The Weinstein conjecture in the presence of submanifolds having a Legendrian foliation
TTHE WEINSTEIN CONJECTURE IN THE PRESENCE OF SUBMANIFOLDSHAVING A LEGENDRIAN FOLIATION
KLAUS NIEDERKR ¨UGER AND ANA RECHTMAN
Abstract.
Helmut Hofer introduced in ’93 a novel technique based on holomorphic curves toprove the Weinstein conjecture. Among the cases where these methods apply are all contact3–manifolds (
M, ξ ) with π ( M ) (cid:54) = 0. We modify Hofer’s argument to prove the Weinsteinconjecture for some examples of higher dimensional contact manifolds. In particular, we are ableto show that the connected sum with a real projective space always has a closed contractibleReeb orbit. Introduction
Let (
M, ξ ) be a contact manifold with contact form α . The associated Reeb field R α is theunique vector field that satisfies the equations α ( R α ) = 1 and ι R α dα = 0everywhere. Weinstein conjecture.
Let ( M, ξ ) be a closed contact manifold, and choose any contact form α with ξ = ker α . The Reeb field R α associated to α always has a closed orbit. In his seminal paper [Hof93], Helmut Hofer found a strong relation between the dynamics ofthe Reeb field and holomorphic curves in symplectizations. Initially Hofer proved the Weinsteinconjecture using these methods in three cases, namely the conjecture holds for a closed contact3–manifold (
M, ξ ), if M is diffeomorphic to S , if ξ is overtwisted or if π ( M ) (cid:54) = 0.A generalization of the second case to higher dimensions has been achieved in [AH09] for contactstructures that have a Plastikstufe. We will try to generalize the third one. In order to justify ourhypothesis, let us recall the key steps in Hofer’s proof. The non-triviality of the second homotopygroup combined with the assumption that the contact structure is tight, allow us to find a non-contractible embedded sphere whose characteristic foliation has only two elliptic singularities andno closed leaves. Near the singularities, an explicit Bishop family of holomorphic disks can beconstructed, and it can be proved that the disks produce a finite energy plane. The existence ofa finite energy plane in a symplectization of the manifold implies the existence of a contractibleperiodic Reeb orbit.Our generalization of Hofer’s theorem for contact (2 n + 1)–manifolds replaces the 2–sphereby an embedded ( n + 1)–submanifold such that the contact structure restricts to an open bookdecomposition. Following Hofer’s ideas, we will prove that if such a submanifold represents anon-trivial homology class then there exists a periodic contractible Reeb orbit.Among the examples where we are able to find such submanifolds, are the connected sum ofany contact manifold M with(a) the projetive space with its standard contact structure(b) certain subcritically fillable manifolds as for example S n × S n +1 or T n × S n +1 . Acknowledgments.
Several people helped us writing this article: In particular we would like tothank Chris Wendl for spotting a mistake in a preliminary version and Janko Latschev for solvingthis problem. We also received helpful comments by Emmanuel Giroux and Leonid Polterovich.The first author would like to thank the University of Chicago, the MSRI, and the CCCI forpartially funding his stay in Chicago, where the initial ideas of this paper have been developed.The second author would like to thank the CONACyT for her postdoctoral fellowship. a r X i v : . [ m a t h . D S ] A p r K. NIEDERKR¨UGER AND A. RECHTMAN The main criteria
Definition. An open book decomposition ( ϑ, B ) of a manifold N consists of(i) a proper codimension 2 submanifold B ⊂ N that has a tubular neighborhood that isdiffeomorphic to B × C , and(ii) a proper fibration ϑ : ( N \ B ) → S such that the map ϑ agrees on the neighborhood B × C with the angular coordinate e iϕ of the C –factor.The submanifold B is called the binding , and the fibers of ϑ are called the pages of the openbook. From the definition it follows that the closure of a page P in N is a compact manifold withboundary B . Remark . Open book decompositions are typically only studied on closed manifolds, in whichcase the binding is also a closed manifold. In this article, we will first restrict to closed manifolds,but then we will study closed manifolds whose universal cover admits an open book decomposition,and we do not want to suppose that the universal cover itself is a closed manifold.Assume (
M, ξ ) is a contact (2 n + 1)–manifold. A submanifold N (cid:44) → M is called maximallyfoliated by ξ if dim N = n + 1, and if the intersection ξ ∩ T N defines a singular foliation on N .The regular leaves of such a foliation are locally Legendrian submanifolds. Definition.
