CC -LIMITS OF LEGENDRIAN SUBMANIFOLDS LUKAS NAKAMURA
Abstract
Laudenbach and Sikorav proved that closed, half-dimensional non-Lagrangian submanifolds of sym-plectic manifolds are immediately displaceable as long as there is no topological obstruction. From thisthey deduced that under certain assumptions the C -limit of a sequence of Lagrangian submanifolds isagain Lagrangian, provided that the limit is smooth.In this note we extend Laudenbach and Sikorav’s ideas to contact manifolds. We prove correspondinglythat certain non-Legendrian submanifolds of contact manifolds can be displaced immediately withoutcreating short Reeb chords as long as there is no topological obstruction. From this it will follow thatunder certain assumptions the C -limit of a sequence of Legendrian submanifolds with uniformly boundedReeb chords is again Legendrian, provided that the limit is smooth. Introduction
The Lagrangian Arnold conjecture [Arn65] implies that a Lagrangian submanifold L of asymplectic manifold M always intersects its image under a Hamiltonian diffeomorphism. Fur-thermore, the number of intersection points should be bounded from below by the Betti numberof L if the intersection is transverse and by the cup-length of L in the general case. For C -smallHamiltonian diffeomorphisms in a cotangent bundle the Arnold conjecture follows easily fromMorse theory. Gromov proved in [Gro85] with the use of pseudo-holomorphic curves that it isimpossible to displace a weakly exact Lagrangian in a geometrically bounded symplectic man-ifold by a Hamiltonian diffeomorphism. Of course, this cannot hold for arbitrary Lagrangiansin arbitrary symplectic manifolds as the example of an embedded circle in R with its standardsymplectic structure shows. But Polterovich [Pol93] showed under the assumptions that L isrational and that M is geometrically bounded that L will always intersect its image under aHamiltonian diffeomorphism ψ as long as ψ is sufficiently small in the Hofer norm. Floer [Flo88]introduced a homology theory for Lagrangian intersections in order to prove the Arnold con-jecture in the case that M is compact and π ( M, L ) = 0. Chekanov [Che98] used Floer’s ideasto prove that the Arnold conjecture holds for all closed Lagrangians in geometrically boundedsymplectic manifolds as long as the Hamiltonian diffeomorphism is sufficiently small in the Hofernorm.Now, the question arises whether non-Lagrangian submanifolds can be rigid as well. To thisend, Laudenbach and Sikorav proved in [LS94] that half-dimensional closed non-Lagrangiansubmanifolds of symplectic manifolds are infinitesimally displaceable as long there is no topo-logical obstruction. Here, infinitesimally displaceable means that there is a Hamiltonian vectorfield nowhere tangent to that submanifold.Similarly to the symplectic case, there are also results about the rigidity of Legendrian sub-manifolds L in a contact manifold M . For example, Rizell and Sullivan ([RS16], [RS18]) provedthat if the contact Hamiltonian H generating a contactomorphism φ H is “sufficiently small”,then there are short (compared to H ) Reeb chords between L and φ ( L ).In this work, we extend Laudenbach and Sikorav’s ideas to contact manifolds. We prove thatunder certain assumptions for a given n -dimensional non-Legendrian submanifold L (wheredim( M ) = 2 n + 1) there exists a contact vector field that is nowhere contained in the sum ofthe tangent space of L and the span of the Reeb vector field along L . E-mail address : [email protected] . a r X i v : . [ m a t h . S G ] A ug C -LIMITS OF LEGENDRIAN SUBMANIFOLDS Laudenbach and Sikorav [LS94] noted that if a sequence { L n } n ∈ N of closed Lagrangian sub-manifolds of a geometrically bounded symplectic manifold C -converges to an embedded sub-manifold L , then the displacement energies of the L n have to be uniformly bounded away fromzero. But if L has vanishing displacement energy, then the sequence of the displacement energiesof the L i has to go to zero. From this they concluded that the limit has to be Lagrangian aswell.In a similar way, it will follow that the limit of a sequence of closed Legendrian submanifoldswith uniformly bounded Reeb chords is again Legendrian (Theorem 3.4). Acknowledgements : This work was carried out as part of the Master’s program “Theoreticaland Mathematical Physics” at the Ludwig-Maximilans-University Munich, and it summarizesthe results of my Master’s thesis. I would first like to thank Thomas Vogel for supervisingthis work and for his numerous helpful remarks about this note. He always found the time toanswer all of my questions. Furthermore, I am grateful to Yang Huang for many interesting andstimulating discussions. Also, I would like to thank Georgios Dimitroglou Rizell for explainingto me some of the results of his joint work with M. Sullivan.2.
