ≤ α Φ − ( γ ) if the puncture is positive, and wind Φ ( e µ ) ≥ α Φ+ ( γ ) if it is negative.Let π λ : T M → ξ denote the natural projection with respect to the splitting T M = R X λ ⊕ ξ and suppose u = ( u R , u M ) : ˙Σ → R × M is a finite energy J –holomorphic curve. Then thecomposition π λ ◦ T u M defines a section of the bundle of complex linear homomorphisms( T ˙Σ , j ) → ( u ∗ ξ, J ). As shown in [HWZ95], this section satisfies a linear Cauchy-Riemanntype equation, and thus is either trivial or has a discrete set of zeros, all of positive order.The former holds if and only if any asymptotic eigenfunction of u vanishes, in which casethey all do: then R ˙Σ u ∗ dλ = 0 and u is a branched cover of a trivial cylinder. Otherwise, (2.4)implies that π λ ◦ T u M has finitely many zeros, and we denote the algebraic count of these bywind π ( u ) ∈ Z . Clearly wind π ( u ) ≥
0, with equality if and only if u M : ˙Σ → M is an immersion. Property ( ⋆ ) and the main results. We now use holomorphic curves to define twotechnical conditions on contact manifolds which imply the results stated in §
1. Property ( ⋆ )and its weak version, introduced below, will serve as obstructions to the existence of non-separating contact embeddings. They are implied by each of the contact topological assump-tions mentioned in Theorem 1, and in fact are more general (see also [Wenf]). Definition 2.5.
A closed three-dimensional contact manifold (
M, ξ ) satisfies property ( ⋆ ) ifthere exists a contact form λ with ker λ = ξ and an admissible R –invariant almost complexstructure J on the symplectization R × M , which admits a finite energy J –holomorphicpunctured sphere u = ( u R , u M ) : ˙Σ = S \ { z , . . . , z N } → R × M with the following properties:(1) u M is an embedding, and the closure of u M ( ˙Σ) ⊂ M is an embedded surface whoseoriented boundary is a union of Reeb orbits, called the “asymptotic orbits” of u .(2) Each asymptotic orbit of u is nondegenerate or Morse-Bott.(3) If T , . . . , T N are the periods of the asymptotic orbits of u , then every Reeb orbit notin the same Morse-Bott submanifold with one of these has period strictly greater than T + . . . + T N .(4) u has no asymptotic orbit that is nondegenerate with Conley-Zehnder index zero,relative to the natural trivialization determined by the image of u M near the puncture.(5) If any asymptotic orbit of u belongs to a 2–dimensional Morse-Bott manifold N ⊂ M disjoint from u M ( ˙Σ), then N is a torus and contains no other asymptotic orbits of u . Remarks. • The fact that Reeb orbits comprise the oriented boundary of u M ( ˙Σ) implies that everypuncture of u is positive. Moreover, each puncture is asymptotic to a distinct Reeborbit, which is simply covered. • The asymptotic formula (2.4) implies that on each cylindrical end of ˙Σ, u M does notintersect the corresponding asymptotic orbit, thus it defines a natural trivializationof ξ along this orbit. One can then show (cf. (2.2)) that relative to this trivialization,the orbit always has nonnegative Conley-Zehnder index if it is nondegenerate—thusour definition requires this index to be anything strictly larger than the minimumpossible value. Definition 2.6.
We say that a closed three-dimensional contact manifold (
M, ξ ) satisfiesweak property ( ⋆ ) if there is a symplectic cobordism ( W, ω ) from (
M, ξ ) to a contact manifold( M ′ , ξ ′ ), such that either ( W, ω ) contains a symplectically embedded sphere of nonnegativeself-intersection number or ( M ′ , ξ ′ ) satisfies property ( ⋆ ).For example, ( M, ξ ) satisfies weak property ( ⋆ ) if it admits a symplectic cap containing anonnegative symplectic sphere, or if it can be made to satisfy property ( ⋆ ) after a sequenceof contact ( − ⋆ ) implies weakproperty ( ⋆ ), and it’s plausible that the converse may also be true, though this is presumablyhard to prove.We can now state some more technical results that imply Theorem 1. These will be provedin §
5, using the machinery of § Theorem 7.
Let ( W, ω ) be a closed and connected symplectic 4-manifold which contains aclosed contact type hypersurface M ⊂ W satisfying weak property ( ⋆ ) . Then M separates W . N NON-SEPARATING CONTACT HYPERSURFACES IN SYMPLECTIC 4–MANIFOLDS 13
Theorem 8.
Let ( W, ω ) be a compact and connected symplectic 4-manifold with convex bound-ary containing a connected component M ⊂ ∂W that satisfies weak property ( ⋆ ) . Then ∂W is connected. Theorem 9.
Let ( W, ω ) be a compact and connected 4-manifold with convex boundary ( M, ξ ) satisfying weak property ( ⋆ ) . Then any closed contact type hypersurface H in W \ M separates W into a convex filling of H and a symplectic cobordism from H to M . In particular, H alsosatisfies the weak ( ⋆ ) property. Remark 2.7.
A compact connected symplectic manifold with convex boundary can nevercontain a symplectic sphere of nonnegative self-intersection. This follows easily from thearguments we will use to prove the above results: otherwise one would find a family ofembedded holomorphic spheres foliating the positive end of the symplectization of the convexboundary, and thus violating the maximum principle.
Remark 2.8.
Note that property ( ⋆ ) depends only on the contact structure: we do not assume in any of these theorems that the contact form induced on M by a Liouville vectorfield is the same one which appears in Definition 2.5.We will show in § M, ξ ) with Giroux torsion satisfies Prop-erty ( ⋆ ). It turns out that this is also true for a contact fiber sum of open books ( M, ξ ) = Φ ( M i , ξ i ) whenever any of the summands ( M i , ξ i ) is planar. This follows from an importantrelationship between open books and holomorphic curves: namely, it is shown in [Abb, Wend]that if the open book on ( M i , ξ i ) is planar, one can take its pages to be projected imagesof embedded index 2 holomorphic curves. A minor variation on this construction in [Wene]extends it to the blown up manifold ( c M i , ˆ ξ i ): the difference here is that each holomorphic pageis asymptotic to a different orbit in a Morse-Bott family foliating the boundary. Moreover,one can easily arrange the contact form in this construction so that all the asymptotic orbitsare either elliptic or Morse-Bott and have much smaller period than any other Reeb orbitin Φ ( M i , ξ i ). It follows that Φ ( M i , ξ i ) satisfies property ( ⋆ ) if any of its constituent openbooks is planar. 3. Giroux torsion
Following a construction in ([Wenc]) but being more careful about periods, we now establishthe following.
Proposition 3.1.