In the situation above, we say that
N (cid:44) → ( M, ξ ) carries a
Legendrian open book ,if the maximal foliation on N defines an open book decomposition of N , i.e., the singular set (cid:8) p ∈ N (cid:12)(cid:12) T p N ⊂ ξ p (cid:9) is the binding of an open book on N , and each regular leaf of the foliationcorresponds to a page of the open book.The following notion is extensively studied in [MNW] as a filling obstruction, here we will onlyuse it as a sufficient condition for the existence of a closed contractible Reeb orbit. Definition.
Let N be a compact submanifold of ( M, ξ ) that is maximally foliated by ξ , and hasnon-empty boundary ∂N that can be written as a product manifold ∂N ∼ = S × L . We say that N carries a Legendrian open book with boundary , if the following conditions are satisfiedby the foliation:(i) The singular set is the union of the boundary ∂N and a closed (not necessarily connected)codimension 2 submanifold B ⊂ N \ ∂N with trivial normal bundle.(ii) There exists a submersion ϑ : N \ B → S that restricts on ∂N ∼ = S × L to the projection onto the first factor.(iii) The regular leaves of the Legendrian foliation ξ ∩ T N are the fibers of ϑ intersected withthe interior of N .(iv) The neighborhood of B has a trivialization B × C for which the angular coordinate e iϕ on C agrees with the map ϑ . Remark . There are two common definitions of the overtwisted disk; according to one versionthe boundary is a regular compact leaf of the foliation, but there is a second version where thefoliation is singular along the boundary of the disk. This second definition is an example of aLegendrian open book with boundary. By a small perturbation it is always possible to move fromone version to the other one, so that both definitions are equivalent. Similarly, it is possible todeform a plastikstufe to obtain a Legendrian open book with boundary, so that the definitionabove includes
P S –overtwisted manifolds.
Theorem 1.
Let ( M, ξ ) be a closed contact manifold, and let N be a compact submanifold. (i) If ξ induces a Legendrian open book on N (without boundary), and if ξ admits a contactform α without closed contractible Reeb orbits, then it follows that N represents the trivialhomology class in H n +1 ( M, Z ) . (ii) If ξ induces a Legendrian open book with boundary on N , then every contact form α on ( M, ξ ) has a closed contractible Reeb orbit. EINSTEIN CONJECTURE IN HIGHER DIMENSIONS 3
Remark . In the situation of Theorem 1.(ii), it also follows that (
M, ξ ) does not admit a (semi-positive) strong symplectic filling in general, and under some cohomological condition it evenexcludes the existence of a weak filling. The proof of this fact is given in [MNW]. We will callsuch a contact structure
P S –overtwisted . Remark . It should be possible to strengthen the conclusions of the theorem. For example, if both N and the moduli space used in the proof of (i) are orientable, the coefficients for the homologygroup can be taken in Z . Proof of Theorem 1.
Following Hofer’s idea for 3–manifolds, we will study a moduli space of holo-morphic disks in the symplectization of (
M, α ). To prove (i), we will then show that the union ofthese holomorphic disks represents a chain in M whose boundary is homologous to the subman-ifold N . The proof of (ii) is based on a contradiction to Gromov compactness as in [AH09], andwe will only discuss it briefly at the end. (i) First, we have to choose a suitable almost complex structure J on the symplectization (cid:0) R × M, d ( e t α ) (cid:1) . We embed (
M, α ) as the 0–level set { } × M , and define J first in a neighborhood of the binding B in { } × N , before extending it over all of R × M . It was shown in [Nie06, Section 3] that the germof the contact form in a neighborhood of B is completely determined by the foliation on N , orsaid otherwise, there is a neighborhood U around the binding B that is strictly contactomorphicto a neighborhood (cid:101) U of the 0–section in (cid:16) R × T ∗ B, dz + 12 ( x dy − y dx ) + λ can (cid:17) , where ( x, y, z ) are the standard coordinates on R , and λ can is the canonical 1–form on T ∗ B . Theset U ∩ N corresponds in this model to the intersection of (cid:101) U with the submanifold { ( x, y, } × B .We will now study the following model for the symplectization of U : Let W = C be theStein manifold with standard complex, and symplectic structures, and with the plurisubharmonicfunction h ( z , z ) = | z | + | z | . To find a Weinstein structure on T ∗ B choose a Riemannianmetric g on the binding B , then the cotangent bundle W = T ∗ B carries an induced Riemannianmetric (cid:101) g , and an exact symplectic structure dλ can given by the differential of the canonical 1–form λ can := − p d q . There is a unique almost complex structure J g on W that is compatible with dλ can and with the metric (cid:101) g . The function h ( q , p ) = (cid:107) p (cid:107) / J g –plurisubharmonic and satisfies dh ◦ J g = − λ can (see also [Nie06, Appendix B]).The product manifold W = W × W = C × T ∗ B is a Weinstein manifold with almost complexstructure J (cid:48) = i ⊕ J g , and plurisubharmonic function h = h + h . Its contact type boundary M (cid:48) := h − (1) contains the submanifold (cid:110)(cid:0)(cid:113) − | z | , z ; q , (cid:1) (cid:12)(cid:12)(cid:12) | z | < ε (cid:111) ∼ = D × B .