Displacing non-Legendrian submanifolds
As mentioned in the introduction, closed Lagrangian submanifolds of many symplectic man-ifolds are rigid. Let us describe the following rather weak rigidity property. Let ( M n , ω ) bea symplectic manifold and L ⊆ M a closed Lagrangian submanifold. The restriction of anyfunction H : M → R to L has a critical point x ∈ L because L is closed, i.e. dH ( x ) | T x L = 0.For the Hamiltonian vector field X H associated to H , defined by i X H ω = − dH , this impliesthat X H ( x ) ∈ T x L ⊥ ω = T x L since L is Lagrangian. In other words, there exists no Hamiltonianvector field on M that is nowhere tangent to L .Now let L n be a closed non-Lagrangian submanifold of M and we ask whether there existsa Hamiltonian vector field nowhere tangent to L . Of course, there might not exist any vectorfield that is nowhere tangent to L as the self-intersection number of L might be non-zero. Butunder the additional assumption that there is no such topological obstruction, Laudenbach andSikorav proved the affirmative answer. Theorem 2.1. [LS94] Let ( M n , ω ) be a symplectic manifold and L a closed, connected sub-manifold of dimension n such that(i) L is non - Lagrangian , i.e. there exists a point x ∈ L such that T x L is not a Lagrangiansubspace of T x M ,(ii) the normal bundle ν of L ⊆ M has a nowhere vanishing section.Then there exists a Hamiltonian vector field on M that is nowhere tangent to L . Remark . Clearly, the generalization of Theorem 2.1 to non-coisotropic submanifolds fails ingeneral as such manifolds may contain closed Lagrangian submanifolds. However, G¨urel [G¨ur08]noted that Theorem 2.1 extends to nowhere coisotropic manifolds. Also, one can prove thateven the parametric and a relative version of the h-principle for Hamiltonian vector fields thatare nowhere tangent to L holds.Analogously to the Lagrangian case, Legendrians obey the following rigidity result. Let( M, ξ = ker α ) be a cooriented contact manifold and L ⊆ M a closed Legendrian submanifold.Let H : M → R be an arbitrary function. Then H | L has a critical point x ∈ L . From dH ( x ) | T x L = 0 it follows that X H ( x ) ∈ T x L ⊥ dα ⊕ (cid:104) R α ( x ) (cid:105) = T x L ⊕ (cid:104) R α ( x ) (cid:105) . Here, X H denotes They consider the cases M = R n and π ( M, L ) = 0 but their proof easily extends to general geometricallybounded symplectic manifolds, cf. Theorem 3.3 below. -LIMITS OF LEGENDRIAN SUBMANIFOLDS 3 the contact vector field associated to H that is defined by(1) i X H dα | ξ = − dH | ξ and α ( X H ) = H, ( · ) ⊥ dα denotes the dα | ξ complement in ξ , and R α denotes the Reeb vector field on M . Thismeans that for a closed Legendrian submanifold there exists no contact vector field that isnowhere contained in T L ⊕ R α .We now also consider non-Legendrian submanifolds. Below, we will apply the proof of The-orem 2.1 in [LS94] in order to show that, as in the symplectic case, there exist contact vectorfields nowhere tangent to T L ⊕ R α as long as there is no topological obstruction, at least for ageneric non-Legendrian submanifold. Note that the flow of such a contact vector field displacesthe non-Legendrian submanifold L in such a way that there are no short Reeb chords between L and its image under the flow. Theorem 2.3.
Let ( M n +1 , ξ = ker α ) be a cooriented contact manifold. Denote its Reeb vectorfield by R α . Let L ⊆ M be a closed, connected submanifold of dimension n such that(i) R α ( x ) / ∈ T x L for all x ∈ L ,(ii) L is non - Legendrian , i.e. there exists a point x ∈ L with T x L (cid:54)⊆ ξ x ,(iii) there exists a nowhere vanishing section of the normal bundle of the subvector bundle T L ⊕ (cid:104) R α | L (cid:105) ⊆ T M | L .Then there exists a contact vector field X such that X ( x ) / ∈ T x L ⊕ (cid:104) R α ( x ) (cid:105) for all x ∈ L .Remark . For a generic n -dimensional submanifold L ⊆ ( M, ker α ), R α will be nowheretangent to L . Hence, Theorem 2.3 describes the generic case. With basically the same proofone can show that a similar statement also holds if we require that the Reeb vector field iseverywhere tangent to L . Remark . Similarly to G¨urel’s result [G¨ur08] that was mentioned in Remark 2.2, Theorem2.3 also holds for submanifolds that have a dimension different from n if one requires that( πT x L ) ⊥ dα (cid:54)⊆ T x L holds for all x ∈ L . Here, π : T M = ξ ⊕ (cid:104) R α (cid:105) → ξ denotes the projectiononto the first factor. Also, the relative and a parametric h-principle hold in the setting of The-orem 2.3 and in this case.Laudenbach and Sikorav deduced Theorem 2.1 from the following more general statement. Theorem 2.6. [LS94] Let M be a manifold, L a closed connected submanifold, and E asubbundle of T M | L with rk( E ) = dim( L ) such that(i) E (cid:54) = T L , i.e. there exists a point x ∈ L with E x (cid:54) = T x L ,(ii) there exists a nowhere vanishing section of E .Then there exists a function H on M such that dH | E x is non-zero for all x ∈ L . Proof of Theorem 2.3.