Let ( M, ξ ) be a closed contact manifold having Giroux torsion. Then ( M, ξ ) satisfies property ( ⋆ ) . Proof . By definition, Giroux torsion means that (
M, ξ ) contains a subset T that can beidentified with a thickened torus S × S × [0 , ξ has the form ξ = ker [cos(2 πθ ) dx + sin(2 πθ ) dy ] (3.1)in coordinates ( x, y, θ ) ∈ S × S × [0 , ξ = ker λ for some contact form λ that is Morse-Bott outside of T , and in T has the form λ = f ( θ ) dx + g ( θ ) dy for smoothfunctions f, g : [0 , → R with γ ( θ ) := ( f ( θ ) , g ( θ )) = h ( θ ) e πiθ ∈ R , where h ( θ ) > h ( θ ) = 1 for θ near 0 and 1. The path γ is thus closed and bounds astar-shaped region in R , and we will show that λ has the desired properties if γ bounds asuitably oblong oval. The Reeb vector field of λ on T is given by X λ = 1 D ( θ ) ( g ′ ( θ ) ∂ x − f ′ ( θ ) ∂ y ) , (3.2)where D ( θ ) := f ( θ ) g ′ ( θ ) − f ′ ( θ ) g ( θ ) >
0. Since this has no ∂ θ component, each torus N ( θ ) := { ( x, y, θ ) | ( x, y ) ∈ S × S } ⊂ T is invariant under the Reeb flow. Moreover, the Reeb flowon each N ( θ ) is linear and has closed orbits if and only if dx ( X λ ) /dy ( X λ ) ∈ Q ∪ {∞} . From(3.2), this ratio is − g ′ ( θ ) /f ′ ( θ ) = − slope( γ ′ ( θ )), so N ( θ ) has closed orbits precisely whenslope( γ ′ ( θ )) is rational or infinite. In this case every orbit in N ( θ ) is closed and representsthe same class in H ( N ( θ )) = Z , which we will denote by a pair of integers ( p ( θ ) , q ( θ )) withgcd( | p ( θ ) | , | q ( θ ) | ) = 1 and p ( θ ) q ( θ ) = − slope( γ ′ ( θ )) ∈ Q ∪ {∞} . (3.3)Since dλ vanishes on N ( θ ), all closed simply covered orbits in N ( θ ) have the same period,which we will denote by T ( θ ) >
0. If σ : R / Z → N ( θ ) parametrizes such an orbit, we compute T ( θ ) = Z σ ∗ λ = p ( θ ) f ( θ ) + q ( θ ) g ( θ ) . (3.4) Lemma 3.2.
Fix ǫ > γ ( θ ) = h ( θ ) e πiθ bounds a convex set symmetric about both axes, h (1 /
4) = h (3 /
4) = ǫ and γ ′ ( θ )and γ ′′ ( θ ) are always linearly independent. Then:(1) λ is Morse-Bott.(2) X λ = ǫ ∂ y on N (1 /
4) and − ǫ ∂ y on N (3 / T (1 /
4) = T (3 /
4) = ǫ , and T ( θ ) > / θ at which N ( θ ) has closed orbits. Proof . It follows by straightforward computation from the assumption that γ ′ ( θ ) and γ ′′ ( θ )are linearly independent that each N ( θ ) with closed orbits is a Morse-Bott submanifold. Thesecond claim follows immediately from (3.2) since symmetry requires g ′ (1 /
4) = g ′ (3 /
4) = 0,and it is then clear that T (1 /
4) = T (3 /
4) = ǫ .To show that all other values of θ have T ( θ ) > /
4, observe first that by symmetry, wecan always assume g ′ and − f ′ have the same sign as f and g respectively. Thus sign( p ) =sign( dx ( X λ )) = sign( g ′ ) = sign( f ) and sign( q ) = sign( dy ( X λ )) = sign( − f ′ ) = sign( g ), soformula (3.4) becomes T ( θ ) = | p ( θ ) || f ( θ ) | + | q ( θ ) || g ( θ ) | . (3.5)Let ∆ denote the diamond shaped region in the xy –plane for which | x | + | y | ≤ / Case γ ( θ ) ∈ ∆ : In this region, outside of the special values θ = 1 / , / < | slope( γ ′ ( θ )) | < ǫ , and by convexity, | g ( θ ) | > ǫ/
2. With the slope nonzero, it follows from(3.3) that both p and q are nonzero: in particular | p | ≥
1. Then from the previous inequality, | q | = | q || p | | p | = 1 | slope( γ ′ ( θ )) | | p | > ǫ | p | ≥ ǫ , and using (3.5), T ( θ ) ≥ | q ( θ ) || g ( θ ) | > ǫ ǫ = 1 / . Case γ ( θ ) / ∈ ∆ : After verifying explicitly that T (0) = T (1) = 1, we can exclude these twocases and assume once more that both p ( θ ) and q ( θ ) are nonzero. Then (3.5) gives T ( θ ) ≥ | f ( θ ) | + | g ( θ ) | > / N NON-SEPARATING CONTACT HYPERSURFACES IN SYMPLECTIC 4–MANIFOLDS 15
PSfrag replacements ∆ − ǫ − ǫ γ Figure 3.
The curve γ and (shaded) region ∆ in Lemma 3.2.by the definition of ∆. (cid:3) Using the lemma, we can arrange λ in T without changing it in M \ T so that T (1 /
4) = T (3 /
4) = ǫ is less than half the period of every other periodic orbit in M . Now copyingthe construction in [Wenc, Example 2.11], we construct a family of embedded J –holomorphiccylinders in R × T that foliate the region between N (1 /
4) and N (3 / u : R × S → R × M : ( s, t ) ( α ( s ) + a , x , t, ρ ( s )) , where a ∈ R and x ∈ S are arbitrary constants, α : R → R is a fixed function that goesto + ∞ at both ends and ρ : R → (1 / , /
4) is a fixed orientation reversing diffeomorphism.Any of these cylinders satisfies the requirements of property ( ⋆ ). (cid:3) Fredholm theory, intersection numbers and compactness
In this section, assume (
W, ω ) is a connected (and possibly noncompact) symplectic 4–manifold with convex boundary ∂W = M . The boundary need not be connected or nonempty;for simplicity we will assume that it is compact, though we will later be able to relax thisassumption. Choosing a Liouville vector field Y and a smooth function f : M → R , we definea contact form λ on M by ι Y ω | M = e f λ and denote by ξ = ker λ the induced contact structure.We can then use the reverse flow of Y to identify a neighborhood of ∂W symplectically witha neighborhood of the boundary of ( { ( t, m ) ∈ R × M | t ≤ f ( m ) } , d ( e t λ )). Thus we cansmoothly attach the cylindrical end E + := ( { ( t, m ) ∈ R × M | t ≥ f ( m ) } with symplectic form d ( e t λ ), forming an enlarged symplectic manifold ( W ∞ , ω ) which natu-rally contains ([ T, ∞ ) × M, d ( e t λ )) for sufficiently large T . Assumption 4.1.
With (
W, ω ) as described above, assume either of the following:(1) (
W, ω ) contains a symplectically embedded sphere u : S → W with self-intersectionnumber zero.(2) ( M, ξ ) satisfies property ( ⋆ ).In the first case, we can define ˙Σ := S with the standard complex structure, choose anyadmissible R –invariant almost complex structure J + on ([ T, ∞ ) × M, d ( e t λ )) and extend it to an ω –compatible almost complex structure J on W ∞ such that u is (after reparametrization)a J –holomorphic curve. In the second case, we can (by appropriate choice of the function f ) take λ and J + to be the particular contact form and almost complex structure arisingfrom Definition 2.5, and again extend J + to an ω –compatible structure J on W ∞ . After asufficiently large R –translation, the J + –holomorphic curve given by Definition 2.5 may thenbe regarded as a J –holomorphic curve u = ( u R , u M ) : ˙Σ → [ T, ∞ ) × M ⊂ W ∞ , where ˙Σ = S \ { z , . . . , z N } with the standard complex structure of S .Given any smooth function ϕ : R → (0 , ∞ ) that is monotone increasing and satisfies ϕ ( t ) = e t for t ≤ T , we can define a new symplectic form on W ∞ by ω ϕ = ( ω in W , d ( ϕλ ) in E + . (4.1)Observe that J is also compatible with ω ϕ . Definition 4.2.
