Some remarks on the size of tubular neighborhoods in contact topology and fillability
SSOME REMARKS ON THE SIZE OF TUBULAR NEIGHBORHOODS INCONTACT TOPOLOGY AND FILLABILITY
KLAUS NIEDERKR ¨UGER AND FRANCISCO PRESAS
Abstract.
The well-known tubular neighborhood theorem for contact submanifolds states thata small enough neighborhood of such a submanifold N is uniquely determined by the contactstructure on N , and the conformally symplectic structure of the normal bundle. In particular,if the submanifold N has trivial normal bundle then its tubular neighborhood will be contacto-morphic to a neighborhood of N × { } in the model space N × R k .In this article we make the observation that if ( N, ξ N ) is a 3–dimensional overtwisted sub-manifold with trivial normal bundle in ( M, ξ ), and if its model neighborhood is sufficiently large,then (
M, ξ ) does not admit an exact symplectic filling.
In symplectic geometry many invariants are known that measure in some way the “size” of asymplectic manifold. The most obvious one is the total volume, but this is usually discarded,because one can change the volume (in case it is finite) by rescaling the symplectic form withoutchanging any other fudamental property of the manifold. The first non-trivial example of aninvariant based on size is the symplectic capacity [Gro85]. It relies on the fact that the size ofa symplectic ball that can be embedded into a symplectic manifold does not only depend on itstotal volume but also on the volume of its intersection with the symplectic 2–planes.Contact geometry does not give a direct generalization of these invariants. The main difficultiesstem from the fact that one is only interested in the contact structure, and not in the contact form,so that the total volume is not defined, and to make matters worse the whole Euclidean space R n +1 with the standard structure can be compressed by a contactomorphism into an arbitrarilysmall open ball in R n +1 .A more successful approach consists in studying the size of the neighborhood of submani-folds. This can be considered to be a generalization of the initial idea since contact balls are justneighborhoods of points. In the literature this idea has been pursued by looking at the tubu-lar neighborhoods of circles. Let ( N, α N ) be a closed contact manifold. The product N × R k carries a contact structure given as the kernel of the form α N + (cid:80) kj =1 ( x j dy j − y j dx j ), where( x , . . . , x k , y , . . . , y k ) are the coordinates of the Euclidean space. If ( N, α N ) is a contact sub-manifold of a manifold ( M, α ) that has trivial (as conformal symplectic) normal bundle, thenone knows by the tubular neighborhood theorem that N has a small neighborhood in M that iscontactomorphic to a small neighborhood of N × { } in the product space N × R k .The contact structure on a solid torus V in S × R depends in an intricate way on the radiusof V [Eli91]. Later, examples of transverse knots in 3–manifolds were found whose maximalneighborhood is only contactomorphic to a small disk bundle in S × R [EH05]. This is provedby measuring the slope of the characteristic foliation on the boundary of cylinders.A different approach has been taken in [EKP06]. There it has been shown that a solid torusaround S × { } in S × R k of radius R cannot be “squeezed” into a solid torus of radius r , if k ≥ R > r <
2. However note that squeezing in the context of [EKP06] is different fromthe expected definition, and refers to the question of whether one subset of a contact manifold canbe deformed by a global isotopy into another one.The observation on which the present article is based is that sufficiently large neighborhoods of N × { } in N × R k contain a generalized plastikstufe (for a definition of the GPS see Section 3), if N is an overtwisted 3–manifold. The construction of a GPS in a tubular neighborhood is explainedin Section 4. In Section 5, we show that the existence of a GPS implies nonfillability, and so itfollows in particular that an overtwisted contact manifold that is embedded into a fillable manifoldcannot have a “large” neighborhood. a r X i v : . [ m a t h . S G ] D ec K. NIEDERKR¨UGER AND F. PRESAS
Unfortunately, the definition of “large” is rather subtle and does not lead to a numerical in-variant, because such an invariant would depend on the contact form on the submanifold. Onecould simply multiply any contact form α N + (cid:80) kj =1 ( x j dy j − y j dx j ) by a constant λ >
0, and thenrescale the radii in the plane by a transformation r j (cid:55)→ r j / √ λ to change the numerical invariant. Acknowledgments.
K. Niederkr¨uger works at the
ENS de Lyon funded by the project
Symplexe of the
Agence Nationale de la Recherche (ANR).Several people helped us writing this article: Many ideas and examples are due to EmmanuelGiroux (in particular Example 1). Example 5 was found in discussions with Hansj¨org Geiges.We thank Ana Rechtman Bulajich for helping us clarifying the general ideas of the paper, andMohammed Abouzaid and Pierre Py for extremely valuable discussions about holomorphic curves.Paolo Ghiggini pointed out to us known results about neighborhoods of transverse knots.1.
Examples
First we give an easy example that shows that embedding an overtwisted 3–manifold into afillable contact manifold does not pose a fundamental problem in positive codimension.
Example . Let M be an arbitrary orientable closed 3–manifold. Its unit cotangent bundle S (cid:0) T ∗ M (cid:1) ∼ = M × S has a contact structure defined by the canonical 1–form. The cotangentbundle T ∗ M together with the form dλ can is an exact symplectic filling (and in fact, it can evenbe turned into a Stein filling).Any contact manifold ( M, α ) can be embedded into (cid:0) S (cid:0) T ∗ M (cid:1) , λ can (cid:1) just by normalizing α sothat (cid:107) α (cid:107) = 1. This defines a section in σ : M → S (cid:0) T ∗ M (cid:1) with σ ∗ λ can = α . This means that every(and in particular also every overtwisted overtwisted one) contact 3–manifold can be embeddedinto a Stein fillable contact 5–manifold.Embedding a contact 3–manifold into a contact manifold of dimension 7 or higher restricts byusing the h –principle and a general position argument to a purely topological question.Our second and third example show that contact submanifolds can have infinitely large tubularneighborhoods. Example . Let (
M, α ) be an arbitrary contact manifold, and let ( S n − , ξ ) be the standardsphere. If dim M ≥ n −
1, then it is easy to give a contact embedding (cid:16) S n − × R k , α + k (cid:88) j =1 ( x j dy j − y j dx j ) (cid:17) (cid:44) → (cid:0) M, α (cid:1) . The proof works in two steps. For the embedding (cid:16) S n − × R k , α + k (cid:88) j =1 ( x j dy j − y j dx j ) (cid:17) (cid:44) → (cid:0) S n +2 k − , α (cid:1) simply use the map ( z , . . . , z n ; x , y , . . . , x k , y k ) (cid:55)→ √ (cid:107) x (cid:107) + (cid:107) y (cid:107) ( z , . . . , z n , x + iy , . . . , x k + iy k ). Since ( S N − , α ) with one point removed is contactomorphic to R N − with standardcontact structure (see for example [Gei06, Proposition 2.13]) and since it is possible to embed thewhole R N − into an arbitrary small Darboux chart (see for example [CvS08, Proposition 3.1]), itfollows that a general ( M, α ) contains embeddings of (cid:0) S n − × R k , α + (cid:80) kj =1 ( x j dy j − y j dx j ) (cid:1) . Example . A generalization is obtained by choosing a contact manifold (
N, α N ) that has anexact symplectic filling ( W, ω = dλ ). The Lioville field X L is globally defined (see Section 2.1),and we can use its negative flow for finding an embedding of the lower half of the symplectization( −∞ , × N where λ pulls back to e t α N . The manifold S × W is together with the 1–form dϑ + λ a contact manifold.The standard model ( N × R , α N + r dϕ ) can be glued outside the 0–section onto S × W , andthis construction yields a closed contact manifold that contains the embedding of N × R . Thisexample can be seen as an open book with binding N , page W , and trivial monodromy. IZE OF NEIGHBORHOODS IN CONTACT TOPOLOGY 3
Not much is known about the different contact structures on R n +1 for n ≥
2. There existsthe standard contact structure ξ , and many different constructions to produce structures thatare not isomorphic to the standard one (for example [BP90, Mul90, Nie06]). Unfortunately wedo not have the techniques to decide whether these exotic contact structures are different fromeach other. A contact structure ξ on R n +1 is called standard at infinity [Eli93], if there existsa compact subset K of R n +1 such that (cid:0) R n +1 − K, ξ (cid:1) is contactomorphic to (cid:0) R n +1 − D R , ξ (cid:1) for a closed disk of an arbitrary radius R . A contact structure ξ on R n +1 only admits a one-point compactificaton to a contact structure on the sphere, if ξ is standard at infinity. For mostexotic contact structures it is not known whether they are standard at infinity or not. The onlyexception known to us so far was given in [KN07], where by removing one point from the sphere,we obtained an exotic contact structure ξ P S on R n +1 that is standard at infinity (but see alsoExample 5). A rather degenerate way of producing a contact structure that is not standard atinfinity consists in taking the standard structure on R n +1 , and do the connected sum at everypoint (0 , . . . , , k ) ∈ R n +1 with k ∈ Z with the sphere ( S n +1 , ξ P S ).Corollary 5 below yields a very explicit way to construct an exotic contact structure that is notstandard at infinity.