We will show how Theorem 2.6 implies Theorem 2.3.The tangent bundle
T M of M splits as T M = ξ ⊕(cid:104) R α (cid:105) . As above, let π denote the projectiononto the first factor. In order to apply Theorem 2.6, we define the vector bundle(2) E := ( πT L ) ⊥ dα on L . Since R α is nowhere tangent to L , this indeed defines a vector bundle with rk( E ) =dim( L ). Because E ⊆ ξ , it follows that E = T L if and only if L is Legendrian. Thus, condition( ii ) in Theorem 2.3 is precisely condition ( i ) in Theorem 2.6.It is convenient to consider a complex structure J : ξ → ξ on the contact distribution suchthat(3) g J ( v, w ) := dα ( v, J w ) , v, w ∈ ξ x , x ∈ L, C -LIMITS OF LEGENDRIAN SUBMANIFOLDS defines a metric on ξ . Such complex structures exist because dα | ξ defines a symplectic structureon ξ (cf. [MS17], Proposition 2.6.4). Then we can extend g J to a metric on M in such a waythat the Reeb vector field R α is orthogonal to ξ .By assumption, there exists a vector field X that is orthogonal to T L ⊕ (cid:104) R α (cid:105) at every pointof L . Especially, X is tangent to ξ along L . Now it is easy to check that J X defines a nowherevanishing section of E .Therefore, Theorem 2.6 implies that there exists a function H : M → R such that dH | E x isnon-zero for all x ∈ L . For any x ∈ L , we have that(4) 0 = dH | E x = dH | ( πT x L ) ⊥ dα ⇔ X H ( x ) ∈ T x L ⊕ (cid:104) R α ( x ) (cid:105) . Hence, Theorem 2.3 follows. (cid:3) C -limits of Legendrian submanifolds Let (
M, ω ) be a symplectic manifold. Eliashberg [Eli87] proved that the group of symplecto-morphisms of M is C -closed as a subset of the group of diffeomorphisms of M . This theoremcan be stated equivalently in terms of graphs of diffeomorphisms of M . For this, recall that adiffeomorphism of M is a symplectomorphism if and only if its graph in ( M × M, pr ∗ ω − pr ∗ ω )is Lagrangian. Then Eliashberg’s result states that the C -limit of a sequence of smooth, La-grangian graphs in M × M is again Lagrangian, provided that it is a smooth graph.Now one can also consider the closure of the symplectomorphism group of M inside thegroup of homeomorphisms of M . A homeomorphism that is a C -limit of symplectomorphismsis called a C -symplectomorphism. Humili`ere, Leclercq and Seyfaddini [HLS15] generalizedElishberg’s Theorem: If a C -symplectomorphism maps a coisotropic submanifold to a smoothmanifold, then the image will be coisotropic as well.In these statements it is assumed that the C -limits of the Lagrangian (or coisotropic) sub-manifolds are induced by C -limits of symplectomorphisms. But Laudenbach and Sikoravshowed that this assumption is not necessary in general. Theorem 3.1. [LS94] Let ( M n , ω ) be a symplectic manifold and L n a closed manifold. Let f i : L → M be a sequence of Lagrangian embeddings of L into M that C -converges to anembedding f : L → M . If(i) ( M, ω ) is geometrically bounded and π ( M, f ( L )) = 0, or(ii) ( M, ω ) = ( R n , ω ),then f is a Lagrangian embedding.Recall the following definition. Definition 3.2. (cf. [Gro85], [AL94]) Let (
M, ω ) be a symplectic manifold. It is geometricallybounded if there exists an almost complex structure J such that g J ( · , · ) := ω ( · , J · ) defines acomplete Riemannian metric for which there exists an upper bound on the sectional curvatureand a positive lower bound on the injectivity radius of ( M, g J ).We will show that Theorem 3.1 even holds if we replace the conditions ( i ) and ( ii ) by themore general condition that ( M, ω ) is geometrically bounded.
Theorem 3.3.