The energy of a J –holomorphic curve u : ˙Σ → W ∞ is E ( u ) = sup ϕ ∈T Z ˙Σ u ∗ ω ϕ , where ω ϕ is as defined in (4.1) and T is the set of all smooth functions ϕ : R → (0 , ∞ ) thatsatisfy ϕ ′ > ϕ ( t ) = e t for t ≤ T and sup ϕ ≤ e T .This is equivalent to the definition of energy given in [BEH + § J –holomorphic curves in W ∞ are proper and thus have no removable punctures:then they also satisfy the asymptotic formula (2.4) and thus have well defined asymptoticeigenfunctions and transversal convergence rates at each puncture.Denote by M ∗ the moduli space of all proper, somewhere injective finite energy J –ho-lomorphic curves in W ∞ , with arbitrary conformal structures on the domains and any twocurves considered equivalent if they are related by a biholomorphic reparametrization thatpreserves each puncture. We assign to M ∗ the natural topology defined by C ∞ –convergenceon compact subsets and C –convergence up to the ends, and denote by M ∗ ⊂ M ∗ theconnected component containing u . Observe that since R u ∗ ω ϕ depends only on ϕ andthe relative homology class represented by u , the energy E ( u ) is uniformly bounded for all u ∈ M ∗ .We shall now define special subsets M c ⊂ M ∗ and M c ⊂ M ∗ , consisting of J –holomorphiccurves that satisfy asymptotic constraints. If u has no punctures, we can simply set M c = M ∗ and M c = M ∗ . Otherwise, let us fix the following notation: for each puncture z ∈ Γof u , denote the corresponding asymptotic orbit of u by γ z , with asymptotic operator A z ,asymptotic eigenfunction e z and transversal convergence rate − µ z , so µ z ∈ σ ( A z ). Chooseany unitary trivialization Φ for ξ along each of the orbits γ z . We will define a new partitionΓ = Γ C ∪ Γ U in terms of the asymptotic behavior of u , calling these the constrained and unconstrained punctures respectively. Namely, define z ∈ Γ to be in Γ C if and only if γ z is either nondegen-erate or belongs to a Morse-Bott submanifold N ⊂ M that intersects u M ( ˙Σ). N NON-SEPARATING CONTACT HYPERSURFACES IN SYMPLECTIC 4–MANIFOLDS 17
Lemma 4.3. If γ z belongs to a Morse-Bott submanifold N ⊂ M of dimension at least 2,then N intersects u M ( ˙Σ) if and only if wind Φ ( e z ) <
0, where Φ is the unique trivialization inwhich the nontrivial sections in ker A z have zero winding. Proof . It is obvious from the asymptotic formula (2.4) that u M intersects N if wind Φ ( e z ) <
0. To prove the converse, observe first that since u M is embedded, it cannot intersect its ownasymptotic orbits. One then has to show that if u intersects any trivial cylinder R × γ ′ over anorbit γ ′ in N , then it also has an “asymptotic intersection” with R × γ z , which cannot be trueif wind Φ ( e z ) = 0. This follows easily from the intersection theory of punctured holomorphiccurves, see [Sie, SW] for details. (cid:3) Lemma 4.4.
For each z ∈ Γ C , there exists a number c z < c z σ ( A z ), α Φ − ( γ z − c z ) = wind Φ ( e z ) and α Φ+ ( γ z − c z ) = wind Φ ( e z ) + 1. Proof . Choose Φ so that wind Φ ( e z ) = 0; in the language of Definition 2.5, this is the specialtrivialization determined by the asymptotic behavior of u M near z . Then α Φ − ( γ z ) ≥
0, and if γ z is nondegenerate, (2.2) implies µ ΦCZ ( γ z ) ≥
0, with equality if and only if α Φ − ( γ z ) = α Φ+ ( γ z ) = 0.The latter is therefore excluded by the condition µ ΦCZ ( γ z ) = 0 from Definition 2.5. It followsthat if µ ∈ σ ( A z ) is the largest eigenvalue with wind Φ ( µ ) = wind Φ ( e z ), then µ < c z to be any number slightly larger than µ .For the case where γ z is Morse-Bott, the fact that u M intersects the Morse-Bott submanifoldmeans 0 = wind Φ ( e z ) < wind Φ (0) due to Lemma 4.3. Thus the eigenvalue µ defined above isagain negative and we can choose c z to be slightly larger. (cid:3) In the following, let c z < z ∈ Γ C , and for z ∈ Γ U set c z := ǫ > , ǫ ) never intersects σ ( A z ). Definition 4.5.
The constrained moduli space M c consists of all curves u ∈ M ∗ having atmost z ∈ Γ C that is a puncture of u , the asymptotic orbit of u is γ z , with transversal convergencerate strictly greater than | c z | . Let M c ⊂ M c denote the connected component containing u . Proposition 4.6.
Every curve u ∈ M c is embedded. Proof . By Definition 2.5, each asymptotic orbit for the curves in M c is either fixed or allowedto vary in a Morse-Bott torus that contains no other asymptotic orbits, thus the orbits of each u ∈ M c are all distinct and simply covered. It follows that embedded curves form an opensubset of M c , which is also non-empty since it contains u . By positivity of intersections, itis also closed, so the claim follows from the assumption that M c is connected. (cid:3) Topologically, M c is a closed subspace of M ∗ . Recall that M ∗ can locally be identified(up to symmetries) with the zero set of the nonlinear Cauchy-Riemann operator ¯ ∂ J , regardedas a smooth section of a certain Banach space bundle. The same is true for M c , but withBanach spaces of maps whose behavior at the ends satisfies exponential weighting constraintsdetermined by the numbers c z . We refer to [Wenb, Weng] for details on the general analyticalsetup, and [HWZ99, Wena, Weng] for the exponential weights. A given curve u ∈ M c is calledFredholm regular if the linearization of ¯ ∂ J at u is surjective. In general, this linearization isa Fredholm operator, whose index (with correction terms for the dimensions of Teichm¨ullerspace and the automorphism group) defines the “virtual dimension” of the moduli space near u . We’ll denote this virtual dimension by ind ( u ; c ), and call it the (constrained) indexof u . If u is Fredholm regular, then the implicit function theorem implies that M c near u isa smooth manifold, whose dimension is given by the index. Theorem 10.
Every u ∈ M c is Fredholm regular and has ind ( u ; c ) = 2 . Moreover, aneighborhood of u in M c forms a smooth –parameter family { u τ } τ ∈ D , with u = u , suchthat: (1) The images u τ ( ˙Σ) foliate a neighborhood of u ( ˙Σ) in W . (2) For any puncture z ∈ Γ U , the set of all curves { u τ } τ ∈ D that approach the same orbitas u at z is a smooth –dimensional submanifold. Proof . We first verify the claim that ind ( u ; c ) = 2. For the case where u is a closedembedded sphere with self-intersection zero, this follows immediately from the adjunctionformula: 0 = u • u = c ( u ∗ T W ∞ ) −