Example . The contact manifold (cid:0) R × C k , α − + k (cid:88) j =1 r j dϑ j (cid:1) , where ( r j , ϑ j ) are polar coordinates on the j –th factor of C k , does not embed into the standardsphere, and is hence not contactomorphic to the standard contact structure on R k +3 . Let K ⊂ R × C k be an arbitrary compact subset. By the same argument, it is easy to see that (cid:0) R × C k − K, α − + (cid:80) j r j dϑ j (cid:1) still contains a GPS, so in particular it cannot be embedded into a“punctured” set U − { p } ⊂ (cid:0) R k +3 , α (cid:1) with the standard contact structure. It follows that (cid:0) R × C k , α − + (cid:80) j r j dϑ j (cid:1) is “non standard at infinity”.Let ( M, α ) be a closed contact manifold that contains a contact submanifold N of codimension 2with trivial normal bundle. A k –fold contact branched covering over M consists of a closedmanifold (cid:102) M , and a smooth surjective map f : (cid:102) M → M such that the map f is a smooth k –foldcovering over M − N , and there is an open neighborhood (cid:101) U ⊂ (cid:102) M of f − ( N ) diffeomorphic to N × D ε , and a neighborhood U ⊂ M of N diffeomorphic to N × D ε k such that the map f takesthe form f : N × D ε → N × D ε k , ( p, z ) (cid:55)→ ( p, z k ) , when restricted to (cid:101) U (see [Gon87]).Using the branched covering, it is easy to define a contact structure on (cid:102) M . First isotope α insuch a way that it becomes α | T N + r dϕ on a subset N × D δ ⊂ N × D ε k for some δ >
0. Thepull-back (cid:101) α := f ∗ α defines on (cid:102) M a 1–form that satisfies away from f − ( N ) everywhere the contactproperty. Over the branching locus f − ( N ), there is a subset N × D k √ δ in (cid:101) U where (cid:101) α evaluatesto α | T N + kr k dϕ .Remove the fiber N × { } from (cid:101) U and glue in N × D δ via the map F : ( p, re iϕ ) (cid:55)→ ( p, k √ r e iϕ )along N × (cid:0) D k √ δ − { } (cid:1) . The pull-back F ∗ (cid:101) α yields α | T N + kr dϕ on the punctured disk bundle,which we can easily extend to the whole patch we are gluing in. We denote this slightly modifiedcontact form again by (cid:101) α . By using a linear stretch map on the disk, we finally obtain that thesubmanifold f − ( N ) ∼ = N has with respect to the model form (cid:0) N × R , α | T N + r dϕ (cid:1) a neighborhood inside ( (cid:102) M , (cid:101) α ) that is at least of size √ k δ . K. NIEDERKR¨UGER AND F. PRESAS
Example . There is an interesting contact structure on the odd dimensional spheres S n − ⊂ C n given as the kernel of the 1–forms α − = i n (cid:88) j =1 (cid:0) z j d ¯ z j − ¯ z j dz j (cid:1) − i (cid:0) f d ¯ f − ¯ f df (cid:1) with f ( z , . . . , z n ) = z + · · · + z n . This form is compatible with the open book with binding B = f − (0), and fibration map ϑ = ¯ f / | f | . In abstract terms, this is the open book with page P ∼ = T ∗ S n − and monodromy map corresponding to the negative Dehn-Seidel twist.An interesting feature of these spheres is that they can be stacked into each other via the naturalinclusions S (cid:44) → S (cid:44) → S (cid:44) → · · · respecting the contact form, and that ( S , α − ) is overtwisted.We find a contact branched cover f : S → ( S , α − ) given by f ( z , z , z ) = ( z ,z ,z k ) (cid:107) ( z ,z ,z k ) (cid:107) that isbranched along S . By choosing k large enough, we will obtain with the construction describedabove a contact structure on S that contains an embedding of ( S , α − ) with an arbitrary largeneighborhood. According to Corollary 5 and Theorem 4, this contact structure will not admit anexact symplectic filling.This result is unsatisfactory, since we do not get an explicit value for k . In fact, we expect that( S , α − ) already has a large neighborhood in any of the spheres ( S n − , α − ) so that taking k = 1(that means not taking any branched covering at all) should already be sufficient.2. Preliminaries
Fillability.
In this section, we will briefly present some standard definitions and propertiesregarding fillability and J –holomorphic curves. Definition. A Liouville field X L is a vector field on a symplectic manifold ( W, ω ) for which L X L ω = ω holds.If ( W, ω ) is a symplectic manifold with boundary M := ∂W , and if X L is a Liouville field on W that is transverse to M , then the kernel of the 1–form α := ω ( X L , − ) | T M defines a contact structure on M . Definition.
Let (
M, ξ ) be a closed contact manifold. A compact symplectic manifold (
W, ω )with boundary ∂W = M is called a strong (symplectic) filling of ( M, ξ ), if there exists aLiouville field X L in a neighborhood of the boundary M pointing outwards through M such that X L defines a contact form for ξ . If the vector field X L is defined globally on W , we speak of an exact symplectic filling . Remark . In a symplectic filling, we can always find a neighborhood of M that is of the form( − ε, × M by using the negative flow of X L to define( − ε, × M → W, ( p, t ) (cid:55)→ Φ t ( p ) . Denote the hypersurfaces { t } × M by M t , and the 1–form ω ( X L , − ) by (cid:98) α . It is clear that (cid:98) α defineson every hypersurface M t a contact structure. The Reeb field X Reeb is the unique vector field on( − ε, × M that is tangent to the hypersurfaces M t , and satisfies both ω ( X Reeb , Y ) = 0 for every Y ∈ T M t , and ω ( X L , X Reeb ) = 1. This field restricts on any hypersurface M t to the usual Reebfield for the contact form (cid:98) α | T M t .Below we will show that the “height” function h : ( − ε, × M → R , ( t, p ) (cid:55)→ t is plurisubhar-monic with respect to certain almost complex structures.In the context of this article we will use the term “adapted almost complex structure” in thefollowing sense. IZE OF NEIGHBORHOODS IN CONTACT TOPOLOGY 5
Definition.
Let (
W, ω ) be a symplectic filling of a contact manifold (
M, α ). An almost complexstructure J is a smooth section of the endomorphism bundle End( T W ) such that J = − . Wesay that J is adapted to the filling , if it is compatible with ω in the usual sense, which meansthat for all X, Y ∈ T p W ω ( JX, JY ) = ω ( X, Y )holds, and g ( X, Y ) := ω ( X, JY )defines a Riemannian metric. Additionally, we require J to satisfy close to the boundary M = ∂W the following properties: For the two vector fields X L and X Reeb introduced above, J is definedas JX L = X Reeb and JX Reeb = − X L , and J leaves the subbundle ξ t = T M t ∩ ker (cid:98) α invariant. Proposition 1.
Let V be an open subset of C , and let u : V → W be a J –holomorphic map. Thefunction h ◦ u : V → R is subharmonic.Proof. A short computation shows that (cid:98) α = − dh ◦ J , and then we get0 ≤ u ∗ ω = u ∗ dι X L ω = u ∗ d (cid:98) α = u ∗ d (cid:0) − dh ◦ J (cid:1) = − u ∗ dd c h = − dd c ( h ◦ u ) = (cid:16) ∂ h ◦ u∂x + ∂ h ◦ u∂y (cid:17) dx ∧ dy . (cid:3) Corollary 2.
By the strong maximum principle and the boundary point lemma (e.g. [GT01] ), any J –holomorphic curve u : (Σ , ∂ Σ) → ( W, ∂W ) is either constant or it touches M = ∂W only atits boundary ∂ Σ , and this intersection is transverse. Furthermore, if u is non constant, then theboundary curve u | ∂ Σ has to intersect the contact structure ξ on ∂W in positive Reeb direction. In the rest of the article, we will denote the half space (cid:8) z ∈ C (cid:12)(cid:12) Im z ≥ (cid:9) by H . Let ϕ : N (cid:35) M be an immersion of a manifold N in M . We define the self-intersection set of N as N (cid:71) := (cid:8) p ∈ N (cid:12)(cid:12) ∃ p (cid:48) (cid:54) = p with ϕ ( p ) = ϕ ( p (cid:48) ) (cid:9) . Tubular neighborhood theorem for contact submanifolds.
Let N be a contact sub-manifold of ( M, α ). The contact structure ξ = ker α can be split along N into the two subbundles ξ | N = ξ N ⊕ ξ ⊥ N , where ξ N is the contact structure on N given by ξ N = T N ∩ ξ | N = ker α | T N , and ξ ⊥ N is thesymplectic orthogonal of ξ N inside ξ | N with respect to the form dα . Note that ξ ⊥ N carries aconformal symplectic structure given by dα , but neither ξ ⊥ N nor the conformal symplectic structuredo depend on the contact form chosen on M . The bundle ξ ⊥ N can be identified with the normalbundle of N .A well known neighborhood theorem states that ξ ⊥ N determines a small neighborhood of N completely. Theorem 3.