The natural contact structure ker( dh ◦ J (cid:48) ) on M (cid:48) induces a singular foliation on this submanifoldthat is diffeomorphic to the neighborhood of the binding of an open book, so that in fact theneighborhood of this submanifold in W is symplectomorphic to a neighborhood of { } × B in thesymplectization, and the plurisubharmonic function h coincides with e t on R × M .The pull-back of J (cid:48) = i ⊕ J g to the symplectization defines thus an almost complex structurein a neighborhood of the binding { } × B in R × M , which we can easily extend to an almostcomplex structure on ( − ε, ε ) × M that is compatible with the symplectic form d ( e t α ), and forwhich dt ◦ J = α . Unfortunately this almost complex structure is not t –invariant, but we canextend J to an almost complex structure that is tamed by d ( e t α ) everywhere, restricts to ξ , andis t –invariant below a certain level set {− C } × M in the symplectization.With the chosen almost complex structure J , it is easy to explicitly write down a Bishop familyof holomorphic disks in a neighborhood of { } × B , and to use an intersection argument to excludethe existence of other holomorphic disks in this neighborhood. Namely, the Bishop family will begiven in the model C × T ∗ B by the intersection of the 2–planes E t , q := (cid:8) ( t , z ; q , ) (cid:12)(cid:12) q ∈ B, t < , z ∈ C (cid:9) K. NIEDERKR¨UGER AND A. RECHTMAN with h − (cid:0) (1 − ε, (cid:1) . The result gives for every point q of the binding B a 1–dimensional familyof round disks attached with their boundary to the foliated submanifold. The radius of the diskdecreases as t →
1, and in the limit the disks collapse to the point q ∈ B . All of the disksare pairwise disjoint, and if we look at the space of parameterized disks, we obtain thus a smooth( n + 3)–dimensional manifold.To exclude the existence of other disks close to the binding, we use an intersection argumentwith the local foliation given by ( i ⊕ J g )–holomorphic codimension 2 submanifolds S z := (cid:8) ( z , z ) (cid:12)(cid:12) z ∈ C (cid:9) × T ∗ B with Re z <
1. For more details see [Nie06, Section 3].We will now look at the moduli space of holomorphic disks given as follows: Denote N \ B by ◦ N , and let (cid:102) M be the space of all J –holomorphic maps u : (cid:0) D , ∂ D (cid:1) → (cid:16) ( −∞ , × M, { }× ◦ N (cid:17) , whose boundary u (cid:0) ∂ D (cid:1) intersects every page of the open book on N exactly once. For simplicitywe will restrict to the component of (cid:102) M that contains the Bishop family (for every component ofthe binding there is an independent Bishop family, but one result of our assumptions will be thatall these families lie in the same component of the moduli space).Before producing a moduli space by taking a quotient of (cid:102) M , we will briefly discuss Gromovcompactness. We claim that there is a uniform energy bound for all curves u ∈ (cid:102) M . The energyof a holomorphic curve u in a symplectization is defined as E α ( u ) := sup ϕ ∈F (cid:90) u d (cid:0) ϕα (cid:1) , where F is the set of smooth functions ϕ : R → [0 ,
1] with ϕ (cid:48) ≥
0. Here we identify R with the R –factor of the symplectization.Using Stokes’ Theorem, we easily obtain for any holomorphic disk u ∈ (cid:102) M that E α ( u ) = (cid:90) ∂u α . There is a continuous function f : N → [0 , ∞ ) such that α | T N = f dϑ , where ϑ : N \ B → S is thefibration of the open book, and because the boundary of the curves u (cid:0) ∂ D (cid:1) crosses every page ofthe open book on N exactly once, we obtain the energy bound E α ( u ) ≤ π max x ∈ N f ( x ) , proving the claim.Let ( u k ) k ⊂ (cid:102) M be a sequence of holomorphic maps. The only disks that may intersect a smallneighborhood of the binding { } × B are the ones that lie in the Bishop family, and hence we willassume that all maps u k stay at finite distance from the binding { } × B , because otherwise itfollows that the u k collapse to a point in B . Proposition 1.