2, thus c ( u ∗ T W ∞ ) = 2 and ind ( u ) = − c ( u ∗ T W ∞ ) =2. In the case where u arises from property ( ⋆ ), it suffices to prove that ind ( u ; c ) = 2 with u regarded as a J + –holomorphic curve in R × M . Recall from [Wena] that one can associatewith u an integer c N ( u ; c ), called the (constrained) normal Chern number, which satisfies2 c N ( u ; c ) = ind ( u ; c ) − g + ( c ) , (4.2)where g is the genus of ˙Σ (in this case zero) and Γ ( c ) is the subset of punctures z ∈ Γ atwhich p ( γ z − c z ) = 0. It also satisfies c N ( u ; c ) = wind π ( u ) + X z ∈ Γ (cid:2) α Φ − ( γ z − c z ) − wind Φ ( e z ) (cid:3) . (4.3)By Lemma 4.4 and the fact that u M : ˙Σ → M is an embedding, the right hand side of (4.3)vanishes, implying c N ( u ; c ) = 0. We claim also that ( c ) = 0, i.e. all punctures satisfy p ( γ z − c z ) = 1; for z ∈ Γ C this already follows from Lemma 4.4. For unconstrained punctures z ∈ Γ U , Lemma 4.3 implies that e z has the same winding number as a nontrivial section inker A z : these also span the two eigenspaces of A z − c z = A z − ǫ with negative eigenvaluesclosest to zero. It follows that every positive eigenvalue of A z − ǫ has strictly larger winding,thus p ( γ z − ǫ ) = 1 as claimed. Now (4.2) implies ind ( u ; c ) = 2.The remainder of the proof consists of minor generalizations of well established results from[HWZ99, Wen05], so we shall merely sketch the main ideas. Since u ∈ M c is embedded, theregularity question can be reduced to the study of the normal Cauchy-Riemann operator D Nu as in [HLS97, HWZ99, Wenb]. The domain of D Nu is an exponentially weighted Banach spaceof sections of the normal bundle N u → ˙Σ, and the sections in ker D Nu have only positive zeroes,whose algebraic count is bounded in general by c N ( u ; c ), cf. [Wenb]. In our case c N ( u ; c ) = c N ( u ; c ) = 0, thus every section in ker D Nu is zero free; a simple linear independence argumentthen shows that dim ker D Nu ≤ D Nu , hence D Nu is surjective. This shows that M c is asmooth 2–manifold near u , and T u M c is identified with a space of smooth nowhere vanishingsections ker D Nu ⊂ Γ( N u ), implying the claim that the curves near u foliate a neighborhood.Finally we note that for each z ∈ Γ U , one can apply an additional constraint to studysubspaces of curves in M c that fix the position of the asymptotic orbit. In the linearizationthis amounts to replacing c z = ǫ by c z = − ǫ ; this idea is explained in detail in [Wen05, Weng].The problem with the additional constraint then has index 1 and is again regular by anargument using the formal adjoint of D Nu , as in [Wenb]. (cid:3) N NON-SEPARATING CONTACT HYPERSURFACES IN SYMPLECTIC 4–MANIFOLDS 19
Note that in the above proof, Fredholm regularity does not require any genericity as-sumptions, rather it comes for free due to “automatic” transversality (cf. [Wenb]). As aconsequence, u can be deformed with sufficiently small perturbations of J and λ so thatTheorem 10 still applies. After such a perturbation (using Lemma 2.2), we can thereforeassume the following from now on:(1) All orbits of period less than some large constant C > J is generic outside of [ T, ∞ ) × M , so that in particular every curve u ∈ M c that isn’twholly contained in [ T, ∞ ) × M has ind ( u ; c ) ≥ J are somewhat delicate and specific to theapplication we have in mind; this will be explained in Lemma 5.2 in §
5. Note that the purposeof this assumption has nothing to do with the curves in M c , which are already regular—ratherwe will see below that genericity is needed to gain control over the degenerations that canoccur in the natural compactification of M c .Due to the Morse-Bott assumption, the compactness theorem of [BEH +
03] now applies toany sequence of J –holomorphic curves in W ∞ that satisfy a suitable C –bound and energybound: in particular, such a sequence has a subsequence that converges to a nodal holomorphicbuilding, typically with multiple levels. In our situation, the bottom level will be a nodal J –holomorphic curve in W ∞ , and all levels above this are nodal J + –holomorphic curves in R × M . Theorem 11.
Suppose u k ∈ M c is a sequence whose images are all contained in W ∪ E + forsome compact subset W ⊂ W . Then a subsequence of u k converges to one of the following: (1) another smooth curve in M c , (2) a holomorphic building with empty bottom level and one nontrivial upper level thatconsists of a smooth, embedded J + –holomorphic curve in R × M satisfying the condi-tions of property ( ⋆ ) , or (3) a nodal J –holomorphic curve in W ∞ with exactly two components, both in M c andboth embedded with (constrained) index .Moreover the set of index curves that can appear as components of nodal curves in the thirdcase is finite. Before we prove the theorem we state the following important corollary. For this, we denoteby S ⊂ W ∞ the set through which the finitely many limit curves from part (3) of Theorem 11pass, and let C ⊂ W ∞ \ S consist of all points that are contained in curves from M c . Corollary 12.
In addition to the assumptions of Theorem 11, assume that the images of allcurves in M c are contained in W ∪ E + for some compact subset W ⊂ W . Then C = W ∞ \ S ,and thus W is compact. Proof . We claim that C is a non-empty, open and closed subset of W ∞ \ S . It is clearlynon-empty since M c also is, by construction. Openness is a direct consequence of Theorem 10part (1). To prove that C is closed, we choose a sequence ( p n ) ⊂ C with p n → p ∗ ∈ W ∞ \ S .Then by definition, there exist curves u n ∈ M c with p n ∈ im ( u n ). A subsequence of u n converges to a holomorphic building u ∗ , which by Theorem 11 is either a smooth curve ora nodal curve with one level. Since p ∗ is in the image of u ∗ and p ∗ S , we conclude that u ∗ ∈ M c and p ∗ ∈ im u ∗ ⊂ C .Now, since S is a finite union of images of holomorphic curves, W ∞ \ S is connected andit follows from the above claim that C = W ∞ \ S . Since by assumption C ⊂ W ∪ E + , weconclude that W is compact. (cid:3) In proving Theorem 11, we will make use of a few concepts from the intersection theory ofpunctured holomorphic curves; this theory is developed in detail in the papers [Sie, SW], andthe last section of [Wenb] also contains a summary. Assume v , v ∈ M c . Then there is analgebraic intersection number i ( v ; c | v ; c ) ∈ Z which has the following properties:(1) i ( v ; c | v ; c ) is unchanged under continuous variations of v and v in M c .(2) If v and v are not both covers of the same somewhere injective curve, then i ( v ; c | v ; c ) ≥ , and the inequality is strict if they intersect.Unlike the usual homological intersection theory applied to closed holomorphic curves,the last statement is not an “if and only if”: it is possible in general for v and v to bedisjoint even if i ( v ; c | v ; c ) >
0, though this phenomenon is in some sense non-generic. Theintersection number can also be defined for curves in the symplectization R × M , possibly withboth positive and negative punctures. In this case one has invariance under R –translation,so if i ( v ; c | v ; c ) = 0 then the projected images of v and v in M never intersect. Lemma 4.7. i ( u ; c | u ; c ) = 0. Proof . Since u has only simply covered Reeb orbits and all of them are distinct, it satisfiesthe following somewhat simplified version of the adjunction formula from [Sie, SW], i ( u ; c | u ; c ) = 2 δ ( u ) + c N ( u ; c ) . (4.4)Here δ ( u ) is the algebraic count of double points and singularities of u (see [MS04]), whichvanishes since u is embedded. As we saw in the proof of Theorem 10, c N ( u ; c ) also vanishes,so the claim follows. (cid:3) Lemma 4.8. If v ∈ M c is contained in [ T, ∞ ) × M ⊂ W ∞ , then its projection to M isembedded. Proof . Write v = ( v R , v M ) : ˙Σ → [ T, ∞ ) × M . By assumption, v can be deformed contin-uously to u through M c , thus i ( v ; c | v ; c ) = i ( u ; c | u ; c ) = 0 by the previous lemma,and c N ( v ; c ) = c N ( u ; c ) = 0. Now (4.3) implies that wind π ( v ) = 0, thus v M is immersed,and the vanishing self-intersection number implies that v has no intersections with any of its R –translations, so v M is also injective. (cid:3) Proof of Theorem 11.