Let ( N, ξ N ) be a contact submanifold of both ( M , ξ ) and ( M , ξ ) . Assume that thetwo normal bundles ( ξ ) ⊥ N and ( ξ ) ⊥ N are isomorphic as conformal symplectic vector bundles. Thenthere exists a small neighborhood of N in M that is contactomorphic to a small neighborhood of N in M . If N has a trivial conformal symplectic normal bundle ξ ⊥ N , then we call the product space N × R k with contact structure α N + (cid:80) kj =1 ( x j dy j − y j dx j ) the standard model for neighborhoods of N . K. NIEDERKR¨UGER AND F. PRESAS The generalized plastikstufe (GPS)
Definition.
Let (
M, α ) be a (2 n + 1)–dimensional contact manifold, and let S be a closed ( n − generalized plastikstufe ( GPS ) is an immersionΦ : S × D (cid:35) M, (cid:0) s, re iϕ (cid:1) → Φ (cid:0) s, re iϕ (cid:1) , such that the pull-back Φ ∗ α reduces to the form f ( r ) dϕ with f ≥ r = 0,and r = 1. Furthermore there is an ε > s, re iϕ ), and ( s (cid:48) , r (cid:48) e iϕ ) with r, r (cid:48) ∈ ( ε, − ε ) that have equal ϕ –coordinate.Finally there must be an open set joining S × { } with S × ∂ D that does not contain any self-intersection points.We call S × { } (or also its image) the core of the GPS, and S × ∂ D (or again the image) its boundary . We denote S × (cid:0) D − { } − ∂ D (cid:1) by GPS ∗ and call it the interior of the GPS. Remark . The regular leaves of the GPS are given by the sets { ϕ = const } . We are hencerequiring that self-intersections only happen between points lying on the same leaf. A differentway to state this requirement consists in saying that there is a continuous map ϑ : Φ (cid:0) GPS ∗ (cid:1) → S such that ϑ (cid:0) Φ( s, r, ϕ ) (cid:1) = ϕ . Theorem 4.
A closed contact manifold ( M, α ) that contains a GPS does not have an exactsymplectic filling.Remark . Using a more precise analysis of bubbling (as in [IS02]) should make it possible toprove that a GPS is an obstruction to finding even a (semipositive) strong symplectic filling. InRemark 4, we sketch how the proof would have to be modified. Note though that [IS02] requiresthat the self-intersections of the GPS are clean.4.
Constructing immersed plastikstufes in neighborhoods of submanifolds
Local construction in codimension two.
The most prominent example of an overtwistedcontact manifold in the literature is R with the structure induced by the contact form α − = cos r dz + r sin r dϕ , written in cylindrical coordinates ( r, ϕ, z ) such that x = r cos ϕ , y = r sin ϕ , and z = z . Anyplane { z = const. } contains an overtwisted disk centered at the origin with radius r = π . Fromthe classification in [Eli93], it follows that ( R , α − ) is up to contactomorphism the unique con-tact structure on R that is overtwisted at infinity, and hence any sufficiently small contractibleneighborhood of an overtwisted disk in a contact 3–manifold is contactomorphic to ( R , α − ).The Reeb field X Reeb associated to α − is given by X Reeb = 1 r + sin r cos r (cid:0) sin r ∂ ϕ + (sin r + r cos r ) ∂ z (cid:1) . Its flow Φ t is linear, because r remains constant, and the coefficients of the z – and the ϕ –coordinateonly depend on the r –coordinate. The Reeb field is tangent to the overtwisted disk on the circle ofradius r such that r = − tan r ( r ≈ . X Reeb has a positive z –component,outside it has a negative one. This means the overtwisted disk D OT and its translation by theReeb flow Φ t ( D OT ) for a time t (cid:54) = 0 only intersect along the circle of radius r (see Fig. 1). Moreprecisely, the Reeb field reduces on the circle of radius r to X Reeb = 1 /r sin r ∂ ϕ , so that itdefines a rotation with period T = 2 πr sin r ≈ . R with R , and define on R × R the contact form α − + R dϑ ,where ( R, ϑ ) are polar coordinates of R . This is a contact fibration, and we will use the firststep of the construction in [Pre07], namely we will trace a closed path γ : S → R that has theshape of a figure-eight, with the double point at the origin, and such that both parts of the eighthave equal area with respect to the standard area form 2 R dR ∧ dϑ . Start at the origin of thedisk, at γ (1) = 0 on this closed loop, and regard the overtwisted disk D OT in the fiber R × { } described above. By using the parallel transport of D OT along the path γ , we are able to describe IZE OF NEIGHBORHOODS IN CONTACT TOPOLOGY 7 D OT Φ t (cid:0) D OT (cid:1) Figure 1.
The overtwisted disk and its image under the Reeb flow only intersectalong a circle of radius r .an immersed plastikstufe. The parallel transport reduces in the fibers to the flow of the vectorfield − c X Reeb with c = (cid:107) γ (cid:107) dϑ ( γ (cid:48) ), so that the monodromy of a closed loop is just given by theReeb flow Φ T for a time T that is equal to the area that has been enclosed by the loop, where wehave to count with orientation. The total area of the figure-eight γ vanishes, because on one partof the eight, we are turning in positive direction, on the other in the opposite one, and the areaof both parts was chosen to be equal. xy R Figure 2.
Parallel transport of the overtwisted disk along a figure-eight pathyields an immersed plastikstufe.We will describe the construction more explicitely to better understand the self-intersection set.The parallel transport of the overtwisted disk defines an immersion D OT × S → R × R , (cid:0) ( x, y, , e iϑ (cid:1) (cid:55)→ (cid:0) Φ T ( ϑ ) ( x, y, , γ ( e iϑ ) (cid:1) , where T ( ϑ ) = (cid:82) γ r dϕ . The map is well defined, because T ( ϑ + 2 π ) = T ( ϑ ). It is also easy to seethat this map is an immersion.The only self-intersection points may lie over the crossing γ (1) = γ ( −
1) in the figure-eight, andin fact, since the Reeb flow moves the interior of the overtwisted disk up, and the outer part down,self-intersections only happen between the two circles (cid:8) ( r cos ϕ, r sin ϕ, (cid:9) × {− , } ⊂ D OT × S . Denote the area enclosed by one of the petals of the figure-eight path by A . The images of any pairof points (cid:0) ( r cos ϕ, r sin ϕ, , (cid:1) and (cid:0) ( r cos( ϕ − t ) , r sin( ϕ − t ) , , − (cid:1) , with t = A/r sin r are identical.Note that if γ is chosen such that A = 2 πr sin r , then the pair of points that intersect eachother always lie on the same ray of the overtwisted disk, and we have in fact constructed a GPS.The figure-eight path has to enclose a sufficiently large area, and we cannot realize such a path γ in a disk D R of radius R < √ r sin r ≈ . Higher codimension.
Use now the same contact structure on ( R , α − ) as above, and extendit to a contact structure on R × C k with contact form α − + k (cid:88) j =1 r j dϕ j , K. NIEDERKR¨UGER AND F. PRESAS where ( r j , ϕ j ) are polar coordinates for the j –th C –factor in C k .Now we will take the k –fold product of figure-eight loops of different sizes, and group them intoa map Γ : T k (cid:35) C k , ( e iϑ , . . . , e iϑ k ) (cid:55)→ (cid:0) γ ( e iϑ ) , / γ ( e iϑ ) , . . . , ( k − / γ ( e iϑ k ) (cid:1) . This map is an immersion with self-intersection setΓ (cid:71) = (cid:8) ( e iϑ , . . . , e iϑ k ) ∈ T k (cid:12)(cid:12) at least one of the ϑ j lies in π Z (cid:9) . Define functions T j ( ϑ ) := 2 j − (cid:82) ϑ γ ∗ (cid:0) r dϕ (cid:1) , and T ( e iϑ , . . . , e iϑ k ) = (cid:80) kj =1 T j ( ϑ j ). Then theimmersion D OT × T k → R × C k , (cid:0) ( x, y, e iϑ , . . . , e iϑ k (cid:1) (cid:55)→ (cid:0) Φ T ( ϑ ,...,ϑ k ) ( x, y, e iϑ , . . . , e iϑ k ) (cid:1) , where Φ t denotes the Reeb flow, is a GPS. Obviously the self-intersection points of this mapare contained in the preimage of the self-intersection set Γ (cid:71) downstairs. Consider two points( e iϑ , . . . , e iϑ k ) and ( e iψ , . . . , e iψ k ) that have the same image under Γ. It follows for each pair( ϑ j , ψ j ) that either ϑ j = ψ j or that ϑ j , ψ j ∈ π Z . The disks lying over such points are given byΦ T ( ϑ ) ( D OT ) and Φ T ( ψ ) ( D OT ) respectively, where D OT = (cid:8) ( x, y, (cid:12)(cid:12) (cid:107) ( x, y, (cid:107) ≤ π (cid:9) . The Reebflow is ϕ –invariant and preserves the distance of the points ( x, y,
0) from the z –axis. Hence in theinterior and the exterior of the circle of radius r , Φ T ( ϑ ) ( x, y,
0) can only be equal to Φ T ( ψ ) ( x (cid:48) , y (cid:48) , T ( ϑ ) = T ( ψ ), because the flow changes the z –coordinate, and by the coefficients chosen in Γfor the paths, T is injective on ( πa , . . . , πa k ) with all a j ∈ { , } . Additionally then we have( x, y,
0) = ( x (cid:48) , y (cid:48) , (cid:107) ( x, y, (cid:107) (cid:54) = r .Self-intersections of the GPS can hence only exist for points where the distance of ( x, y ) from theorigin is equal to r , but by the size condition on the figure-eight loops the holonomy will alwayscorrespond to a rotation by a multiple of 2 π so that all conditions of a GPS are satisfied by thismap.4.3. Application to contact submanifolds.