Let ( u k n ) n be a sequence of holomorphic maps whose image is bounded away from { } × B . There there is a subsequence ( u k n ) n and a family of biholomorphisms ϕ n ∈ Aut( D ) ,such that the reparameterized maps ( u k n ◦ ϕ n ) n converge uniformly in C ∞ to a map u ∞ ∈ (cid:102) M .Proof. Assume the conclusion is false, then the gradient of the reparameterized sequence is blowingup, and this would either lead to the existence of a holomorphic sphere, a finite energy plane, or adisk bubbling off. Symplectizations never contain holomorphic spheres, and since by our assump-tion (
M, α ) does not have closed contractible Reeb orbits, we also have excluded the existence offinite energy planes. Finally, bubbling of disks is not allowed because the maps in (cid:102) M cross everypage of the open book on N exactly once, and this implies that the boundary of the disks areundecomposable. (cid:3) EINSTEIN CONJECTURE IN HIGHER DIMENSIONS 5
With the limit behavior of the maps in (cid:102) M understood, we will now study the moduli space M := (cid:102) M × D / ∼ , where we identify pairs ( u, z ) , ( u (cid:48) , z (cid:48) ) ∈ (cid:102) M × D , if and only if there is a M¨obius transformation ϕ ∈ Aut( D ) such that ( u, z ) = (cid:0) u (cid:48) ◦ ϕ − , ϕ ( z (cid:48) ) (cid:1) . Note that the action of Aut( D ) on (cid:102) M is properand free because every map u ∈ (cid:102) M is injective along its boundary, and the identity is the onlybiholomorphism of D that keeps the boundary of the disk pointwise fixed. It follows that M isa non-compact smooth ( n + 2)–dimensional manifold with boundary. The boundary correspondsto equivalence classes [ u, z ] ∈ M with z ∈ ∂ D . Proposition 1 above allows to understand thatthe compactification of M is in fact a smooth compact manifold with boundary: If (cid:0) [ u k , z k ] (cid:1) k is a sequence of elements in M , and if the image of the maps u k stays at a finite distance fromthe binding { } × B , then we know that there is a subsequence (cid:0) [ u k n , z k n ] (cid:1) n and a family ofreparameterizations ϕ n ∈ Aut( D ) such that u k n ◦ ϕ − n converges locally uniformly to a map u ∞ ∈ (cid:102) M . The subsequence (cid:0) [ u k n ◦ ϕ − n , ϕ n ( z k n )] (cid:1) n contains a further subsequence that convergesto a proper element [ u ∞ , z ∞ ] of the moduli space M .If the image of a map u k intersects a small neighborhood U of the binding in the symplectization,then it is up to reparameterization an element of the Bishop family. Thus, when the image of themaps u k gets close to the binding { }× B , we can find a subsequence (cid:0) [ u k n , z k n ] (cid:1) n such that all the u k n lie in the Bishop family. Here, we can describe M and its closure explicitly. The E t , q –planesare all pairwise disjoint, hence we have that there is exactly one disk [ u, z ] ∈ M with u ( z ) = p forevery p in the symplectization lying in the image of the Bishop family (cid:8) ( t, z ; q , ) (cid:12)(cid:12) q ∈ B, t < , z ∈ C , | z | ≤ − t (cid:9) . Then the compactification of the Bishop family is naturally diffeomorphicto the smooth manifold with boundary (cid:8) ( t, z ; q , ) (cid:12)(cid:12) q ∈ B, t ≤ , z ∈ C , | z | ≤ − t (cid:9) . There is a well-defined smooth evaluation mapev :
M → R × M, [ u, z ] (cid:55)→ u ( z )from the compactification of the moduli space into the symplectization. Definition.