By [BEH + u k has a subsequence converging to some holomorphicbuilding, which we’ll denote by u . Our first task is to show that unless u is a 2–level buildingwith empty bottom level as described in case (2), it can have no nontrivial upper levels. Thisis already clear in the case where u is closed, as convexity prevents u k from venturing intothe region [ T, ∞ ) × M at all. Let us therefore assume that u k has punctures and that u hasnontrivial upper levels. If no component in these upper levels has any negative punctures, thenthere must be only one nontrivial level, which consists of one or more connected components v , . . . , v N attached to each other by nodes. All of these components have punctures, sincethe symplectic form in R × M is exact; moreover, the positive ends of each v i correspond tosome subset of the positive ends of u , and since these are all simply covered and distinct, N NON-SEPARATING CONTACT HYPERSURFACES IN SYMPLECTIC 4–MANIFOLDS 21 each v i is somewhere injective and satisfies the asymptotic constraints defined by c . Now(4.2) and (4.3) give 0 ≤ π ( v i ) ≤ c N ( v i ; c ) = ind ( v i ; c ) − , hence ind ( v i ; c ) ≥
2. Since ind ( u ; c ) = 2 as well, we conclude that u can have at most oneconnected component, with no nodes, i.e. it is a smooth J + –holomorphic curve in R × M withonly positive punctures. Up to R –translation, u can therefore be identified with some smoothcurve in M c whose image is contained in [ T, ∞ ) × M , and the projection into M is embeddeddue to Lemma 4.8. It follows that this curve satisfies the conditions of property ( ⋆ ).Alternatively, suppose u has nontrivial upper levels and the top level contains a J + –holomorphic curve u + in R × M which is not the trivial cylinder over an orbit and hasboth positive and negative punctures. Repeating the above argument about behavior at thepositive ends, u + is somewhere injective. Applying Stokes’ theorem to R u ∗ + dλ ≥
0, the neg-ative asymptotic orbits of u + have total period bounded by the total period of the positiveorbits, implying that all of the negative orbits belong to the same Morse-Bott manifolds asthe orbits of u . We claim that after some R –translation, u + intersects u . This will implya contradiction almost immediately, as positivity of intersections then gives an intersectionof u k with some R –translation of u for sufficiently large k , contradicting Lemma 4.7 since i ( u k ; c | u ; c ) = i ( u ; c | u ; c ) = 0.To prove the claim, it suffices to show that the projected images of u + and u in M intersect each other. Suppose γ is an asymptotic orbit of u that lies in the same Morse-Bott submanifold N ⊂ M as one of the negative asymptotic orbits γ ′ of u + . Denote thecorresponding asymptotic eigenfunctions by e and e ′ respectively. We consider the followingcases: Case 1: N is a circle. Then γ is nondegenerate and γ ′ is the k –fold cover of γ forsome k ∈ N . Choose a trivialization Φ along γ so that wind Φ ( e ) = 0. By Lemma 4.4, A γ has two eigenvalues (counting multiplicity) µ < Φ ( µ ) = 0. Then the k –foldcovers of their eigenfunctions are eigenfunctions of A γ ′ with negative eigenvalues and zerowinding, implying that every positive eigenvalue of A γ ′ has strictly positive winding. Thuswind Φ ( e ′ ) ≥ α Φ+ ( γ ′ ) >
0, forcing the projections of u and u + in M to intersect each othernear N . Case 2: N is a torus disjoint from u M . Now γ ′ can be deformed through a 1–parameter family of orbits to a k –fold cover of γ for some k ∈ N . Choose a trivialization Φalong every simply covered orbit in N so that sections in the 0–eigenspaces have zero winding.By Lemma 4.3, A γ has an eigenvalue µ < Φ ( e ) = wind Φ ( µ ) = 0, and taking k –fold covers of eigenfunctions, we similarly find eigenfunctions of A γ ′ that have zero windingand eigenvalues kµ < Φ ( e ′ ) ≥ α Φ+ ( γ ′ ) >
0, which forces theprojection of u + in M to intersect N , i.e. u + intersects a trivial cylinder R × γ for some orbit γ ⊂ N . Then by the homotopy invariance of the intersection number, u + also intersects R × γ . This intersection is transverse unless it occurs at a point where π λ ◦ T u + = 0, but thesimilarity principle implies that there are finitely many such points (see [HWZ95]). Thus ifnecessary we can use Theorem 10 to perturb u and thus move γ to a nearby orbit, so thatthe intersection of R × γ with u + is transverse. This implies a transverse intersection of theprojected image of u + in M with γ , and therefore an intersection of the projections of u + and u nearby. Case 3: N is a Morse-Bott manifold intersecting u M . The argument is similar tocase 2, only now we use the intersection of u M with N to show that u M intersects γ ′ and thusalso the projected image of u + near γ ′ .We’ve shown now that u cannot have any nontrivial upper level except in case (2), so it musttherefore be a 1–level building in W ∞ , i.e. a nodal J –holomorphic curve. The deduction ofcase (3) now proceeds almost exactly as in the proof of [Wenc, Theorem 7]. To summarize, theconnected components of u are all either punctured curves with positive ends at distinct simplycovered orbits (and thus somewhere injective), or closed curves (which must be nonconstantby an index argument). The latter could in general be multiple covers, but if v is a k –fold branched cover of some closed somewhere injective curve v , then we find ind ( v ) = k · ind ( v ) + 2( k − u becomes more than 2 unless there isat most one node connecting two components, and in this case both components must besomewhere injective. The adjunction formula (4.4) can now be used to show that these twocomponents, v and v , are both embedded, satisfy i ( v i ; c | v i ; c ) = − i ( v i ; c | u ; c ) = 0 and i ( v ; c | v ; c ) = 1; moreover, they are both Fredholm regular and have (constrained) index 0.There’s one minor point to address which was irrelevant in [Wenc]: if there are no punctures,we haven’t ruled out the possibility that u is a smooth multiple cover, i.e. u = v ◦ ϕ for someclosed somewhere injective sphere v and holomorphic branched cover ϕ : S → S . Since c ( u ∗ T W ∞ ) = 2, this is allowed numerically only if c ( v ∗ T W ∞ ) = 1 and ϕ has degree 2. Butthen we get a simple contradiction using the adjunction formula: since u • u = 0, the sameholds for v , thus 0 = v • v = 2 δ ( v ) + c ( v ∗ T W ∞ ) − δ ( v ) − δ ( v ) is the algebraic count of double points and singularities. The right hand side isodd; in particular it can never be zero.It remains to show that the set of all index 0 curves arising from nodal degenerations of u k is finite. Indeed, suppose v k is a sequence of finite energy J –holomorphic curves in W ∞ withuniform energy and C –bounds such that(1) The punctures of v k are identified with a subset of Γ and satisfy the asymptoticconstraints of Definition 4.5.(2) i ( v k ; c | u ; c ) = 0.(3) ind ( v k ; c ) = 0.Then we claim that v k has a convergent subsequence. The argument is familiar: we rule outnontrivial upper levels exactly as before by showing that any nontrivial component v + in sucha level must intersect u . Thus the only remaining possible non-smooth limit is a nodal curvein W ∞ , but the same index argument now implies that there is at most one component, thusno nodes, and the limit is somewhere injective. It follows that this set of curves is a compactsmooth 0–dimensional manifold, i.e. a finite set. (cid:3) Proofs of the main results
Proofs of Theorems 6 and 7.
We consider a closed and connected symplectic 4–manifold (
W, ω ) which contains a closed contact type hypersurface M such that W \ M is connected. Under the assumptions of Theorem 6 or 7, we will construct from this anoncompact symplectic manifold with convex boundary to which Corollary 12 applies, givinga contradiction. The general idea of the construction is outlined in Figure 4. N NON-SEPARATING CONTACT HYPERSURFACES IN SYMPLECTIC 4–MANIFOLDS 23
To start with, we compactify W \ M by adding to each end a copy of M , obtaining acompact and connected symplectic manifold ( W , ω ) with one convex boundary component M + and an identical concave boundary component M − . Inductively, we define the compactsymplectic manifold W n by W n := W n − ∪ M − = M + W , denoting the symplectic form on W n again by ω . Note that W n − is a compact symplectic submanifold of W n in a natural way.Thus the set ( W , ω ) := [ n ≥ ( W n , ω ) (5.1)is a noncompact symplectic manifold with convex boundary M corresponding to the convexboundary of W .Assume that W contains a symplectically embedded sphere S ⊂ W with S • S = N ≥ ω is exact on M , Stokes’ theorem implies that S cannot be contained entirely in M .We can thus blow up W at N distinct points in S that are not in M , modifying both W and S so that S • S = 0 without loss of generality. Now we claim that S can be “lifted” toa symplectic sphere e S in ( W , ω ) with e S • e S = 0. To see this, construct a symplectic infinitecover ( f W , e ω ) of ( W, ω ) by gluing together a sequence of copies { ( W j , ω ) } j ∈ Z of ( W , ω ), withthe concave boundary of W j attached to the convex boundary of W j +11 for each j ∈ Z . Sincethe sphere is simply connected, S has a lift e S ⊂ f W , and moreover, ( f W , e ω ) naturally contains( W , ω ), which we may assume contains e S without loss of generality.Similarly, if M with its induced contact structure satisfies weak property ( ⋆ ), then afterattaching a symplectic cobordism to the convex boundary of ( W , ω ), we may assume withoutloss of generality that either ( W , ω ) contains a symplectic sphere of zero self-intersection (afterblowing up) or property ( ⋆ ) holds for ∂ W .In either case, ( W , ω ) now satisfies Assumption 4.1. As explained in §
4, we can then attachto ∂ W a cylindrical end E + that contains ([ T, ∞ ) × M, d ( e t λ )) for sufficiently large T ∈ R and a suitable contact form λ , obtaining an enlarged symplectic manifold ( W ∞ , ω ), with an ω –compatible almost complex structure J that is admissible and R –invariant on [ T, ∞ ) × M ,and a non-empty moduli space M c ⊂ M c of J –holomorphic curves in W ∞ . Moreover forsome n ∈ N , we can assume that J belongs to the following set. Definition 5.1.