Let ( M n , ω ) be a geometrically bounded symplectic manifold and L n a closedmanifold. Let f i : L → M be a sequence of Lagrangian embeddings of L into M that C -converges to an embedding f : L → M . Then f is a Lagrangian embedding. Now consider a cooriented contact manifold ( M, ker α ). Correspondingly to Eliashberg’sresult, M¨uller and Spaeth [MS14] showed that the group of contactomorphism of M is C -closedas a subset of the group of diffeomorphisms. Again, we also obtain a statement about the graphs -LIMITS OF LEGENDRIAN SUBMANIFOLDS 5 of contactomorphisms as follows. Consider the projections pr , pr : E := M × M × R → M ontothe first and second factor, respectively. A section of pr : ( E, e z pr ∗ α − pr ∗ α ) → M is Legendrianif and only if it is of the form x (cid:55)→ ( x, ψ ( x ) , g ( x )) for some contactomorphism ψ : M → M .Here, z denotes the coordinate on R and g is the conformal factor of ψ defined by ψ ∗ α = e g α .If we now apply M¨uller and Spaeth’s Theorem to a sequence of contactomorphisms for whichtheir respective conformal factors converge uniformly (cf. also [MS15]), then it follows that the C -limit of a sequence of smooth Legendrian sections of pr : E → M is again Legendrian aslong as it is a smooth section.Still under the assumption that the conformal factors converge uniformly, Rosen and Zhang[RZ18] proved a result analogous to the Humili`ere-Leclercq-Seyfaddini Theorem, namely, thatsmooth images of coisotropic submanifolds (i.e. ( T L ∩ ξ ) ⊥ dα ⊆ T L , cf. [Hua15]) under homeo-morphisms that are C -limits of contactomorphisms are again coisotropic. Usher [Ush20] showedthat the conclusion of this statement is still true if the conformal factors are only required tobe uniformly bounded from below.Now we want to examine the question under which conditions smooth C -limits of Legen-drian submanifolds are again Legendrian, even if the limit in not induced by a C -limit ofcontactomorphisms. It is well-known that any n-dimensional submanifold of a contact manifold( M n +1 , ξ ) can be C -approximated by Legendrian submanifolds as long as there is no topo-logical obstruction (see [EM02], 16.1.3), but we will show that under certain conditions suchapproximations must have short Reeb chords. Theorem 3.4.
Let ( M n +1 , ξ = ker α ) be a cooriented contact manifold and L n a closed man-ifold. Let f i : L → M be a sequence of Legendrian embeddings of L into M that C -converge toan embedding f = f ∞ : L → M . Assume that there exists ε > such that for all i ∈ N there areno Reeb chords of length less than ε going from f i ( L ) to itself. If one of the following conditionsis satisfied, then f is a Legendrian embedding.(a) The Reeb vector field is nowhere tangent to f ( L ) and there exist real numbers a, b ∈ R , a Φ t ( ∂P ). Remark . (1) Because the question whether the C -limit is Legendrian does not depend onthe contact form, the theorem should be read as, “If there exists a contact form such that thereis a positive uniform lower bound on the length of the Reeb chords of the f ( L i ), then the limitis Legendrian”.(2) It is known that a Liouville manifold is always geometrically bounded. Therefore, ( c )immediately implies ( b ) in the case that M is the contactization of a Liouville manifold. Weexplicitly stated that part of ( b ) nonetheless because the proofs of ( b ) and ( c ) rely on differentresults about Legendrian and non-Legendrian submanifolds.(3) An embedding i : M × [ a, b ] → N as in ( a ) exists if ( M, α ) is a boundary component of acompact symplectic manifold with boundary of contact type.(4) The fact that non-Legendrian submanifolds can be C -approximated by Legendrian sub-manifolds also shows that the closedness condition on L in Theorem 3.3 cannot be removed.Indeed, a C -converging sequence of Legendrian submanifolds lifts in the symplectization to In fact, we only have to require that the contact form on M = P × R is equal to the standard contact formon P × R outside of a compact set. C -LIMITS OF LEGENDRIAN SUBMANIFOLDS a C -converging (in the weak topology) sequence of cylindrical Lagrangian submanifolds andthe limit of the latter sequence is Lagrangian if and only if the limit of the former sequence isLegendrian. Remark . Now let us consider C -approximations of paths in R with its standard contactstructure ξ = ker( dz − ydx ).On the one hand, if the path is induced by the Reeb flow, then it is easy to see that it can be C -approximated by Legendrians that do not have any Reeb chords. For example, if L is theinterval(5) L := I = { (0 , , z ) ∈ R | z ∈ [0 , } , then L can be C -approximated by Legendrians, whose Lagrangian projection looks like a spiral(Figure 1). Figure 1.