Let (
N, α N ) be an overtwisted contact 3–manifold.We will show that the product manifold (cid:0) N × C k , α N + (cid:80) kj =1 r j dϑ j (cid:1) , where ( r j , ϑ j ) are polarcoordinates on the j –th factor of C k contains a GPS.Consider a small contractible neighborhood of an overtwisted disk D OT in N . This neighborhoodis contactomorphic to ( R , α − ), because it is overtwisted at infinity. Choose a large ball B in R (so large that the Reeb flow for α − restricted to the overtwisted disk exists for long enoughtimes), then there is a function f : N → R such that the chosen ball B can be embedded by astrict contactomorphism (that means preserving the contact form) into ( N, f α N ). The contactform f α N + (cid:80) kj =1 f r j dϑ j on the product manifold N × C k can be transformed by the map( p ; z , . . . , z k ) (cid:55)→ ( p ; z / √ f , . . . , z k / √ f ) into (cid:0) N × C k , f α N + k (cid:88) j =1 r j dϑ j (cid:1) . This contains a subset of the form (cid:0) B × C k , α − + (cid:80) kj =1 r j dϑ j (cid:1) in which we can perform thecontruction explained above. Corollary 5.
Let ( M, α ) be a closed contact (2 n +1) –manifold that contains an overtwisted contactsubmanifold N of dimension that has trivial contact normal bundle. There is a neighborhood of N that is contactomorphic to a neighborhood U of N × { } in the product space (cid:0) N × C k , α N + (cid:80) kj =1 r j dϑ j (cid:1) .If the neighborhood U contains a sufficiently large disk bundle of N × { } , then it follows that M does not admit an exact symplectic filling.Proof. By the construction just described (cid:0) N × C k , α N + (cid:80) kj =1 r j dϑ j (cid:1) contains a GPS. Since theGPS is compact, it is contained in some disk bundle around N × { } . If the neighborhood of N is contactomorphic to this disk bundle, then ( M, α ) contains a GPS, and hence cannot have anexact symplectic filling. (cid:3)
IZE OF NEIGHBORHOODS IN CONTACT TOPOLOGY 9 Proof of Theorem 4
Sketch of the proof.
The proof is based on [Nie06] (which in turn is ultimately based on[Eli88, Gro85]), and it is very helpful to have a good understanding of this first article. Assume that(
M, α ) has an exact symplectic filling (
W, ω ). We choose an adapted almost complex structure J on W that has in a neighborhood of the core S × { } the special form described in [Nie06, Section 3],and in a neighborhood of the boundary S × ∂ D the particular form described in Section 5.3 below.The chosen complex structure allows us to write down explicitely the members of a Bishopfamily around the core of the GPS, so that we find a non-empty moduli space M of holomorphicdisks u : ( D , ∂ D ) → ( M, GPS ∗ ) with a marked point z ∈ ∂ D . The boundary of each holomorphicdisk u intersects every regular leaf of the GPS exactly once, or expressed differently the followingcomposition defines a diffeomorphism ϑ ◦ u | ∂ D : S → S on the circle. The Bishop family iscanonically diffeomorphic to a neighborhood of the core S × { } via the evaluation mapev z : M →
GPS ∗ , u (cid:55)→ u ( z ) . We can now apply similar intersection arguments for the boundary S × ∂ D of the GPS (Sec-tion 5.3), and for the core ([Nie06, Section 3]), showing that there exists a neighborhood of ∂ GPSthat cannot be penetrated by any holomorphic disk, and that the only disks that come close tothe core are the ones lying in the Bishop family.Choose now a smooth generic path γ ⊂ S × D that avoids the self-intersection points of theGPS, and that connects the core S × { } with the boundary of the GPS. In Section 5.2, we definethe moduli space M γ := ev − z ( γ ), and show that it is a smooth 1–dimensional manifold. From nowon, we will further restrict M γ to the connected component of the moduli space that contains theBishop family. Then in fact M γ has to be diffeomorphic to an open interval. The compactificationof one of the ends of the interval simply consists in decreasing the size of the disks in the Bishopfamily until they collapse to a single point at γ (0) on the core of the GPS.Our aim will be to understand the possible limits at the other end of the interval M γ , andto deduce a contradiction to the fillability of M . The energy of all disks u ∈ M γ is bounded by2 π max f , where α = f ( r ) dϕ on the GPS. By requiring that the GPS has only clean intersections,we could apply the compactness theorem in [IS02] to deduce even a contradiction for the existenceof a semipositive filling (see Remark 4). Instead of merely referring to that result, we have decidedto give a full proof of compactness in our situation (see Section 5.4). This way we can dropthe stringent conditions on the self-intersections of the GPS, and the required proof is in factsignificantly simpler than the full proof of the compactness theorem.It then follows that for any sequence of disks u k ∈ M γ , we find a family of reparametrizations ϕ k : D → D such that u k ◦ ϕ k contains a subsequence converging uniformly with all derivativesto a disk u ∞ ∈ M γ . This means that M γ is compact, but since at the same time we know thatthe far-most right element u ∞ in M γ has a small neighborhood in M γ homeomorphic to an openinterval, it follows that u ∞ is not a boundary point of M γ . Compactness contradicts thus theexistence of the filling. Remark . We will briefly sketch how [IS02] could be used to prove the non-existence of even asemipositive filling, if the GPS is cleanly immersed.The limit of a sequence of holomorphic disks can be described as the union of finitely manyholomorphic spheres u S , . . . , u KS and finitely many holomorphic disks v , . . . , v N . The holomorphicdisks v j : ( D , ∂ D ) → ( W, GPS) are everywhere smooth with the possible exception of boundarypoints that lie on self-intersections of the GPS. Here v j will still be continuous though (As anexample of such disks, take a figure-eight path in the complex plane C . By the Riemann mappingtheorem, there is a holomorphic disk enclosed into each of the loops, but obviously these diskscannot be smooth on their boundary at the self-intersection point of the eight).We will now first prove that the limit curve of a sequence in M γ is only composed of a singledisk, which then necessarily has to be smooth. Assume we would have a decomposition into severaldisks v , . . . , v N . The boundary of each of these disks v j is a continuous path in GPS ∗ , we canhence combine the disks with the projection ϑ defined in Remark 2 to obtain a continuous map ϑ ◦ v j | ∂ D : S → S . Thus we can associate to each of the disks v j a degree. In fact it follows that deg ϑ ◦ v j | ∂ D >
0, because almost all points on the boundary of v j are smooth, and for them v j has to intersect, by Corollary 2, all leaves of the foliation of the GPS in positive direction.Finally assume that there are still several disks, each one necessarily with deg ϑ ◦ v j | ∂ D ≥
1. Thismeans that the composition of the maps ϑ ◦ v j | ∂ D will cover the circle several times, but this isnot possible for the limit of injective maps ϑ ◦ u k | ∂ D . There is hence only a single disk in thelimit. Using Theorem 9 below it finally also follows that this disk is smooth, and has a boundarythat lifts to a smooth loop in S × D .The reason why there are no holomorphic spheres as bubbles is a genericity argument, sincethe disk and all spheres are regular smooth holomorphic objects, we can compute the dimensionof the bubble tree in which our limit object would lie. By the assumption of semi-positivity, itfollows that the dimension would be negative.5.2. The moduli space.