Let J per be the space of compatible almost complex structures on ( W ∞ , ω )which match J on ([ T, ∞ ) × M, d ( e t λ )) and whose restrictions to W ∼ = W n +1 \ W n ⊂ W ∞ are independent of n for n ≥ n ( J ). Such a J will be called periodic. Lemma 5.2.
For a generic J ∈ J per , all J –holomorphic curves in M c are Fredholm regular. Proof . Recall that the J –holomorphic curves in M c are somewhere injective, see §
4. Theproof of transversality is a small variation on the standard technique, as in [MS04]: the keyis to show that the universal moduli space { ( u, J ) | u is J –holomorphic } is a smooth Banachmanifold for periodic J and u satisfying the relevant conditions. This will use the fact thata perturbation of J can be localized at an injective point of u without interfering at otherpoints in the image of u . Then regular values of the projection ( u, J ) J are generic by theSard-Smale theorem, and for these, all J –curves are Fredholm regular.Assume J ∈ J per and u ∈ M c is not fully contained in [ T, ∞ ) × M . If u also intersects W n ∪ E + , then it suffices to perturb J only in this region and thus preserve periodicity of J . Thus it remains only to show that J per permits sufficient perturbations of J when theimage of u is contained in W \ W n , in which case u must be a somewhere injective closedcurve. Since J is required to be periodic, the only danger not present in the standard case is that u may have periodic points , in the following sense. Recall that W ∞ contains infinitelymany identical copies of a certain manifold V , in the form c W n := W n +1 \ W n . Thus eachpoint x ∈ V appears infinitely often in W ∞ , and we call these different points translates of x . Then z ∈ ˙Σ is a periodic point of u if a translate of u ( z ) is contained in the image im ( u )of u . In this case a periodic perturbation of J cannot be localized in the image of u .We claim that for any somewhere injective closed holomorphic curve in W \ W n , the setof injective points which are not periodic is open and dense. To see this, we can consider thecovering space π : f W → W which was constructed above Definition 5.1. Since J is periodic,the projection π ◦ u is a holomorphic curve in W . It will suffice to show that also π ◦ u issomewhere injective, since then the set of injective points of π ◦ u is open and dense, andinjective points of π ◦ u give rise to non-periodic injective points of u . Denote by τ : f W → f W the deck transformation that maps f W n to f W n +1 . Then if π ◦ u is multiply covered, thefact that u is somewhere injective implies (using unique continuation) that u and τ k ◦ u areequivalent curves for some integer k = 0. But then u is also equivalent to τ nk ◦ u for any n ∈ Z , implying that the image of u in f W is unbounded. Since u was assumed to be closed,this is a contradiction and shows that π ◦ u is indeed somewhere injective.With this, the usual proof that the universal moduli space is a smooth Banach manifoldgoes through unchanged. (cid:3) For the remainder of this section we assume that the almost complex structure J (formerlycalled J ) is periodic and generic. Proposition 5.3.
There exists N ∈ N such that for all u ∈ M ∗ we have im ( u ) ⊂ W N ∪ E + . (5.2) Proof . We denote the convex boundary of W n ⊂ W by M + and the concave boundary by M − n . Recall that M + is the same for all W n . Then we claim that there exists a positiveconstant c > u ∈ M ∗ with im ( u ) ∩ M + and im ( u ) ∩ M − n both nonemptyhave energy E ( u ) ≥ c n . (5.3)This follows from the monotonicity lemma (see Lemma 5.4 below) and the fact that the almostcomplex structure is periodic. Indeed, we fix a copy of W in W n and denote for the momentits convex and concave boundary by ∂W + and ∂W − respectively. We claim that there exists˜ c > v with v − ( ∂W + ) = ∅ and v − ( ∂W − ) = ∅ has at leastenergy E ( v ) ≥ ˜ c . To see this we observe that each such v has to pass through a point in W with distance ǫ > ∂W + ∪ ∂W − of W . Thus we conclude from Lemma 5.4that E ( v ) ≥ Cǫ for each v , where C and ǫ only depend on the almost complex structure J .Since J is periodic, and a map u ∈ M ∗ with im ( u ) ∩ M + = ∅ and im ( u ) ∩ M − n = ∅ passesthrough the boundaries of n copies of W , equation (5.3) follows. Using the uniform energybound for u ∈ M ∗ , this implies the proposition in the case where u has punctures, as every u ∈ M ∗ is then either confined to E + or passes through M + .A small modification is required for the case without punctures: here u ∈ M ∗ is a sphere,and we can choose its lift from W to W ∞ so that without loss of generality, the image of u intersects W (i.e. the first copy). Then we claim that every u ∈ M ∗ intersects W .Otherwise, the fact that M ∗ is connected implies the existence of some holomorphic spherein M ∗ that touches M − tangentially from inside W \ W , and this is impossible by convexity. N NON-SEPARATING CONTACT HYPERSURFACES IN SYMPLECTIC 4–MANIFOLDS 25
We conclude that every u ∈ M ∗ which escapes from W ∪ E + must also pass through M − ,so the above argument goes through by using M − in place of M + . (cid:3) For the sake of completeness, we include here the monotonicity lemma, see [Hum97] for aproof.
Lemma 5.4.
For any compact almost complex manifold (
W, J ) with Hermitian metric g ,there are constants ǫ and C >
S, j ) is a compactRiemann surface, possibly with boundary, and u : S → W is a pseudoholomorphic curve.Then for every z ∈ Int( S ) and r ∈ (0 , ǫ ) such that u ( ∂S ) ∩ B r ( u ( z )) = ∅ , the inequalityArea ( u ( S ) ∩ B r ( u ( z ))) ≥ Cr holds.Since W N is compact, Proposition 5.3 allows us to apply Corollary 12. But this impliesthat W is compact, and is thus a contradiction, concluding the proof of Theorems 6 and 7.5.2. Proof of Theorem 8.
Theorem 8 follows immediately from Theorem 7 and Exam-ple 1.2, since a symplectic semifilling with disconnected boundary can always be turned intoa closed symplectic 4–manifold containing non-separating contact hypersurfaces. One cannonetheless give a slightly easier proof as follows.Assume that the boundary ∂W is disconnected and contains a component M satisfyingproperty ( ⋆ ). Thus W satisfies Assumption 4.1, and after attaching cylindrical ends, weobtain a moduli space M c of J –holomorphic curves that fill the enlarged manifold W ∞ .Moreover, all J –holomorphic curves have positive punctures going to the end correspondingto M . Since they fill W ∞ , some of these curves must therefore touch ∂W \ M tangentially,which is impossible if ∂W is convex.5.3. Proof of Theorem 9.