Lagrangian projection of a Legendrian submanifold that is C -approximatingthe interval. On the other hand, if an embedded path γ : [0 , → R is not Legendrian and if the Reebvector field is nowhere collinear to its velocity vector, then it cannot be C -approximated bya Legendrian path without Reeb chords. In order to see this, let γ : [ − , → R be such anembedded non-Legendrian path and let η : [ − , → R be a Legendrian embedding withoutReeb chords that is ε -close to γ for some ε >
0. Let π : R → R denote the Lagrangianprojection. We write πγ for π ◦ γ and πη for π ◦ η . πγ : [ − , → R is an immersed path. Let (cid:101) γ : [ − , → R be the unique Legendrian lift of πγ to R such that (cid:101) γ (0) = γ (0). Since γ isnot Legendrian, we can assume that, after possibly restricting to a subinterval of [ − , πγ isan embedding and that (cid:101) γ (1) (cid:54) = γ (1). Let z , (cid:101) z and z η denote the z -coordinates of γ , (cid:101) γ and η ,respectively. Define C := | z (1) − (cid:101) z (1) | >
0. Note that (cid:101) z (and, in fact, the z -coordinate of anyLegendrian path) satisfies(6) (cid:101) z (1) − (cid:101) z (0) = (cid:90) (cid:101) γ | [0 , ydx. For any κ >
0, let U κ denote the closed κ -neighbourhood of πγ ([0 , ε , we can assume that there exists a closed ball-shaped neighbourhood V ε of πγ ([0 , U ε ⊆ V ε ⊆ U ε . We define t − := inf { t ∈ [ − , | πγ ( s ) ∈ V ε ∀ s ∈ [ t, } , (7) s − := inf { t ∈ [ − , | πη ( s ) ∈ V ε ∀ s ∈ [ t, } , (8)and similarly we define t + := sup { t ∈ [1 , | πγ ( s ) ∈ V ε ∀ s ∈ [1 , t ] } , (9) s + := sup { t ∈ [1 , | πη ( s ) ∈ V ε ∀ s ∈ [1 , t ] } . (10) We assume that γ is defined on the interval [ − ,
2] instead of [0 ,
1] in order to make it easier to write downthe argument below. -LIMITS OF LEGENDRIAN SUBMANIFOLDS 7 Since γ and η are embedded paths, it follows that t − , s − → t + , s + → ε →
0. Nowchoose ε so small and V ε in such a way that the following conditions are satisfied:(a) ε < C (b) t − , s − > − , t + , s + < (cid:107) γ ( s − ) − γ (0) (cid:107) < C and (cid:107) γ ( s + ) − γ (1) (cid:107) < C ,(d) (cid:107) (cid:101) γ ( t − ) − (cid:101) γ (0) (cid:107) < C and (cid:107) (cid:101) γ ( t + ) − (cid:101) γ (1) (cid:107) < C ,(e) For any four points x , x − , x , x − ∈ ∂V ε with (cid:107) x − − x − (cid:107) < ε and (cid:107) x − x (cid:107) < ε andfor any two embedded paths σ , σ : [0 , → V ε with σ i (0) = x i − and σ i (1) = x i + for i ∈ { , } ,we have that(11) (cid:12)(cid:12)(cid:12) (cid:90) σ ydx − (cid:90) σ ydx (cid:12)(cid:12)(cid:12) < C (f) πγ ( − ε (cid:107) ( πγ ) (cid:48) (0) (cid:107) ) (cid:54)∈ U ε and πγ (1 + ε (cid:107) ( πγ ) (cid:48) (0) (cid:107) ) (cid:54)∈ U ε .(g) For all t ∈ (cid:104) − ε (cid:107) ( πγ ) (cid:48) (0) (cid:107) , (cid:105) we have that (cid:107) πγ ( t ) − πγ (0) (cid:107) < ε , and for all t ∈ (cid:104) , ε (cid:107) ( πγ ) (cid:48) (0) (cid:107) (cid:105) we have that (cid:107) πγ ( t ) − πγ (1) (cid:107) < ε .It is clear that (a)-(d) will be satisfied if ε is sufficiently small.To see that (e) can be satisfied, choose ε so small and choose V ε in such a way that for anytwo points y , y ∈ ∂V ε with (cid:12)(cid:12) y − y (cid:12)(cid:12) < ε there exists an embedded path χ : [0 , → V ε with χ (0) = y and χ (1) = y such that (cid:12)(cid:12)(cid:12)(cid:82) χ ydx (cid:12)(cid:12)(cid:12) < C . Furthermore, we assume that area( V ε ) < C .Let x , x − , x , x − ∈ ∂V ε be four points and σ , σ : [0 , → V ε be two paths as in (e).By our assumptions, there exist two embedded paths χ − and χ + with χ − (0) = x − , χ − (1) = x − , χ + (0) = x and χ + (1) = x such that (cid:12)(cid:12)(cid:12)(cid:82) χ ± ydx (cid:12)(cid:12)(cid:12) < C . Now let λ − , λ + , λ , λ : [0 , → ∂V ε be four paths that are embeddings when restricted to (0 ,