The degree deg f ∈ Z of a continuous map f : X → Y between two closed n –manifolds X and Y is defined as the element A ∈ Z such that f [ X ] = A [ Y ] ∈ H n ( Y, Z ).For smooth maps it is easy to compute deg f , because it suffices to take a regular value y ∈ Y of f , and count [Eps66] deg f = f − ( y ) mod 2 . Hence, it follows immediately that the restriction of the evaluation map to the boundary ∂ M ofthe moduli space is a smooth map ev | ∂ M : ∂ M → { } × N of degree 1 (as can be easily seen by using that close to the binding { } × B there is for every p ∈ { } × N a unique disk [ u, z ] ∈ ∂ M with u ( z ) = p ). In particular by combining the trivialidentity ev ◦ ι ∂ M = ι N ◦ ev | ∂ M for the standard inclusions ι ∂ M : ∂ M (cid:44) → M and ι N : N (cid:44) → R × M , with the fact that ∂ M is null-homologous in H n +1 ( M , Z ), and using that ev ◦ ι ∂ M induces the trivial map on H n +1 ( ∂ M , Z ),we obtain that (cid:0) ι N (cid:1) : H n +1 ( N, Z ) → H n +1 ( M, Z )vanishes, because (cid:0) ev | ∂ M (cid:1) is an isomorphism. It follows that N represents a trivial ( n + 1)–classin H n +1 ( M, Z ) as we wanted to show. (ii) If N carries a Legendrian open book with boundary, we will proceed as follows: Chooseclose to the binding { } × B on the symplectization the almost complex structure described abovethat allows us to find the Bishop family of holomorphic disks. K. NIEDERKR¨UGER AND A. RECHTMAN
In [NP10, Section 5.3], it was shown that we can find a specific almost complex structure on aneighborhood of the boundary { } × ∂N ∼ = S × L that prevents any holomorphic disk to enterthis area. After choosing these two almost complex structures, close to the binding B and to theboundary ∂N , extend them to a global almost complex structure J on R × M that is compatiblewith the symplectic form d ( e t α ), and for which dt ◦ J = α . Additionally, we require J to be t –invariant below a certain level set {− C } × M in the symplectization.Denote now N \ ( B ∪ ∂N ) by ◦ N , and study the space (cid:102) M of J –holomorphic maps u : (cid:0) D , ∂ D (cid:1) → (cid:16) ( −∞ , × M, { }× ◦ N (cid:17) , whose boundaries u (cid:0) ∂ D (cid:1) transverse every page of the open book on N exactly once. If we assumethat ( M, α ) does not have any contractible periodic Reeb orbits, then the compactness argumentfor sequences in (cid:102) M works as above, because there is an area around ∂N where no holomorphiccurves are allowed to enter.The moduli space, we will study now is given by M := (cid:102) M × S / ∼ , where we identify pairs ( u, z ) , ( u (cid:48) , z (cid:48) ) ∈ (cid:102) M × S , if and only if there is a M¨obius transformation ϕ ∈ Aut( D ) such that ( u, z ) = (cid:0) u (cid:48) ◦ ϕ − , ϕ ( z (cid:48) ) (cid:1) . By the arguments above, M is a smooth( n + 1)–dimensional manifold with a smooth evaluation mapev : M → { } × N, [ u, z ] (cid:55)→ u ( z ) . If we choose a generic (differentiable) path γ : [0 , → N that connects a binding component of B with a component of the boundary ∂N , and is such that γ (cid:0) ]0 , (cid:1) ⊂ ◦ N , then the evaluation mapis transverse to γ . The pre-image ev − ( γ ) is a non-empty 1–dimensional smooth submanifold of M . We only consider the component M of ev − ( γ ) that contains elements of the Bishop family.The closure of the submanifold M has one end that corresponds to the disks that collapse toa point on the binding, and so M cannot be a circle, but must be instead an interval. The otherend of the interval exists by Gromov compactness, but by our assumptions this limit curve will bea regular element of M , so that in fact it is not the end of the interval leading to a contradiction,which implies the existence of a closed contractible Reeb orbit. (cid:3) We can generalize Theorem 1 by changing open books to covered open books, let us start withthe definition.
Definition.
Let N be a closed manifold with universal cover π : (cid:101) N → N . A pair ( ϑ, B ) consistingof a closed codimension 2 submanifold B of N , and a proper fibration ϑ : ( N \ B ) → S , is calleda k –fold covered open book decomposition of N if it induces an open book decompositionon the universal cover (cid:101) N . More precisely, we require that there is an open book decomposition( (cid:101) ϑ, (cid:101) B ) on (cid:101) N , where the binding (cid:101) B is π − ( B ), and where the fibration (cid:101) ϑ : (cid:101) N \ (cid:101) B → S commuteswith π and z (cid:55)→ z k according to the following diagram: (cid:101) N \ (cid:101) B S N \ B S (cid:45) (cid:101) ϑ (cid:63) π (cid:63) z (cid:55)→ z k (cid:45) ϑ Definition.