Let (
W, ω ) be a compact connected 4-manifold with convexboundary (
M, ξ ) satisfying the weak ( ⋆ ) property. After attaching a symplectic cobordism to ∂W , we may without loss of generality remove the word “weak”. Now assume that H ⊂ W \ M is a non-separating contact hypersurface. Thus we can cut W open along H and compactifyto obtain a connected symplectic cobordism W with two convex boundary components H + and M , and one concave boundary component H − .Now we can repeat the construction in the proof of Theorems 6 and 7, namely we glueinfinitely many copies of W along H , obtaining a noncompact symplectic manifold W withone convex boundary component H and infinitely many convex boundary components whichare copies of M . From here, we proceed exactly as in the previous proofs, using the modulispace of holomorphic curves arising from property ( ⋆ ) on the first copy of M . The only newfeature is that ∂ W is not compact, but since it consists of copies of the same compact andconvex components, the results of § M c from ever approaching the other copies of M . In particular, Corollary 12 applies andagain yields a contradiction. Appendix A. Relative nondegeneracy of contact forms
Our main argument uses holomorphic curves asymptotic to Morse-Bott families of periodicorbits. We prefer not to assume from the start that the contact form is globally
Morse-Bott.Thus, we need a perturbation result that preserves a given Morse-Bott submanifold and makes
PSfrag replacements (
W, ω ) (
M, ξ )( W n , ω ) W n ∪ E + Y Y ( W , ω ) Figure 4.
The compact symplectic manifold (
W, ω ) contains the non-separating contact hypersurface (
M, ξ ). W \ M is compactified to produce( W , ω ), which has two boundary components contactomorphic to M , oneconvex and one concave. Successively attaching n copies of W to itself pro-duces ( W n , ω ). Then property ( ⋆ ) gives rise to a moduli space of finite energycurves which, due to the monotonicity lemma, cannot escape from W n ∪ E + if n is sufficiently large. λ nondegenerate everywhere else. For this, it suffices to show that one can perturb λ in someprecompact subset to make all orbits that pass through that subset nondegenerate. N NON-SEPARATING CONTACT HYPERSURFACES IN SYMPLECTIC 4–MANIFOLDS 27
Theorem 13.
Suppose M is a (2 n − –dimensional manifold with a smooth contact form λ , and U ⊂ M is an open subset with compact closure. Then there exists a Baire subset Λ reg ( U ) ⊂ { f ∈ C ∞ ( M ) | f > and f | M \U ≡ } such that for each f ∈ Λ reg ( U ) , every periodic orbit of X fλ passing through U is nondegenerate. Proof . We give a proof in two steps, first showing that a generic choice of the function f makes all simply covered orbits of X fλ passing through U nondegenerate. Then we extendthis to multiple covers by a further perturbation.The first step is an adaptation of the standard Sard-Smale argument. Let ξ = ker λ , andfor some large k ∈ N , define the Banach space C k U ( M ) = n f ∈ C k ( M, R ) (cid:12)(cid:12) f | M \U ≡ o and Banach manifoldΛ k ( U ) = { f ∈ C k ( M, R ) | f > f − ∈ C k U ( M ) } , whose tangent space at any f ∈ Λ k ( U ) can be identified with C k U ( M ). We will consider thenonlinear operator σ ( x, T, f ) := ˙ x − T X fλ ( x )as a section of a Banach space bundle over H ( S , M ) × (0 , ∞ ) × Λ k ( U ) whose fiber at ( x, T, f )is L ( x ∗ T M ). Since X fλ depends on the first derivative of f , it is of class C k − and the section σ is therefore of class C k − . Choosing any symmetric connection ∇ on M , the linearizationof σ at ( x, T, f ) ∈ σ − (0) with respect to the first variable defines the operator D x : H ( x ∗ T M ) → L ( x ∗ T M ) : ˆ x
7→ ∇ t ˆ x − T ∇ ˆ x X fλ . (A.1)Since ˙ x = T X fλ ( x ), we can identify the normal bundle of x with x ∗ ξ and thus define asplitting x ∗ T M = T S ⊕ x ∗ ξ . A short calculation then allows us to rewrite D x with respectto the splitting in the block form D x = (cid:18) ∂ t D Nx (cid:19) , (A.2)where D Nx : H ( x ∗ ξ ) → L ( x ∗ ξ ) is defined again by (A.1), and is a Fredholm operator ofindex 0. The orbit x is nondegenerate if and only if D Nx is an isomorphism.The total linearization of σ at ( x, T, f ) ∈ σ − (0) is now Dσ ( x, T, f )(ˆ x, ˆ T , ˆ f ) = D x ˆ x − ˆ T X fλ ( x ) − T b X ( x ) , where we define the vector field b X := ∂ τ X ( f + τ ˆ f ) λ | τ =0 . It follows from the definition of theReeb vector field that b X takes the form − ˆ f X fλ + V ˆ f where V ˆ f ∈ Γ( ξ ) is uniquely determinedby the condition d ( f λ )( V ˆ f , · ) (cid:12)(cid:12)(cid:12) ξ = d ˆ f (cid:12)(cid:12)(cid:12) ξ . (A.3)We define the universal moduli space of parametrized Reeb orbits as M := σ − (0), and let M ∗ ⊂ M denote the open subset consisting of triples ( x, T, f ) for which x is simply coveredand x ( S ) ∩ U 6 = ∅ . Similarly, denote M ∗ ( f ) = { ( x, T ) | ( x, T, f ) ∈ M ∗ } . We claim that Dσ ( x, T, f ) is surjective whenever ( x, T, f ) ∈ M ∗ , hence M ∗ is a C k − –smoothBanach manifold. To see this, note that one can always find η ∈ H ( T S ) and ˆ T ∈ R so that T x ( ∂ t η ) − ˆ T X fλ ( x ) takes any desired value in L ( x ∗ ( R X fλ )), thus it suffices to show that the“normal part” H ( x ∗ ξ ) ⊕ C k U ( M ) → L ( x ∗ ξ ) : (ˆ x, ˆ f ) D Nx ˆ x − T V ˆ f is surjective. If it isn’t, then there exists a section η = 0 ∈ L ( x ∗ ξ ) such that h D Nx ˆ x, η i L = 0for all ˆ x ∈ H ( x ∗ ξ ) and h V ˆ f , η i L = 0 for all ˆ f ∈ C k U ( M ) vanishing outside of U . Thefirst relation implies that η is in the kernel of the formal adjoint of D Nx , a first order lineardifferential operator, hence η is smooth and nowhere vanishing. But then if x ( t ) ∈ U , thenusing (A.3), ˆ f can be chosen near x ( t ) so that the second relation requires η to vanish on aneighborhood of t , giving a contradiction.Now applying the Sard-Smale theorem to the natural projection M ∗ → Λ k ( U ) : ( x, T, f ) f , we find a Baire subset Λ k reg ( U ) ⊂ Λ k ( U ) for which every simply covered Reeb orbit passingthrough U is nondegenerate.For the second step, denote by dist( , ) the distance functions resulting from any choice ofRiemannian metrics on S and M , and define for each positive integer N ∈ N a subset M N ( f ) ⊂ M ∗ ( f )consisting of Reeb orbits ( x, T ) that satisfy the following conditions:(1) T ≤ N .(2) There exists t ∈ S such thatinf t ′ ∈ S \{ t } dist( x ( t ) , x ( t ′ ))dist( t, t ′ ) ≥ N . (3) There exists t ∈ S such that dist( x ( t ) , M \ U ) ≥ /N .Moreover, let Λ reg ,N ( U ) ⊂ Λ ∞ ( U ) denote the space of all smooth functions f ∈ Λ k ( U ) forwhich all covers of orbits in M N ( f ) up to multiplicity N are nondegenerate. Since nonde-generacy is an open condition and any sequence ( x k , T k ) ∈ M N ( f k ) with f k → f in C ∞ hasa convergent subsequence by the Arzel`a-Ascoli theorem, Λ reg ,N ( U ) is an open set. We claimit is also dense. Indeed, any f ∈ Λ ∞ ( U ) has a perturbation f ǫ ∈ Λ k ( U ) for which all thesimple orbits in M N ( f ǫ ) are nondegenerate due to step 1. In this case M N ( f ǫ ) is a smoothcompact 1–manifold, i.e. a finite union of circles, which are the parametrizations of finitelymany distinct nondegenerate orbits, and the space is stable under small perturbations of f ǫ .Thus by a further perturbation, we can make f ǫ smooth and arrange that none of the orbitsin M N ( f ǫ ) have a Floquet multiplier that is a k th root of unity for k ∈ { , . . . , N } . The lattercan be achieved using a normal form for f ǫ λ as in [HWZ96a, Lemma 2.3] near each individualorbit: in particular, we can perturb so that each orbit remains unchanged but the linearizedreturn map changes arbitrarily within the space of symplectic linear maps. This proves thatΛ reg ,N ( U ) is dense in Λ ∞ ( U ), and we can now construct Λ reg ( U ) as a countable intersectionof open dense sets: Λ reg ( U ) = \ N ∈ N Λ reg ,N ( U ) . (cid:3) Acknowledgments
We would like to thank Klaus Mohnke for bringing the question of non-separating contacthypersurfaces to our attention, John Etnyre for providing Example 1.2, Klaus Niederkr¨uger,
N NON-SEPARATING CONTACT HYPERSURFACES IN SYMPLECTIC 4–MANIFOLDS 29
Paolo Ghiggini and Slava Matveyev for enlightening discussions and Yasha Eliashberg for somehelpful comments on an earlier draft of the paper. We began work on this article while BB wasvisiting ETH Z¨urich, and we’d like to thank ETH for its stimulating working environment.PA is partially supported by NSF grant DMS-0805085. CW is partially supported by NSFPostdoctoral Fellowship DMS-0603500.