The aim of this section is to define the moduli space of holomorphic disksand to prove that it is a smooth manifold. Care has to be taken, because the boundary conditionconsidered in this article is not a properly embedded, but only an immersed submanifold. Themain idea is to restrict to those holomorphic curves whose boundary lies locally always on a singleleaf of the immersed submanifold. We can then easily adapt standard results.Let (
W, J ) be an almost complex manifold, and let L be a compact manifold with 2 dim L =dim W . Definition. An immersed totally real submanifold is an immersion ϕ : L (cid:35) W such that (cid:0) Dϕ · T x L (cid:1) ⊕ (cid:0) J · Dϕ · T x L (cid:1) = T ϕ ( x ) W at every x ∈ L .Let ϕ : L (cid:35) W be a totally real immersed submanifold with self-intersection set L (cid:71) . Choosea (not necessarily connected) submanifold A (cid:44) → L that is disjoint from L (cid:71) . Let Σ be a Riemannsurface with N boundary components ∂ Σ , . . . , ∂ Σ N , and choose on each boundary component amarked point z j ∈ ∂ Σ j . Then define B (Σ; L ; A ) to be the set of maps u : (Σ , ∂ Σ) (cid:55)→ (cid:0) W, ϕ ( L ) (cid:1) for which the boundary circles u | ∂ Σ can be lifted to continuous loops c : ∂ Σ → L such that ϕ ◦ c = u | ∂ Σ , and c ( z j ) ∈ A .Note that with our conditions the lift of the boundary circles u | ∂ Σ is unique, because if therewere two different loops c, c (cid:48) : ∂ Σ j → L with ϕ ◦ c = ϕ ◦ c (cid:48) , and c ( z j ) , c (cid:48) ( z j ) ∈ A , it follows thatthe set (cid:8) z ∈ ∂ Σ j (cid:12)(cid:12) c ( z ) = c (cid:48) ( z ) (cid:9) contains the point z j , and is hence non-empty. Furthermore thisset is closed, because it is the preimage of the diagonal (cid:52) W := { ( p, p ) | p ∈ W } under the map c × c (cid:48) : ∂ Σ j × ∂ Σ j → W × W intersected with the diagonal (cid:52) ∂ Σ j . Finally, L can be covered byopen sets on each of which the immersion ϕ is injective, and hence if c ( z ) = c (cid:48) ( z ) there is also anopen neighborhood of z on which both paths coincide. It follows that c and c (cid:48) are equal.We have to prove that B (Σ; L ; A ) is a Banach manifold by finding a suitable atlas. To definea chart around a map u ∈ B (Σ; L ; A ), construct first a Banach space B u by considering thespace of sections in E := u − T W satisfying the following boundary condition: Choose the uniquecollection of loops c that satisfy Φ ◦ c = u | ∂ Σ . We can define a subbundle F ≤ E | ∂ Σ over theboundary of the surface by pushing T c ( z ) L with Dϕ into E . We require the sections σ : Σ → E tolie along the boundary ∂ Σ in the subbundle F , and to be at the marked points z j ∈ ∂ Σ j tangentto A .Our aim will be to map these sections in a suitable way into B (Σ; L ; A ). For this, we firstchoose a Riemannian metric on L for which A is totally geodesic. Then we extend it to a productmetric on ∂ Σ × L . There is an ε > ϕ restricted to any ε –disk centered at an c ( z )is an embedding. Furthermore, we find an ε > d (cid:0) c ( z ) , c ( z (cid:48) ) (cid:1) < ε / z, z (cid:48) ∈ ∂ Σ such that d ( z, z (cid:48) ) < ε . Let ε be smaller than min { ε / , ε } , and let U ε ( c ) bethe ε –neighborhood of the loops (cid:8)(cid:0) z, c ( z ) (cid:1)(cid:9) ⊂ ∂ Σ × L , i.e., the collection of all points ( z (cid:48) , x (cid:48) ) thatlie at most at distance ε from the set of loops. The restriction of the immersionid × ϕ : ∂ Σ × L (cid:35) Σ × W, ( z, x ) (cid:55)→ (cid:0) z, ϕ ( x ) (cid:1) IZE OF NEIGHBORHOODS IN CONTACT TOPOLOGY 11 to U ε ( c ) defines an embedded submanifold of Σ × W , because any two points ( z, x ) , ( z (cid:48) , x (cid:48) ) ∈ U ε ( c )for which (cid:0) z, ϕ ( x ) (cid:1) = (cid:0) z (cid:48) , ϕ ( x (cid:48) ) (cid:1) , obviously satisfy z = z (cid:48) , and as we will show x and x (cid:48) both lie inan ε –disk around c ( z ) such that x = x (cid:48) . Let (cid:0) z , c ( z ) (cid:1) be a point for which d (cid:0)(cid:0) z , c ( z ) (cid:1) , ( z, x ) (cid:1) <ε , then using the triangle inequality we get d (cid:0)(cid:0) z, c ( z ) (cid:1) , ( z, x ) (cid:1) ≤ d (cid:0)(cid:0) z , c ( z ) (cid:1) , ( z, c ( z )) (cid:1) + d (cid:0)(cid:0) z , c ( z ) (cid:1) , ( z, x ) (cid:1) < ε , which shows that ( z, x ) lies closer than ε to ( z, c ( x )).Now we can push the metric from U ε ( c ) forward and extend it to one on Σ × W , so that (cid:0) id × ϕ (cid:1)(cid:0) U ε ( c ) (cid:1) will be totally geodesic.Let σ ∈ B u be one of the sections of E described above. If σ is sufficiently small, then applyingthe geodesic exponential map to the section (0 , σ ) in T (Σ × E ), and then projecting to the W –component gives a map that lies in the space B (Σ; L ; A ). The construction described gives abijection between small sections and maps in B (Σ; L ; A ) close to u . The reason is that there is asmooth map that allows us to regard any manifold in M × M tangent to M × { x } at ( x , x )as a graph over M × { x } in a neighborhood of that point.Since we do not see locally the other intersection branches it follows that the Cauchy Rie-mann equation defines a Fredholm operator on B (Σ; L ; A ). For a generic adapted almost complexstructure J , it follows that the moduli space (cid:102) M (Σ; L ; A ) = (cid:8) u ∈ B (Σ; L ; A ) (cid:12)(cid:12) ¯ ∂ J u = 0 (cid:9) is a smooth manifold. In our case, we then have that M γ := (cid:102) M ( D ; GPS; γ ) /G is a 1–dimensionalmanifold.5.3. The boundary of the GPS.
The standard definition of the plastikstufe requires the bound-ary ∂ PS ( S ) to be a regular leaf of the foliation [Nie06]. That way, PS ( S ) − S × { } is a totallyreal manifold, and gives thus a Fredholm boundary condition for regarding holomorphic disks, atthe same time smooth holomorphic disks in the moduli space have to be transverse to the foliationso that they cannot touch the boundary.In our definition of the GPS, we want the contact form instead to vanish on the boundary S × ∂ D . In this section, we will show by an intersection argument that there is a neighborhood ofthe boundary which blocks any holomorphic curve from entering it. Our definition thus impliesat this point effectively the same statement as the standard one. Proposition 6.
Let F be a maximally foliated submanifold inside a contact manifold ( M, α ) .Assume one of its boundary components to be diffeomorphic to N ∼ = S × S , with S a closedmanifold, such that the restriction α | T F of the contact form has the following properties on thecollar neighborhood N × [0 , ε ) = { ( s, e iϕ , r ) } (1) α | T F vanishes on N (in particular N is a Legendrian submanifold), and (2) the interior of the collar is foliated and the leaves are S × { e iϕ } × (0 , ε ) , for any fixed e iϕ ∈ S .Then there is a neighborhood of N in M that is contactomorphic to an open subset of (cid:0) R × T ∗ S × S × R , dz + λ can − r dϕ (cid:1) such that N × [0 , ε ) lies in this model in { } × S × S × [0 , ε ) .Proof. First note that it is clear that the restriction of the contact form can be written on thecollar neighborhood as α | T F = f dϕ , with a smooth function f : N × [0 , ε ) → R ≥ which only vanishes on N × { } . The 2–form dα isa symplectic form on the (2 n )–dimensional kernel ξ = ker α , so in particular dα | T F cannot vanishon any point p ∈ N , because otherwise T p F would be an ( n + 1)–dimensional isotropic subspace of( ξ p , dα ). It follows that ∂ r f ( p, >
0, and so the map Φ : N × [0 , ε ) → N × R , ( p, r ) (cid:55)→ ( p, f ( p, r )) is after a suitable restriction a diffeomorphism with inverse Φ − ( p, r ) = ( p, f − p ( r )), where wewrote f p ( · ) := f ( p, · ). The pull-back of α | T F under Φ − gives (cid:0) Φ − (cid:1) ∗ ( f dϕ ) = f ( p, f − p ( r )) dϕ = r dϕ . Thus, we can assume after changing the orientation of S that α is of the form − r dϕ on the collarneighborhood.Consider now the normal bundle of the submanifold N × [0 , ε ) in M . A trivialization can beobtained by realizing first that the Reeb field X Reeb is transverse to N , because T F | N lies in thecontact structure, so that there is a small neighborhood, where X Reeb is transverse to F . Choosenow an almost complex structure J on ξ = ker α that is compatible with dα such that J leavesthe space on N spanned by (cid:104) ∂ r , ∂ ϕ (cid:105) invariant. The submanifolds S ( e iϕ ,r ) := S × { ( e iϕ , r ) } , with( e iϕ , r ) fixed, are all tangent to the contact structure, and it follows that J · T S ( e iϕ ,r ) is transverseto F , because if there was an X ∈ T S , such that JX ∈ T F , then0 < dα ( X, JX ) = − dr ∧ dϕ ( X, JX ) = 0 . With the tubular neighborhood theorem it follows that there is an open set around N × [0 , ε )diffeomorphic to R × T ∗ S × S × ( − ε, ε ), and the set N × [0 , ε ) lies in { } × S × S × [0 , ε ).In the final step, we use a version of the Moser trick explained for example in [Gei06, The-orem 2.24] to find a vector field X t that isotopes the given contact form into the desired one dz + λ can − r dϕ . Let α t , t ∈ [0 , ψ t defined around N such that ψ ∗ t α t = α . The field X t generating this isotopysatisfies the equation L X t α t + ˙ α t = 0 . By writing X t = H t R t + Y t , where H t is a smooth function, R t is the Reeb vector field of α t , and Y t ∈ ker α t , we obtain plugging then R t into the equation above dH t ( R t ) = − ˙ α t ( R t ) . The vector field Y t is completely determined by H t , because Y t satisfies the equations ι Y t α t = 0 ,ι Y t dα t = − dH t − ˙ α t , hence it suffices to find a suitable function H t . Consider the 1-parameter family of Reeb fields R t as a single vector field on the manifold [0 , × (cid:0) R × T ∗ S × S × R (cid:1) . Since R t is transverse to thesubmanifold (cid:101) N := [0 , × (cid:0) { } × T ∗ S × S × R (cid:1) along [0 , × N × [0 , ε ), it is possible to define asolution H t to dH t ( R t ) = − ˙ α t ( R t ), such that H t | e N ≡
0. In fact, because ˙ α | N × [0 ,ε ) = 0, it followsthat dH t | N × [0 ,ε ) = 0, and so the vector field X t = H t R t + Y t vanishes on N × [0 , ε . Hence X t canbe integrated on a small neighborhood of the collar N × [0 , ε ), and N × [0 , ε ) is not moved underthe flow, which finishes the proof of the proposition. (cid:3) We can easily choose a compatible adapted almost complex structure J on the symplectization (cid:16) W = R × ( R × T ∗ S × S × R ) , ω = d (cid:0) e t ( dz + λ can − r dϕ ) (cid:1)(cid:17) , with coordinates { ( t, z ; q , p ; e iϕ , r ) } . Observe that the Reeb field is given by X Reeb = e − t ∂ z , andthat the kernel of α is spanned by the vectors X − λ can ( X ) ∂ z for all X ∈ T ( T ∗ S ), ∂ ϕ + r ∂ z and ∂ r . Choose a metric g on S , and let J be the dλ can –compatible almost complex structure on T ∗ S constructed in [Nie06, Appendix B].With this, we can define a J on W by setting J∂ t = X Reeb , JX Reeb = − ∂ t , J ∂ r = − ∂ ϕ − r ∂ z , J ( ∂ ϕ + r ∂ z ) = ∂ r , and J ( X − λ can ( X ) ∂ z ) = J X − λ can ( J X ) ∂ z . The last two equations can alsobe written as J ∂ ϕ = ∂ r + re t ∂ t and JX = J X − e t λ can ( X ) ∂ t − λ can ( J X ) ∂ z . IZE OF NEIGHBORHOODS IN CONTACT TOPOLOGY 13
As a matrix, the complex structure J takes the form J ( t ; z ; q , p ; e iϕ , r ) = − e t − e t λ can re t e − t − λ can ◦ J − r J −
10 0 0 1 0 . Note that the center row and column represent linear maps from or to T ( T ∗ S ). A lengthy com-putation shows that this structure is compatible with ω . Proposition 7.