Accordingly we say that a maximally foliated submanifold N of a contact manifoldcarries a Legendrian covered open book , if the maximal foliation on N defines a covered openbook decomposition of N .If N is a maximally foliated compact submanifold with boundary in a contact manifold ( M, ξ ),then we say that ξ induces a Legendrian covered open book with boundary if the foliationon the interior of N defines a covered open book, and if it satisfies close to the boundary the sameconditions as a proper Legendrian open book with boundary. EINSTEIN CONJECTURE IN HIGHER DIMENSIONS 7
Example . Note that a proper open book decomposition ( ϑ, B ) of a manifold N is a 1–fold coveredopen book decomposition, as the open book on (cid:101) N will be given by (cid:101) ϑ = ϑ ◦ π , and (cid:101) B = π − ( B ).But this of course does not imply that (cid:101) N is a 1–fold cover of N , as the following example shows:The standard open book decomposition on S (see Fig. 1) induces in an obvious way an open bookdecomposition on the manifold S × S , and its universal cover R × S . Example . The unit sphere S n − = (cid:8) ( x , . . . , x n ) ∈ R n (cid:12)(cid:12) x + · · · + x n = 1 (cid:9) admits an open bookwith binding B = (cid:8) ( x , . . . , x n ) ∈ S n − (cid:12)(cid:12) x = x = 0 (cid:9) , and fibration map ϑ : S n − \ B → S , ( x , . . . , x n ) (cid:55)→ ( x , x ) (cid:112) x + x . The binding is an ( n − n − R P n − can be obtained as the quotient of the unit sphere S n − bythe antipodal map A : S n − → S n − , ( x , . . . , x n ) (cid:55)→ ( − x , . . . , − x n ) . The open book on S n − described above projects onto a covered open book of R P n − with binding B (cid:48) = (cid:8) [0 : 0 : x : · · · : x n ] ∈ R P n − (cid:9) ∼ = R P n − , and fibration map ϑ (cid:48) : R P n − \ B (cid:48) → S , [ x : · · · : x n ] (cid:55)→ ( x − x , x x ) x + x , which is induced by the square of ϑ . The pages of this open book are still ( n − R P n − that is not aproper open book decomposition. Figure 1.
The standard open book on the 2–sphere.
Figure 2.
The induced covered open book on R P . The boundary of the diskis identified under the antipodal map, so that for example the line drawn in redrepresents a single page that touches the binding at both of its boundaries. Definition.
Let N be a closed submanifold of a contact manifold ( M, ξ ), and assume that ξ induces a Legendrian k –fold covered open book on N . We say that N is nucleation free , if everyloop γ : S → N \ B that projects via ϑ : N \ B → S to a generator of π ( S ), represents in π ( M )an element that is at least of order k . K. NIEDERKR¨UGER AND A. RECHTMAN
Remark . The reason for our definition of nucleation free is that it excludes bubbling of certainholomorphic disks in the symplectization R × M . The boundary of every non-constant holomorphicdisk u that is attached with ∂u to the submanifold { } × N will always have positive transverseintersections with the pages of the covered open book. Since ∂u is also clearly contractible in M ,it follows that ϑ ( ∂u ) ⊂ S will make a (positive) multiple of k turns in the covered open book.If we then choose a sequence of holomorphic disks ( u n ) n such that each one intersects everypage of the open book exactly k times, the limit curve of ( u n ) n cannot decompose into severalnon-constant holomorphic disks v , · · · , v N , because the boundary of these curves would describeloops in N \ B that are contractible in M , but that make strictly less than k turns in the coveredopen book. Theorem 2.
Let ( M, ξ ) be a closed contact manifold, and let N be a compact submanifold. (i) If ξ admits a contact form α without closed contractible Reeb orbits, and if it induces aLegendrian covered open book on N that is nucleation free, then N represents the trivialhomology class in H n +1 ( M, Z ) . (ii) If ξ induces on N a Legendrian covered open book with boundary that is nucleation free,then every contact form α of ( M, ξ ) has a closed contractible Reeb orbit.Remark . Note that the conditions in Theorem 2.(ii) do not imply the non-fillability of M .Nonetheless, it implies that there is a loop in N \ B that projects via ϑ onto a positive generatorof S , and represents in the filling W of M an element of π ( W ) of order strictly less than k . Proof.