1) such that λ − (0) = x − , λ − (1) = x − , λ + (0) = x , λ + (1) = x , λ (0) = x − , λ (1) = x , λ (0) = x − and λ (1) = x . As d ( ydx ) = − dx ∧ dy , it follows from Stokes’ Theorem that(12) (cid:12)(cid:12)(cid:12) (cid:90) χ − ydx − (cid:90) λ − ydx (cid:12)(cid:12)(cid:12) ≤ area( V ε ) < C . Similarly, it follows that(13) (cid:12)(cid:12)(cid:12) (cid:90) χ + ydx − (cid:90) λ + ydx (cid:12)(cid:12)(cid:12) < C , (cid:12)(cid:12)(cid:12) (cid:90) σ ydx − (cid:90) λ ydx (cid:12)(cid:12)(cid:12) < C , (cid:12)(cid:12)(cid:12) (cid:90) σ ydx − (cid:90) λ ydx (cid:12)(cid:12)(cid:12) < C . The absolute value of the winding number of the concatenation λ ∗ λ + ∗ λ ∗ λ − is at mostfour. Here, ( · ) denotes the inversion of paths. Therefore, it follows again from Stokes’ Theoremthat(14) (cid:12)(cid:12)(cid:12) (cid:90) λ ∗ λ + ∗ λ ∗ λ − ydx (cid:12)(cid:12)(cid:12) ≤ V ) < C. Combining the above inequalities one easily concludes that(15) (cid:12)(cid:12)(cid:12) (cid:90) σ ydx − (cid:90) σ ydx (cid:12)(cid:12)(cid:12) < C. This proves (e).By looking at the Taylor expansion of πγ around 0 and 1, it can also be seen that (f) and(g) are satisfied if ε is sufficiently small.From now on assume that ε and V ε are such that the conditions (a) - (g) are satisfied.As η is ε -close to γ , (f) implies that πη ( − ε (cid:107) ( πγ ) (cid:48) (0) (cid:107) ) (cid:54)∈ U ε and πη (1 + ε (cid:107) ( πγ ) (cid:48) (0) (cid:107) ) (cid:54)∈ U ε . Since V ε ⊆ U ε , we can conclude from this observation together with (f) that s − , t − > − ε (cid:107) ( πγ ) (cid:48) (0) (cid:107) and s + , t + < ε (cid:107) ( πγ ) (cid:48) (0) (cid:107) . Using (g) and the fact that η is ε -close to γ it follows that(16) (cid:107) πη ( s − ) − πγ ( t − ) (cid:107) = (cid:107) (cid:0) πη ( s − ) − πγ ( s − ) (cid:1) + (cid:0) πγ ( s − ) − πγ (0) (cid:1) + (cid:0) πγ (0) − πγ ( t − ) (cid:1) (cid:107) < ε C -LIMITS OF LEGENDRIAN SUBMANIFOLDS and similarly also(17) (cid:107) πη ( s + ) − πγ ( t + ) (cid:107) < ε. We can see that(18) | z η (0) − z η ( s − ) | = | ( z η (0) − z (0)) + ( z (0) − z ( s − )) + ( z ( s − ) − z η ( s − )) | ≤ C, where in the last inequality we used (a) and (c) together with the assumption that η is ε -closeto γ . In the same way we also obtain(19) | z η ( s + ) − z η (1) | ≤ C. We can conclude that(20) | ( (cid:101) z (1) − (cid:101) z (0)) − ( z η (1) − z η (0)) | (d) , (18) , (19) ≤ | ( (cid:101) z ( t + ) − (cid:101) z ( t − )) − ( z η ( s + ) − z η ( s − )) | + 8 C (6) = (cid:12)(cid:12)(cid:12) (cid:90) π (cid:101) γ | [ t − ,t +] ydx − (cid:90) πη | [ s − ,s +] ydx (cid:12)(cid:12)(cid:12) + 8 C (b) , (16) , (17) , (e) < C, where in the last step we used the assumptions that π (cid:101) γ = πγ and πη are embeddings in orderto apply (e) (recall that η does not have any Reeb chords).Now,(21) | z (1) − z (0) − ( z η (1) − z η (0)) | z (0)= (cid:101) z (0) ≥ | z (1) − (cid:101) z (1) | − | (cid:101) z (1) − (cid:101) z (0) − ( z η (1) − z η (0)) | def. C , (20) > C leads to a contradiction if ε is small enough because η is ε -close to γ .This shows that η must have Reeb chords if ε is sufficiently small. It is also clear that theseReeb chords need to be short because η is contained in the ε -neighbourhood of γ and the Reebvector field is nowhere tangent to γ .Using Darboux charts, it follows that this statement holds in any 3-dimensional contact man-ifold. To the author’s knowledge, it is an open question under which conditions it is possibleor impossible to C -approximate open submanifolds L n ⊆ ( M n +1 , ξ = ker α ) by Legendriansubmanifolds without short Reeb chords in the case n > Remark . If we lower the dimension of L and ask whether the C -limit of isotropic subman-ifolds are isotropic, then the answer is no since there is a C -dense h-principle for subcriticalisotropic embeddings into symplectic and contact manifolds ([EM02], Theorem 12.4.1).Another open question is whether Theorem 3.3 fails if we do not require M to be geomet-rically bounded, and, similarly, whether the assumptions in Theorem 3.4 on M are necessary.Also, one might expect these theorems to hold even for non-compact L if we require the em-beddings to be fixed outside some compact subset. Proof of Theorem 3.3.