The almost complex manifold ( W, J ) can be mapped with a biholomorphism to ( C × T ∗ S × C ∗ , i ⊕ J ⊕ i ) . In this model, the contact manifold M is described by the set M ∼ = (cid:110) ( x + iy ; q , p ; w ) ∈ C × T ∗ S × C ∗ (cid:12)(cid:12)(cid:12) x = − (cid:107) p (cid:107) + (ln | w | ) (cid:111) , and the maximally foliated submanifold F is F ∼ = (cid:110) ( x ; q , w ) ∈ R × S × C ∗ (cid:12)(cid:12)(cid:12) x = − | w | ) , | w | ≥ (cid:111) ⊂ C × T ∗ S × C ∗ . Proof.
The desired biholomorphism isΦ( t, z ; q , p ; e iϕ , r ) = (˜ t, ˜ z ; ˜ q , ˜ p ; ˜ re i ˜ ϕ ) = (cid:18) − e − t − F − r , z ; q , p ; e r e iϕ (cid:19) , with the function F : T ∗ M → R , ( q , p ) (cid:55)→ (cid:107) p (cid:107) . It brings J into standard form with respect to the coordinate pairs (˜ re i ˜ ϕ ), (˜ t, ˜ z ). More explicitely,by pulling back J under the diffeomorphismΦ − (˜ t, ˜ z ; ˜ q , ˜ p ; ˜ re i ˜ ϕ ) = ( t, z ; q , p ; e iϕ , r ) = (cid:18) − ln( − ˜ t − F − (ln ˜ r ) , ˜ z ; ˜ q , ˜ p ; e i ˜ ϕ , ln ˜ r (cid:19) , i.e. by computing D Φ · J · D Φ − , we obtain the matrix D Φ · J · D Φ − = − − λ can − dF ◦ J dF − λ can ◦ J J − / ˜ r r , and since, according to [Nie06, Appendix B], dF ◦ J = − λ can , this gives the desired normalform. (cid:3) Proposition 8.
Let F be a maximally foliated submanifold in a contact manifold ( M, α ) . Let ( W, ω ) be a symplectic filling of M , and assume F to have a boundary component of the typeexplained in Proposition 6. There is a neighborhood U of the boundary with an almost complexstructure, which prevents any holomorphic curve u : (Σ , ∂ Σ) → ( W, F ) that has in F contractibleboundary components, from entering U .Proof. Choose for the neighborhood U of ∂F the model described in Proposition 7 together withthe almost complex structure J given there. This J can be easily extended over the whole filling( W, ω ). Note that the neighborhood is foliated by J -holomorphic codimension 2 manifolds of theform N C := { C } × T ∗ S × C ∗ for any fixed complex number C .Let now u : (Σ , ∂ Σ) → ( W, F ) be a holomorphic curve that has in F contractible boundary, andassume that u intersects the model neighborhood U . Write the restriction of u to V := u − ( U ) ⊂ Σas u | V : V → C × T ∗ S × C ∗ , z (cid:55)→ (cid:0) u ( z ); q ( z ) , p ( z ); u ( z ) (cid:1) . First, we will show that the imaginary part of the first coordinate u is constant. If it was not,then there would be by Sard’s Theorem a regular value c y ∈ R , such that u − ( R + ic y ) consistsof finitely many regular 1–dimensional submanifolds. The real part of u changes along thesesubmanifolds, because u satisfies the Cauchy-Riemann equation. Hence it is possible to find acomplex number c x + ic y ∈ C such that N c x + ic y has finitely many transverse intersection pointswith u . By our assumption, it is possible to cap off the holomorphic curve u by adding disks thatlie inside F . Note that N c x + ic y is the boundary of the submanifold (cid:101) N c x + ic y := (cid:8) x + ic y (cid:12)(cid:12) x ∈ [ c x , ∞ ) (cid:9) × T ∗ S × C ∗ . The intersection of (cid:101) N c x + ic y with M gives a submanifold that is disjoint from F , and this subman-ifold together with N c x + ic y represents the trivial homology class in H n − ( W ).The only intersections between the capped off holomorphic curve and ∂ ( W ∩ (cid:101) N c x + ic y ) lie in thesubsets, where both classes are represented by J -holomorphic manifolds. Hence the intersectionnumber is positive, but since ∂ ( W ∩ (cid:101) N c x + ic y ) represents the trivial class in homology, this is clearlya contradiction.It follows that Im u is constant, and with the Cauchy-Riemann equation, we immediatelyobtain that the real part of u must also be constant. This in turn means that the holomorphiccurve is completely contained in the neighborhood, because the only way that u could not becompletely contained in the neighborhood U is if | u ( z ) | changes sufficiently or if p grows, but inboth cases u will hit the hypersurface M before leaving U . Hence u is contained in U . Considernow the T ∗ S –part of u . Since u sits on F , it follows that the T ∗ S –part has boundary on thezero-section of T ∗ S , and so it has no energy, and is thus constant. So far, it follows that u ( z ) canbe written as u ( z ) = ( c x + ic y ; q , f ( z )), where f : D → C ∗ such that f ( S ) lies in one of thecircles of fixed radius R or 1 /R . But in fact, only the circle of radius R > F , hence allboundary component of Σ are mapped to the circle of radius R , and so by the maximum principle | f | is bounded by R , and by the boundary point lemma the derivative of f along the boundarymay nowhere vanish. (cid:3) Bubbling analysis.
To obtain compactness of our moduli space, we need to distinguishtwo cases: Either the first derivatives of the sequence are uniformly bounded from the beginning,and we have subsequence with a clean limit (after adapting the standard result to the immersedboundary condition), or if the first derivatives explode, we show that we do find a global uniformbound on the derivatives if we reparametrize the disks in a suitable way.
Theorem 9.