We follow the lines of the proof of Theorem 1, but several details have to be adjusted to thenew situation. To find a Bishop family around the binding B , we will construct a model aroundthe binding (cid:101) B in the cover, perform all steps as in Theorem 1, and finally show that π : (cid:101) N → N induces similar results in the base.Note that the fundamental group G := π ( N ) acts by deck transformations on (cid:101) N , and that (cid:101) N /G ∼ = N . We will identify a tubular neighborhood of N in M with a neighborhood U of the0–section in the normal bundle νN . The universal cover of νN is just given by the pull-backbundle π − ( νN ) over (cid:101) N , and so we find a neighborhood (cid:101) U of the 0–section of π − ( νN ) such that (cid:101) U /G = U . We can also pull-back the contact form α | U to a G –invariant contact form (cid:101) α on (cid:101) U .The contact form (cid:101) α induces on (cid:101) N an open book decomposition, and in principle we can use[Nie06, Section 3] to obtain a neighborhood of the binding (cid:101) B strictly contactomorphic to a neigh-borhood of the 0–section in R × T ∗ (cid:101) B with the contact form dz + ( x dy − y dx ) + λ can . We needto be a bit more careful though, because (cid:101) B does not need to be compact. But the constructionof this contactomorphism is based on the Moser trick, and a closer inspection of the proof showsthat not only does this contactomorphism exist, but that it is even G –equivariant: where G actson the T ∗ (cid:101) B –factor by the linearization of the G –action on (cid:101) B , and on the R –factor by lineartransformations leaving the z –direction invariant.The symplectization R × (cid:101) U is the universal cover of the symplectization R × U . The fundamentalgroup G = π ( N ) acts trivially on the R –factor, and thus respects the symplectic form d ( e t (cid:101) α ). Itis also not difficult (though tiresome) to check that the almost complex structure J constructedin [Nie06, Section 3] is also G –invariant.As in the Proof of Theorem 1, we find in R × (cid:101) U a Bishop family of J –holomorphic disks,and also the corresponding family of codimension 2 almost complex submanifolds S z used for theintersection argument. Furthermore since J is G –invariant, it follows that G maps each of thesefamilies into itself, and so we may project the almost complex structure J , and these families intothe symplectization R × M .The boundary of the Bishop disks in R × M intersect each page of the covered open bookexactly k –times, and we supposed that N is nucleation free so that bubbling is not possible. Thisway the rest of the proof is now exactly as the one of Theorem 1. (cid:3) EINSTEIN CONJECTURE IN HIGHER DIMENSIONS 9 Examples and applications
The main difficulty consists in finding a situation where we can apply Theorems 1 and 2 toprove the Weinstein conjecture.Note that it is easy to find examples of submanifolds with an induced Legendrian open book inany Darboux chart. For example, it is easy to see that S n +1 can be embedded into (cid:0) S n +1 , ξ (cid:1) via( x , . . . , x n +1 ) ∈ S n +1 (cid:44) → (cid:0) x + ix , x , . . . , x n +1 (cid:1) ∈ C n +1 such that the standard contact form restricts to x dx − x dx which clearly defines the canonicalopen book on S n +1 with an n –ball as a page, and with trivial monodromy. Another example wasgiven in [Nie06, Section 5.2], where it was shown that we can embed S × S n − in the desired wayinto (cid:0) R n +1 , ξ (cid:1) .On the other hand there are often evident obstructions to the realization of a homology class bya maximally foliated submanifold as an open book. For example, the only closed 2–dimensionalmanifolds that admit a proper or a covered open book decomposition are S and R P . The reasonfor this is that if Σ is a closed surface that admits a (covered) open book, then we can lift therotational vector field ∂ ϕ from S to Σ, and obtain a vector field whose index is positive at eachof its singularities. By the Poincar´e-Hopf theorem it follows that the Euler characteristic of Σ hasto be positive, but the only compact surfaces that have positive Euler characteristic are S and R P . Hence for purely topological obstructions, we obtain that T (or for example a hyperbolic 3–manifold) does not contain any embedded non-nullhomologous 2–sphere or real projective 2–space,because both would have to lift to a non-nullhomologous S in R .But it is also easy to give contact topological obstructions, because there are many contactmanifolds that do not have contractible Reeb orbits as the following examples will show. Example . Let (
M, ξ ) be the unit cotangent bundle S ( T ∗ T n ) of the torus with its canonicalcontact structure. We can identify M with T n × S n − with coordinates ( x , . . . , x n ) ∈ T n and( y , . . . , y n ) ∈ S n − and write the canonical 1–form as λ can = n (cid:88) j =1 y j dx j . The Reeb field for this form is R = (cid:80) j y j ∂ x j , and so it follows that the orbits move in constantdirection along the torus, and hence there will not be any closed contractible Reeb orbits. Inparticular it follows that it is not possible to embed any manifold with a Legendrian open bookinto (cid:0) S ( T ∗ T n ) , λ can (cid:1) that represents a non-trivial class in H n +1 (cid:0) S ( T ∗ T n ) , Z (cid:1) .After having described some of the problems of our method, we will give some positive examples. Example . Let (