References [Abb] C. Abbas,
Holomorphic open book decompositions . Preprint arXiv:0907.3512.[BO] M. Bhupal and K. Ono,
Symplectic fillings of links of quotient surface singularities . PreprintarXiv:0808.3794.[BEH +
03] F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki, and E. Zehnder,
Compactness results in sym-plectic field theory , Geom. Topol. (2003), 799–888 (electronic).[Eli90] Y. Eliashberg, Filling by holomorphic discs and its applications , Geometry of low-dimensional man-ifolds, 2 (Durham, 1989), London Math. Soc. Lecture Note Ser., vol. 151, Cambridge Univ. Press,Cambridge, 1990, pp. 45–67.[Eli04] ,
A few remarks about symplectic filling , Geom. Topol. (2004), 277–293 (electronic).[Etn04] J. B. Etnyre, Planar open book decompositions and contact structures , Int. Math. Res. Not. (2004),no. 79, 4255–4267.[Etn] . Private communication.[EH02] J. B. Etnyre and K. Honda,
On symplectic cobordisms , Math. Ann. (2002), no. 1, 31–39.[Gay06] D. T. Gay,
Four-dimensional symplectic cobordisms containing three-handles , Geom. Topol. (2006), 1749–1759 (electronic).[Gei95] H. Geiges, Examples of symplectic -manifolds with disconnected boundary of contact type , Bull.London Math. Soc. (1995), no. 3, 278–280.[Gei97] , Constructions of contact manifolds , Math. Proc. Cambridge Philos. Soc. (1997), no. 3,455–464.[Gei08] ,
An introduction to contact topology , Cambridge Studies in Advanced Mathematics, vol. 109,Cambridge University Press, Cambridge, 2008.[Gir01] E. Giroux,
Structures de contact sur les vari´et´es fibr´ees en cercles audessus d’une surface , Comment.Math. Helv. (2001), no. 2, 218–262 (French, with French summary).[Gir] . Lecture given at Georgia Topology conference, May 24, 2001, notes available at .[Gro85] M. Gromov, Pseudoholomorphic curves in symplectic manifolds , Invent. Math. (1985), no. 2,307–347.[Gro86] , Partial differential relations , Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Re-sults in Mathematics and Related Areas (3)], vol. 9, Springer-Verlag, Berlin, 1986.[Hof93] H. Hofer,
Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjec-ture in dimension three , Invent. Math. (1993), no. 3, 515–563.[HLS97] H. Hofer, V. Lizan, and J.-C. Sikorav,
On genericity for holomorphic curves in four-dimensionalalmost-complex manifolds , J. Geom. Anal. (1997), no. 1, 149–159.[HWZ95] H. Hofer, K. Wysocki, and E. Zehnder, Properties of pseudo-holomorphic curves in symplectisations.II. Embedding controls and algebraic invariants , Geom. Funct. Anal. (1995), no. 2, 270–328.[HWZ96a] , Properties of pseudoholomorphic curves in symplectisations. I. Asymptotics , Ann. Inst. H.Poincar´e Anal. Non Lin´eaire (1996), no. 3, 337–379.[HWZ96b] , Properties of pseudoholomorphic curves in symplectisations. IV. Asymptotics with degen-eracies , Contact and symplectic geometry (Cambridge, 1994), 1996, pp. 78–117.[HWZ99] ,
Properties of pseudoholomorphic curves in symplectizations. III. Fredholm theory , Topicsin nonlinear analysis, 1999, pp. 381–475.[Hon00] K. Honda,
On the classification of tight contact structures. II , J. Differential Geom. (2000),no. 1, 83–143.[Hum97] C. Hummel, Gromov’s compactness theorem for pseudo-holomorphic curves , Progress in Mathe-matics, vol. 151, Birkh¨auser Verlag, Basel, 1997.[Kan97] Y. Kanda,
The classification of tight contact structures on the -torus , Comm. Anal. Geom. (1997), no. 3, 413–438. [McD90] D. McDuff, The structure of rational and ruled symplectic -manifolds , J. Amer. Math. Soc. (1990), no. 3, 679–712.[McD91] , Symplectic manifolds with contact type boundaries , Invent. Math. (1991), no. 3, 651–671.[MS04] D. McDuff and D. Salamon, J -holomorphic curves and symplectic topology , American MathematicalSociety, Providence, RI, 2004.[Mor03] E. Mora, Pseudoholomorphic cylinders in symplectisations , Ph.D. Thesis, New York University,2003.[OO05] H. Ohta and K. Ono,
Simple singularities and symplectic fillings , J. Differential Geom. (2005),no. 1, 1–42.[Sie] R. Siefring, Intersection theory of punctured pseudoholomorphic curves . Preprint arXiv:0907.0470.[SW] R. Siefring and C. Wendl,
Pseudoholomorphic curves, intersections and Morse-Bott asymptotics .In preparation.[Wel07] J.-Y. Welschinger,
Effective classes and Lagrangian tori in symplectic four-manifolds , J. SymplecticGeom. (2007), no. 1, 9–18.[Wen05] C. Wendl, Finite energy foliations and surgery on transverse links , Ph.D. Thesis, New York Uni-versity, 2005.[Wena] ,
Compactness for embedded pseudoholomorphic curves in 3-manifolds . To appear in JEMS,Preprint arXiv:math/0703509.[Wenb] ,
Automatic transversality and orbifolds of punctured holomorphic curves in dimension four .To appear in Comment. Math. Helv., Preprint arXiv:0802.3842.[Wenc] ,
Strongly fillable contact manifolds and J –holomorphic foliations . To appear in Duke Math.J., Preprint arXiv:0806.3193.[Wend] , Open book decompositions and stable Hamiltonian structures . To appear in Expos. Math.,Preprint arXiv:0808.3220.[Wene] ,
Holomorphic curves in blown up open books . In preparation.[Wenf] ,
Contact fiber sums, monodromy maps and symplectic fillings . In preparation.[Weng] ,
Punctured holomorphic curves with boundary in -manifolds: Fredholm theory and embe-deddness . In preparation.[Zeh03] K. Zehmisch, The Eliashberg-Gromov tightness theorem , Diplom Thesis, Universit¨at Leipzig, 2003.
Peter Albers, Department of Mathematics, ETH Z¨urich
E-mail address : [email protected] Barney Bramham, Max-Planck Institute, Leipzig
E-mail address : [email protected] Chris Wendl, Department of Mathematics, ETH Z¨urich
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