Let Σ be a Riemann surface that does not need to be compact, and may or may nothave boundary. Let Ω k ⊂ Σ be a family of increasing open sets that exhaust Σ , i.e., ∪ k Ω k = Σ and Ω k ⊂ Ω k (cid:48) for k ≤ k (cid:48) .Define ∂ Ω k := Ω k ∩ ∂ Σ . Let ( W, J ) be a compact almost complex manifold that contains a totallyreal immersion ϕ : L (cid:35) W of a compact manifold L .Let u k be a sequence of holomorphic maps u k : (cid:0) Ω k , ∂ Ω k (cid:1) → (cid:0) W, ϕ ( L ) (cid:1) whose derivatives areuniformly bounded on compact sets, i.e., if K ⊂ Σ is a compact set, then there exists a constant C ( K ) > such that (cid:107) Du k ( z ) (cid:107) ≤ C ( K ) for all k and all z ∈ Ω k ∩ K . Additionally assume that the restriction of u k to the boundary ∂ Ω k lifts to a collection of smooth paths u Lk : ∂ Ω k → L such that ϕ ◦ u Lk = u k | ∂ Ω k .Then there exists a subsequence of u k that converges on any compact subset uniformly with allderivatives to a holomorphic curve u ∞ : (Σ , ∂ Σ) → (cid:0) W, ϕ ( L ) (cid:1) , whose boundary lifts to a collectionof smooth paths u L ∞ : ∂ Σ → L , and the boundary paths u Lk also converge locally uniformly to u L ∞ .Proof. The theorem is well-known in case that ∂ Σ = ∅ or that ϕ ( L ) is an embedded totally realsubmanifold (see for example [MS04, Theorem 4.1.1]). In fact, our situation can be reduced toone, where we can apply this standard result. Using Arzel`a-Ascoli it is easy to find a subsequence u k that converges uniformly in C on any compact set to a continuous map u ∞ , and such that thelifts u Lk : ∂ Ω k → L converge in C on any compact set to a lift u L ∞ : ∂ Σ → L with ϕ ◦ u L ∞ = u ∞ | ∂ Σ . IZE OF NEIGHBORHOODS IN CONTACT TOPOLOGY 15
Let K ⊂ Σ be a compact set on which we want to show uniform C ∞ –convergence. If ∂K := K ∩ ∂ Σ is empty, then the uniform converges for the derivatives follows from the standard result.If ∂K is non-empty, then cover u L ∞ ( ∂K ) with a finite collection of open sets V , . . . , V N on eachof which ϕ is injective. We can choose smaller open subsets V (cid:48) j ⊂ V j whose closure V (cid:48) j is alsocontained in V j , and whose union V (cid:48) ∪ · · · ∪ V (cid:48) N still cover u L ∞ ( ∂K ).Cover also K itself with open sets U k that either do not intersect the boundary ∂K or if U k ∩ ∂K (cid:54) = ∅ , then there is a V (cid:48) j such that u L ∞ ( U k ∩ ∂K ) ⊂ V (cid:48) j . Only finitely many U k are neededto cover K . We get for every U k ∩ K uniform C ∞ –convergence, because if U k intersects now ∂K we can use the standard result: For n large enough u n ( U k ∩ ∂K ∩ Ω n ) will be contained in thelarger subset V j , on which ϕ is an embedding. (cid:3) Theorem 10 (Gromov compactness) . Let u n : ( D , ∂ D ) → ( W, GPS) be a sequence of holomorphic disks that represent elements in the moduli space M γ .There exists a family ϕ n : D → D of biholomorphisms such that u n ◦ ϕ n contains a subsequenceconverging uniformly in C ∞ to a holomorphic disk u ∞ : ( D , ∂ D ) → ( W, GPS) that represents again an element in M γ .Proof. Choose an arbitrary J –compatible metric on W , and endow the disk D ⊂ C with thestandard metric g on the complex plane. Denote by (cid:107) Du k ( z ) (cid:107) D the norm of the differential of u k at a point z ∈ D with respect to g on the disk, and the chosen metric on W . If (cid:107) Du k (cid:107) D isuniformly bounded for all k ∈ N and all z ∈ D , then by Theorem 9 we are done.So assume this to be false, then there exists (by going to a subsequence if necessary) a sequence z k ∈ D such that (cid:107) Du k ( z k ) (cid:107) D → ∞ , and in fact by using rotations, we may assume that all z k lie on the interval [0 , H ⊂ C be the upper half plane { z | Im z ≥ } endowed with the standard metric, and denoteby (cid:107) Dv ( z ) (cid:107) H the norm of the differential of a map v : H → W at a point z ∈ H with respect tothe standard metric on the half plane, and the chosen metric on W . We can map the half planeinto the unit disk using the biholomorphismΦ : H → D − {− } , z (cid:55)→ i − zi + z , and use this to pull-back the sequence of disks to u H k := u k ◦ Φ : ( H , R ) → ( W, GPS). The map Φis not an isometry, but on compact sets of the upper half plane, Φ ∗ g is equivalent to the standardmetric. Hence it follows that also (cid:107) Du H k (cid:107) H cannot be bounded on the segment I := { it | t ∈ [0 , } .Let x k ∈ I be a point where (cid:107) Du H k (cid:107) H takes its maximum on I .Apply the Hofer Lemma (see for example [MS04, Lemma 4.6.4]) for fixed k , and δ = 1 /
2, thatmeans, restrict (cid:107) Du H k (cid:107) H to the unit disk D ( x k ) ∩ H . There is a positive ε k with ε k ≤ /
2, and a y k ∈ D / ( x k ) ∩ H such that (cid:107) Du H k ( x k ) (cid:107) H ≤ ε k (cid:107) Du H k ( y k ) (cid:107) H and (cid:107) Du H k ( z ) (cid:107) H ≤ (cid:107) Du H k ( y k ) (cid:107) H for all z ∈ D ε k ( y k ) ∩ H .Set c k := (cid:107) Du H k ( y k ) (cid:107) H . First we will show that for large k , all the disks D ε k ( y k ) intersectthe boundary ∂ H = R of the half plane. Even stricter, there exists a constant K > c k Im( y k ) < K for all k (if the disks intersect the real line, we have Im y k < ε k , multiplying with c k on both sides would still allow the left side to be unbounded). Suppose that such a constant did not exist, so that by going to a subsequence, c k Im y k converges monotonously to ∞ . Define H k := { z ∈ C | Im z ≥ − c k Im y k } , and a sequence of biholomorphisms ϕ k : D ε k c k ∩ H k → D ε k ( y k ) ∩ H , z (cid:55)→ y k + zc k . Pulling back, we find holomorphic maps (cid:98) u k := u H k ◦ ϕ k : D ε k c k ∩ H k → W with (cid:107) D (cid:98) u k (0) (cid:107) = 1,and (cid:107) D (cid:98) u k (cid:107) ≤ (cid:98) u ∞ : C → W . The standard removal of singularity theorem yields then anon-constant holomorphic sphere, which cannot exist in an exact symplectic manifold. Thus thereis a constant K > c k Im( y k ) < K .Now we slightly modify the charts used above to keep the boundary of the reparametrizeddomains on the height of the real line. Set y (cid:48) k := c k Im y k and r k := ε k c k , and consider thefollowing sequence of biholomorphisms ψ k : D r k ( iy (cid:48) k ) ∩ H → D ε k ( y k ) ∩ H , z (cid:55)→ zc k + Re y k . Note that the intersection of D ε k ( y k ) with the real line is given by the interval D ε k ( y k ) ∩ R = D ( x k ) ∩ R ⊂ ( − , . The image of the interval ( − ,
1) under Φ is the segment on the boundary of the unit disk enclosedbetween the angles ( − π/ , π/ (cid:98) u k := u H k ◦ ψ k : D r k ( iy (cid:48) k ) ∩ H → W we have (cid:107) D (cid:98) u k (cid:107) ≤
2, and (cid:107) D (cid:98) u k ( iy (cid:48) k ) (cid:107) = 1. We can also find a subsequence of (cid:98) u k with increasing domains,i.e., D r k ( iy (cid:48) k ) ⊂ D r l ( iy (cid:48) l ) for all l ≥ k , by using that the y (cid:48) k are all bounded while the radii of thedisks r k become arbitrarily large. Then Theorem 9 provides a subsequence of the (cid:98) u k that convergeslocally uniformly with all derivatives to a holomorphic map (cid:98) u ∞ : ( H , R ) → ( W, GPS). To see that (cid:98) u ∞ is not constant, take a subsequence such that y (cid:48) k converges to y (cid:48)∞ . The norm of the derivative of u ∞ at iy ∞ is (cid:107) D (cid:98) u ∞ ( iy (cid:48)∞ ) (cid:107) = 1, because (cid:107) D (cid:98) u ∞ ( iy (cid:48)∞ ) − D (cid:98) u k ( iy (cid:48) k ) (cid:107) ≤ (cid:107) D (cid:98) u ∞ ( iy (cid:48)∞ ) − D (cid:98) u ∞ ( iy (cid:48) k ) (cid:107) + (cid:107) D (cid:98) u ∞ ( iy (cid:48) k ) − D (cid:98) u k ( iy (cid:48) k ) (cid:107) becomes arbitrarily small. The first term is small, because the differentialof (cid:98) u ∞ is continuous, the second can be estimated by using that the convergence of (cid:98) u k to (cid:98) u ∞ isuniform on a small compact neighborhood of iy (cid:48)∞ .Let us come back to the initial family of disks u k : ( D , ∂ D ) → ( W, GPS). The maps ψ k inducereparametrizations of the whole disk by Φ ◦ ψ k ◦ Φ − . The image of a compact subset of D − {− } under Φ − is a compact subset in H , so that we get on any compact subset of D − {− } uniform C ∞ –convergence of u k ◦ ψ k to u ∞ := (cid:98) u ∞ ◦ Φ − . To complete the proof of our compactness theorem,we have to show that the first derivatives of u k ◦ ψ k are also uniformly bounded in a neighborhoodof {− } .Rotate the disk D by multiplying its points by e iπ = − − (cid:0) H − { } , ( −∞ , ∪ (0 , ∞ ) (cid:1) → ( W, GPS) , z (cid:55)→ u ∞ (cid:0) − Φ( z ) (cid:1) has finite energy, and we can apply the removal of singularity theorem in the form describedin Theorem 11. The consequence for u ∞ is that the composition ϑ ◦ u ∞ | ∂ D −{− } extends to acontinuous map S → S that is strictly monotonous. In fact, ϑ ◦ u ∞ | ∂ D −{− } covers the wholecircle with exception of the point e i ϕ ∞ := lim e iϕ →− ϑ ◦ u ∞ ( e iϕ ) , and so for any ε –neighborhood U ε ⊂ S of e i ϕ ∞ , we find a δ > { ϑ ◦ u k ◦ ψ k ( e iϕ ) | ϕ ∈ ( − π/ δ, π/ − δ ) } covers for any sufficiently large k the complement S − U ε of U ε . Let K be the segment { e iϕ | ϕ ∈ ( − π/ , π/ } . Remember that the images of K exhausts ∂ D − {− } ifwe apply the reparametrizations ψ k , and so it follows in particular that the unparametrized disks u k : ( D , ∂ D ) → ( W, GPS) intersect on K for sufficiently large k almost all leaves of the foliationof the GPS. IZE OF NEIGHBORHOODS IN CONTACT TOPOLOGY 17
Assume now that the first derivatives of the u k ◦ ψ k are not uniformly bounded in a neighborhoodof {− } . By the same reasoning, it follows that the u k ◦ ψ k intersect almost all leaves of the GPSon the segment K (cid:48) = { e iϕ | ϕ ∈ ( π/ , π/ } , but this yields a contradiction. (cid:3) In our special situation, we only need the following very weak form of removal of singularity,which states that a holomorphic curve that has a puncture on its boundary approaches the sameleaf of the foliation from both sides of the puncture.