M, ξ ) be a contact manifold that is subcritically Stein fillable, that means itcan be filled by a Stein manifold of the form ( C × W, dx ∧ dy + dλ ), where ( W, dλ ) is a 2 n –dimensional Stein manifold. Then it follows that ( M, ξ ) admits an open book with page W andtrivial monodromy consisting of taking the angular coordinate on the C –factor of C × W as afibration over S .Any properly embedded Lagrangian submanifold L in W gives rise to an ( n +1)–submanifold N of M that is foliated as a Legendrian open book. In fact, N is obtained by taking the intersectionof C × L ⊂ C × W with the convex boundary M . Another way to describe the construction isby saying that we take the product of L with S , and then close this off by adding ∂L × D in aneighborhood of the binding of M .Unfortunately this manifold will often be homologically trivial. We can avoid this problem if W is a Stein manifold with plurisubharmonic Morse function h : W → [0 , ∞ ), and if the highestcritical point p is of index n , because then we can take for L the unstable manifold of p whichwill be a Lagrangian plane which intersects the skeleton of W only in p . This way, we obtain for N a sphere with the standard Legendrian open book decomposition, and the intersection between N and the skeleton of any page is 1, so that [ N ] may not be trivial in H n +1 ( M, Z ), and we canapply our theorem to find a contractible Reeb orbit. The easiest examples that fit into this situation are unit bundles of C ⊕ T ∗ S for any closedmanifold S . To be even more explicit, take the contact structure ξ on T n × S n +1 given by ξ = ker (cid:16) n (cid:88) j =1 y j dx j + 12 (cid:0) y n +1 dy n +2 − y n +2 dy n +1 (cid:1)(cid:17) with ( x , . . . , x n ) the coordinates on T n , and ( y , . . . , y n +2 ) the coordinates on S n +1 . Here, anysphere { x } × S n +1 is foliated by a Legendrian open book, and we obtain, in contrast to Example 3,that ( T n × S n +1 , ξ ) always has a closed contractible Reeb orbit.Similarly the contact structure on S n × S n +1 given by using the trivial open book with page T ∗ S n and trivial monodromy also always has a closed contractible Reeb orbits. Example . The most obvious example, where we find a submanifold with a Legendrian coveredopen book is the real projective space with the standard contact structure (cid:16) R P n − = (cid:8) [ x : · · · : x n : y : · · · : y n ] (cid:12)(cid:12) (cid:88) j ( x j + y j ) = 1 (cid:9) , ξ := ker n (cid:88) j =1 (cid:0) x j dy j − y j dx j (cid:1)(cid:17) given as the quotient of the standard contact sphere S n − by the antipodal map. The submanifold (cid:8) [ x : · · · : x n : x n +1 : 0 : · · · : 0] (cid:9) ∼ = R P n +1 represents the non-trivial class in H n ( R P n − , Z ),and carries the covered open book described in Example 2. It follows from Theorem 2 that anycontact form for ξ admits a contractible closed Reeb orbit.Note that the Weinstein conjecture for Example 5 is well known, because it has already beenproved a long time ago for the standard contact structure on the unit sphere [Rab78]. Similarlythe Weinstein conjecture for subcritically fillable manifolds can be proved in general (that meanswithout imposing the condition on the critical points) by using the technically much more difficultresults from SFT [Yau04].Still, we believe that our results have some value in depending only on local information, forexample it is easy to prove: Lemma 1.
Let ( M , ξ ) and ( M , ξ ) be two closed cooriented contact manifolds with dim M =dim M . If ( M , ξ ) satisfies the conditions of Theorem 1 or 2, then any contact form on theconnected sum (cid:0) M M , ξ ξ (cid:1) has a contractible Reeb orbit. Similar results can be obtained for other surgeries, but they require a more careful analysis ineach situation.
References [AH09] P. Albers and H. Hofer,
On the Weinstein conjecture in higher dimensions , Comment. Math. Helv. (2009), no. 2, 429–436.[Eps66] D. Epstein, The degree of a map , Proc. London Math. Soc. (3) (1966), 369–383.[Hof93] H. Hofer, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture indimension three , Invent. Math. (1993), no. 3, 515–563.[MNW] P. Massot, K. Niederkr¨uger, and C. Wendl,
Weak fillability of higher dimensional contact manifolds , inpreparation.[Nie06] K. Niederkr¨uger,
The plastikstufe - a generalization of the overtwisted disk to higher dimensions. , Algebr.Geom. Topol. (2006), 2473–2508.[NP10] K. Niederkr¨uger and F. Presas, Some remarks on the size of tubular neighborhoods in contact topology andfillability , Geom. Topol. (2010), no. 2, 719–754.[Rab78] P. Rabinowitz, Periodic solutions of Hamiltonian systems , Comm. Pure Appl. Math. (1978), no. 2,157–184.[Yau04] M.-L. Yau, Cylindrical contact homology of subcritical Stein-fillable contact manifolds. , Geom. Topol. (2004), 1243–1280. E-mail address , K. Niederkr¨uger: [email protected]
EINSTEIN CONJECTURE IN HIGHER DIMENSIONS 11 (K. Niederkr¨uger)
Institut de math´ematiques de Toulouse, Universit´e Paul Sabatier – Toulouse III,118 route de Narbonne, F-31062 Toulouse Cedex 9, France
E-mail address , A. Rechtman: [email protected] (A. Rechtman)(A. Rechtman)