As we can apply the theorem to every connected component of L , wecan assume that L is connected.Recall that for a compactly supported Hamiltonian symplectomorphism ψ the Hofer norm(cf. [Hof90]) is defined as(22) (cid:107) ψ (cid:107) := inf H (cid:107) H (cid:107) osc , where the infimum is taken over all time-dependent functions H t on M whose associated Hamil-tonian flow φ Ht satisfies φ H = ψ . Here, (cid:107) H (cid:107) osc denotes the oscillatory energy of H which isdefined as(23) (cid:107) H (cid:107) osc := (cid:90) (cid:18) max x ∈ M H ( x, s ) − min x ∈ M H ( x, s ) (cid:19) ds. -LIMITS OF LEGENDRIAN SUBMANIFOLDS 9 The Hofer norm is used to define the displacement energy e ( U ) of a subset U ⊆ M as(24) e ( U ) := inf {(cid:107) ψ (cid:107)| ψ ( U ) ∩ U = ∅} . Now assume that the conclusion of the theorem is false, i.e. there exists a sequence of La-grangian embeddings f i : L → M that C -converge to an embedding f : L → M , but f is notLagrangian. Let ι : S → T ∗ S denote the zero-section. After possibly replacing L , M , f i and f by L × S , M × T ∗ S , f i × ι and f × ι , respectively, we can assume that f ( L ) ⊆ M admitsa nowhere-vanishing section of its normal bundle. In order to simplify the notation, we willidentify f ( L ) with L and write L i := f i ( L ). By Theorem 2.1 there exists a Hamiltonian vectorfield nowhere tangent to L . Hence, for any ε > L ⊆ M that isdisplaced by this Hamiltonian isotopy from itself in a time less than ε by compactness of L .Since the f i converge uniformly towards f , we can find for any ε > N = N ( ε ) ∈ N such that L k is displaced form itself in a time less than ε for all k ≥ N . This implies that thedisplacement energy of the L i goes to zero as i increases.Chekanov proved in [Che98] that there is a lower bound on the displacement energy of a closedLagrangian submanifold in a geometrically bounded symplectic manifold in terms of the minimalarea of non-constant pseudoholomorphic spheres in M and non-constant pseudoholomorphicdiscs in M with boundary on L . Let N ⊆ M be a compact tubular neighbourhood of L . If L i is sufficiently C -close to L , then f i and f are homotopic as maps into N . For example,one can explicitly define such a homotopy by moving along the shortest geodesic connecting f ( x ) and f i ( x ) for all x ∈ L . Since f : L → N is a homotopy equivalence, this impliesthat f i : L → N is a homotopy equivalence as well if i is sufficiently large. Without loss ofgenerality we assume that this is the case for all f i . A non-constant pseudoholomorphic curvewith boundary on one of the L i has positive symplectic area. Hence, it defines a non-trivialclass is H ( M, L ; R ) ∼ = H ( M, N ; R ).According to Proposition 4 . . V ⊆ M of N such that every pseudoholomorphic curve whose image intersects N liescompletely in V . Let U ⊆ M be a compact submanifold (possibly with boundary) that contains V . Then Lemma 3.8 below shows that the areas of non-constant pseudoholomorphic discs withboundary on one of the L i are bounded away from zero. Together with Chekanov’s energycapacity inequality this implies that the displacement energies of the L i are uniformly boundedaway from zero. This gives the desired contradiction. (cid:3) Lemma 3.8.
Let N be a compact submanifold of a compact manifold U (possibly with boundary).Then there is a constant C > such that any disc representing a non-trivial class in H ( U, N ; R ) has area larger than C .Proof of Lemma. The following proof is an adaptation of the proof of the corresponding lemmain [LS94].By compactness, H ( U, N ; Z ) is finitely generated. Let { a i } i ∈ I be a basis of the free quotientof H ( U, N ; Z ), where I is a finite index set. Then the a i also form a real basis of H ( U, N ; R ).Denote by { α i } i ∈ I the basis of H dR ( U, N ; R ) dual to the a i . Then the homology class of anydisc D with boundary on N can be written in the form D = (cid:80) i ∈ I n i a i , where n i ∈ Z is theintegral of α i over D . Hence, we see that for all i ∈ I ,(25) | n i | = (cid:12)(cid:12)(cid:12) (cid:90) D α i (cid:12)(cid:12)(cid:12) ≤ area( D ) (cid:107) α i (cid:107) C , which implies that(26) area( D ) ≥ max i | n i |(cid:107) α i (cid:107) C ≥ min i (cid:107) α i (cid:107) C , if the homology class of D is non-zero (i.e. if not all of the n i vanish). (cid:3) Proof of Theorem 3.4.