Theorem 11 (Removal of singularity) . Let ( W, ω ) be a compact manifold with exact symplecticform ω = dα , and with convex boundary ∂W = ( M, α ) . Assume that M contains a GPS ϕ : S × D (cid:35) M , and choose an adapted almost complex structure J on W . Assume u : (cid:16) D ε ∩ H − { } , ( − ε, ∪ (0 , ε ) (cid:17) → (cid:0) W, GPS (cid:1) to be a non-constant holomorphic curve that has finite energy. Recall that there was a continuousmap ϑ : GPS ∗ → S that labels the leaves of the foliation on the GPS.We find a continuous path (cid:98) c : ( − ε, ε ) → S with (cid:98) c | ( − ε, ∪ (0 ,ε ) = ϑ ◦ u | ( − ε, ∪ (0 ,ε ) . A more geometric way to state this result is to say that the boundary segments of the holomorphiccurve approach from both sides of asymptotically the same leaf.Proof. One of the basic ingredients in all proofs of this type is the following estimate for the energyof uE ( u ) = (cid:90) D ε ∩ H −{ } u ∗ ω = (cid:90) ε (cid:90) γ r | ∂ ϕ u | r r dr ∧ dϕ ≥ (cid:90) ε (cid:18)(cid:90) γ r | ∂ ϕ u | dϕ (cid:19) dr πr = (cid:90) ε L ( γ r ) πr dr , where γ r is the image (cid:8) u ( re iϕ ) (cid:12)(cid:12) ϕ ∈ [0 , π ] (cid:9) of the half-circle of radius r in the hyperbolic plane,and L ( γ r ) is its length with respect to the compatible metric on W . It is clear that L ( γ r ) cannotbe bounded from below, because the energy E ( u ) is finite.Denote the segments composing the map ϑ ◦ u | ( − ε, ∪ (0 ,ε ) by c − : ( − ε, → S , and c + :(0 , ε ) → S . By Corollary 2, both maps c ± are strictly increasing.It easily follows that the c ± are bounded close to 0 (in the sense that they do not turn infinitelyoften as z → ∞ ), because there is a sequence of radii r k with r k → L ( γ r k ) → D r − D r k ) ∩ H by D ( r , r k ). Then E (cid:0) u | D ( r ,r k ) (cid:1) = (cid:90) ∂D ( r ,r k ) u ∗ α ≥ (cid:90) [ − r , − r k ] ∪ [ r k , r ] u ∗ α − (cid:0) L ( γ r ) + L ( γ r k ) (cid:1) max (cid:107) α (cid:107) → ∞ . It follows that we find continuous extensions (cid:98) c − : ( − ε, → S , and (cid:98) c + : [0 , ε ) → S . If (cid:98) c − (0) = (cid:98) c + (0), we are done, so assume these limits to be different. Choose a small δ >
0, such that the δ –neighborhoods U − , U + ⊂ S around (cid:98) c − (0) and (cid:98) c + (0) respectively do not overlap. There is an ε (cid:48) > , ε (cid:48) ) is contained in (cid:98) c − ( U + ), and ( − ε (cid:48) ,
0] is contained in (cid:98) c − − ( U − ),and all the points in u (cid:0) (0 , ε (cid:48) ) (cid:1) are at distance more than C > u (cid:0) ( − ε (cid:48) , (cid:1) . Inparticular it follows that the length L ( γ r ) for any r ∈ (0 , ε (cid:48) ) is larger than C , and so by the energyinequality at the beginning of the proof, we get a contradiction to (cid:98) c − (0) (cid:54) = (cid:98) c + (0). (cid:3) Outlook and open questions
One obvious application of the observations made in this paper is the definition of a capacityinvariant for contact manifolds. Unfortunately, we were not able to measure the “size” efficientlyin a numerical way so that our invariant is rather rough.To measure the capacity, we choose a contact manifold (
N, ξ N ) that will serve as the “testingprobe”. Then we can define for any contact manifold (
M, ξ ) with dim M = 2 k + dim N , and k ≥
1, aninvariant C ξ N defined as follows C ξ N ( M, ξ ) =
N, ξ N ) cannot be embedded with trivial normal bundle into M ; ∞ N × R k with the standard contact form can be embedded into M ;1 otherwise, that means ( N, ξ N ) can be embedded with trivial normalbundle into M , but not with the full neighborhood.This way, we obtain for the standard sphere (cid:0) S n − , ξ (cid:1) that C ξ ( M, ξ ) = ∞ for any contactmanifold ( M, ξ ). If (cid:0)
N, ξ − (cid:1) is an overtwisted contact 3–manifold, and if ( M, ξ ) is a manifold withexact symplectic filling, then C ξ − ( M, ξ ) < ∞ .The most important problem in this context would be to find examples of contact manifolds thatdo allow the embedding of an overtwisted contact manifold N with the full model neighborhood N × R k , because otherwise it is so far unclear whether the capacity C ξ N is able to distinguishany manifolds. Possible candidates to check are the following: Question 1.
Let ( M, α ) be a closed contact manifold. Bourgeois described in [Bou02] a construc-tion of a contact structure on M × T for which every fiber M ×{ p } with p ∈ T is contactomorphicto the initial manifold. How large is the tubular neighborhood of such a fiber? Question 2.
Giroux conjectures that contact manifolds of arbitrary dimension obtained from thenegative stabilization of an open book should be “overtwisted”. The simplest example of such amanifold is a sphere ( S n − , α − ) constructed by taking the cotangent bundle T ∗ S n − for the pages,and a negative Dehn-Seidel twist as the monodromy map (see Example 5). How large is the tubularneighborhood of ( S , α − ) in a higher dimensional sphere? References [Bou02] F. Bourgeois,
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Reflection principle and J -complex curves with boundary on totallyreal immersions , Commun. Contemp. Math. (2002), no. 1, 65–106.[KN07] O. van Koert and K. Niederkr¨uger, Every contact manifolds can be given a nonfillable contact structure ,Int. Math. Res. Not. IMRN (2007), no. 23, Art. ID rnm115, 22.[MS04] D. McDuff and D. Salamon, J -holomorphic curves and symplectic topology. , Colloquium Publications.American Mathematical Society 52. Providence, RI: American Mathematical Society (AMS)., 2004.[Mul90] M.-P. Muller, Une structure symplectique sur R avec une sph`ere lagrangienne plong´ee et un champ deLiouville complet , Comment. Math. Helv. (1990), no. 4, 623–663.[Nie06] K. Niederkr¨uger, The plastikstufe - a generalization of the overtwisted disk to higher dimensions. , Algebr.Geom. Topol. (2006), 2473–2508.[Pre07] F. Presas, A class of non-fillable contact structures , Geom. Topol. (2007), 2203–2225. IZE OF NEIGHBORHOODS IN CONTACT TOPOLOGY 19
E-mail address , K. Niederkr¨uger: [email protected] (K. Niederkr¨uger) ´Ecole Normale Sup´erieure de Lyon, Unit´e de Math´ematiques Pures et Appliqu´ees,UMR CNRS 5669, 46 all´ee d’Italie, 69364 LYON Cedex 07, France
E-mail address , F. Presas: [email protected] (F. Presas)
Departamento de ´Algebra, Facultad de Matem´aticas, Universidad Complutense de Madrid,Plaza de Ciencias n oo