As before we can assume that L is connected. C -LIMITS OF LEGENDRIAN SUBMANIFOLDS Part ( a ): We will reduce Theorem 3.4 ( a ) to Theorem 3.3 by using the following constructionfrom [Moh01].First note that since R α is nowhere tangent to f ∞ ( L ), we can assume that, after possiblydecreasing ε > ε > L × [0 , ε ] → M ( x, t ) (cid:55)→ ( φ αt ◦ f i ) ( x )(27)is an embedding for all i ∈ N ∪ {∞} , where φ αt denotes the Reeb flow in ( M, α ). Let(28) ( γ , γ ) : S → [0 , ε ] × [ a, b ]be an embedded loop. Consider the embeddings F i : L × S → ( M × [ a, b ] , d ( e s α ))( x, t ) (cid:55)→ (( φ αγ ( t ) ◦ f i )( x ) , γ ( t )) . (29)It is clear that the C -convergence of the f i implies C -convergence of the F i . Furthermore,a straightforward computation shows that F i is a Lagrangian embedding if and only if f i is aLegendrian embedding. Hence, we can apply Theorem 3.3 to conclude that f ∞ is a Legendrianembedding.The proofs of ( b ) and ( c ) are similar to the proof of Theorem 3.3. We will use known rigidityresults for Legendrian and non-rigidity results for non-Legendrian submanifolds to prove thestatement. Again, we identify L with f ( L ) and write L i := f i ( L ). Part ( b ): Assume that L is not Legendrian. After possibly replacing ( M, α ), L , f i and f by( M × T ∗ S , α − pdq ), L × S , f i × ι and f × ι , respectively, we can assume that there exists avector field that is nowhere (along L ) contained in T L ⊕ (cid:104) R α (cid:105) . Here, ι : S → T ∗ S denotes thezero section.Theorem 2.3 implies that there exists a contact vector field X that is nowhere contained in T L ⊕ (cid:104) R α (cid:105) . This implies that its flow φ t := φ Xt displaces L for sufficiently small times such thatthere are no short (compared to the length of the Reeb chords of L ) Reeb chords between L and φ t ( L ) for any t > σ denote the minimallength of Reeb chords of L . Then, for any λ > δ > σ − λ between L and φ t ( L ) for all 0 < t < δ . In this case, φ t also displaces a neighbourhood of L ⊆ M without short Reeb chords by compactness of L .Then for any sufficiently small t >
0, there exists an N ∈ N such that φ t also displaces L i without short Reeb chords for all i ≥ N . This shows that for any η > N ∈ N and a function H : M → R such that (cid:107) H (cid:107) C < η and the contactomorphism associated to H displaces L i without short Reeb chords for all i ≥ N .For any closed Legendrian submanifold N ⊆ M , let σ ( α, N ) denote the minimal length ofReeb chords γ of N and of closed Reeb orbits γ in M satisfying [ γ ] = 0 ∈ π ( M, N ). Rizelland Sullivan proved that if the C -norm of a generic function H on M is small compared to σ ( α, N ), there always exist short (compared to the C -norm of H ) Reeb chords between N and φ H ( N ) ([RS16], Theorem 1.3) if M satisfies the conditions in ( b ). This gives the desiredcontradiction because, after possibly approximating H , we can assume that it is generic. Part ( c ): For a compactly supported contactomorphism ψ on ( M, α ) that is isotopic to theidentity one can define(30) (cid:107) ψ (cid:107) α := inf H (cid:107) H (cid:107) , In fact, they only required that the oscillatory energy of H and the conformal factor of the contact flowassociated to H are sufficiently small. -LIMITS OF LEGENDRIAN SUBMANIFOLDS 11 where the infimum is taken over all time-dependent functions H t whose associated contactisotopy φ Ht satisfies φ H = ψ . Here, (cid:107) H (cid:107) is defined by(31) (cid:107) H (cid:107) := (cid:90) max x ∈ M H ( x, s ) ds. Shelukhin [She17] proved that this defines a (non-degenerate) norm on the group of compactlysupported contactomorphisms isotopic to the identity.Now assume that L is not Legendrian. Note that for a generic contactomorphism φ , φ ( L ) willnot intersect L since dim( L ) = n and dim( M ) = 2 n + 1. Let φ be such a contactomorphism.Then there exists a lower bound C > L and φ ( L ).Theorem 1.9 and Proposition 7.4 in [RZ18] together imply that there exist a sequence φ n ofcontactomorphisms isotopic to the identity such that lim n →∞ (cid:107) φ n (cid:107) α = 0 and φ n ( L ) = φ ( L ) forall n ∈ N . By compactness of L we can find for any n ∈ N and any η > U = U ( n, η ) of L ⊆ M such that there are no Reeb chords of length smaller than C − η between U and φ n ( U ). After possibly perturbing the φ n and choosing a slightly larger η , we can assumethat the φ n are generic and still have the above properties (except, of course, φ n ( L ) = φ ( L )).Since the L i C -converge to L , we can find for any two positive numbers δ, η > n, K ∈ N such that L i ⊆ U ( n, η ) for all i ≥ K and (cid:107) φ n (cid:107) α < δ . In particular, there areno Reeb chords of length smaller than C − η between L i and φ n ( L i ).This is a contradiction to a result of Rizell and Sullivan [RS18] that states that there haveto exist short Reeb chords between L i and φ n ( L i ) in the above setting if (cid:107) φ n (cid:107) α is sufficientlysmall. (cid:3) Remark . In the proof of ( b ) we only had to consider Reeb chords γ that satisfy [ γ ] = 0 ∈ π ( M, f i ( L )). One could seemingly strengthen the assumption in part ( b ) of Theorem 3.4 byonly requiring that there exists a uniform lower bound on the length of the Reeb chords that sat-isfy this condition. But it is easy to see that, in fact, compactness of L and the C -convergenceof the f i imply that there cannot be a sequence of Reeb chords that are non-zero in π ( M, f i ( L ))and whose length converges to zero. Indeed, for sufficiently large i , f i : L → N is a homotopyequivalence between L and a tubular neighbourhood N of f ( L ) and any sufficiently short Reebchord of f i ( L ) is contained in N . Hence, such a Reeb chord is trivial in π ( M, N ) ∼ = π ( M, f i ( L )). References [AL94] Mich`ele Audin and Jacques Lafontaine (eds.),
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