Tight neighborhoods of contact submanifolds

TIGHT NEIGHBORHOODS OF CONTACT SUBMANIFOLDS LUIS HERNÁNDEZ–CORBATO, LUCÍA MARTÍN–MERCHÁN, AND FRANCISCO PRESAS arXiv:1802.07006v1 [math.SG] 20 Feb 2018 Abstract. We prove that any small enough neighborhood of a closed contact submanifold is always tight under a mild assumption on its normal bundle. The non-existence of C 0 –small positive loops of contactomorphisms in general overtwisted manifolds is shown as a corollary. 1. Introduction A contact manifold (M, ξ) is an (2n + 1)–dimensional manifold equipped with a maximally non–integrable codimension 1 distribution ξ ⊂ T M . If we assume that ξ is coorientable, as will be the case in the article, the hyperplane distribution can be written as the kernel of a global 1–form α, ξ = ker(α), and the maximal non–integrable condition reads as α ∧ (dα)n 6= 0. These conditions imply that (ξ, dα) is a symplectic vector bundle over M . However, a contact structure on M cannot be directly recovered from a hyperplane distribution ξ and a symplectic structure ω on the fibers. The formal data (ξ, ω) is called formal contact structure. Let Cont(M ) and FCont(M ) denote the set of contact and formal contact structures, respectively. Gromov proved that if M is open the natural inclusion is a homotopy equivalence. The statement does not readily extend to closed manifolds. In dimension 3, Eliashberg introduced a subclass ContOT (M ) of Cont(M ), the so–called overtwisted contact structures, and proved that any formal contact homotopy class contains a unique, up to isotopy, overtwisted contact structure. Recently, this result has been extended to arbitrary dimension in [2] so the notion of overtwisted contact structure has been settled in general. Prior to [2], different proposals for the definition of the overtwisting phenomenum appeared in the literature. The plastikstufe, introduced in [12], resembled the overtwisted disk in the sense that it provides an obstruction to symplectic fillability. The presence of a plastikstufe has been shown to be equivalent to the contact structure being overtwisted (check [3, Theorem 1.1] and [11] for a list of disguises of an overtwisted structure). One of the corollaries obtained in [3] is a stability property for overtwisted structures: if (M, ker α) is overtwisted then (M × D2 (R), ker(α + r 2 dθ)) is also overtwisted provided R > 0 is large enough, where D2 (R) denotes the open 2–disk of radius R and r 2 dθ denotes the standard radial Liouville form in R2 . 1.1. Statements of the results. This paper explores the other end of the previous discussion, can small neighborhoods of contact submanifolds be overtwisted? We provide a negative answer to the question in several instances. The main result presented in the article is the following: Theorem 1. Let (M, ker α) be a contact manifold. Then there exists ε > 0 such that (M × D2 (ε), ker(α + r 2 dθ)) is tight. This theorem was previously obtained by Gironella [9, Corollary H] in the case of 3–manifolds with a completely different approach. An interesting consequence is stated in the next corollary: Corollary 2. Given any overtwisted contact manifold (M, α), there exists a radius R0 ∈ R+ \{0} such that (M × D2 (R), α + r 2 dθ) is tight if R ∈ (0, R0 ) and is overtwisted if R > R0 . Note that a similar statement was already proven in [13] but in the case of GPS–overtwisted. Theorem 1 can be extended to arbitrary neighborhoods of codimension 2 contact submanifolds M whose normal bundle has a nowhere vanishing section: 2010 Mathematics Subject Classification. Primary 37J10. Secondary: 37C40, 37J55. Key words and phrases. Contactomorphism, overtwisted, orderable, Hamiltonian. 1 Theorem 3. Suppose M is a contact submanifold of the contact manifold (N, ξ). Assume that the normal bundle of M has a nowhere vanishing section. Then, there is a neighborhood of M in N that is tight. is based on Theorem 10. That theorem states that for m The proof of Theorems 1 and 3P 2 large enough (M ×P 2m (ε), ker(α+ m i=1 ri dθi )) admits a contact embedding in a closed contact manifold of the same dimension that is Stein fillable, therefore M × P 2m (ε) is obviously tight. However, Theorems 1 and 3 do not prove such a strong result. Their proof uses [3, Theorem 1.1.(ii)] and some packing lemmas to obtain a contradiction by stabilizing and reducing to Theorem 10. 1.2. Applications. 1.2.1. Remarks about contact submanifolds. We are assuming a choice of contact forms whenever the measure of a radius of the tubular neighborhoods of a contact submanifold is required. 1. Assume that (M, ξ) contact embeds into an overtwisted contact manifold (N, ξOT ) as a codimension 2 submanifold with trivial normal bundle. By Theorem 1, it is clear that the overtwisted disk cannot be localized on arbitrary small neighborhoods of M , even assuming that M itself is overtwisted. This stands in sharp contrast with [11] and [3] in which it is shown that the overtwisted disk can be localized around a very special kind of codimension n submanifold: a plastikstufe [12]. 2. Assume now that (M, ξOT ) is overtwisted and contact embeds into a tight contact manifold (N, ξ) as codimension 2 submanifold with trivial normal bundle. Then we can perform a fibered connected sum of (N, ξ) with itself along (M, ξOT ). The gluing region is M × (−ε, ε) × S1 , for some ε > 0, and coordinates (p, t, θ) can be chosen such that the glued contact structure admits an associated contact form α = αOT + t dθ. It is clear that the contact connection associated to the contact fibration M × (−ε, ε) × S1 → (−ε, ε) × S1 [14] induces the identity when we lift by parallel transport the loop {0} × S1 . The parallel transport of an overtwisted disk of the fiber induces a plastikstufe, see [14] for more details. By [11], the manifold is overtwisted. Call RM > 0 the biggest radius for which M × D2 (RM ) contact √ embeds in N . The connected sum N #M N readily increases the biggest radius to be RN #M N ≥ 2RM : the √ annulus has twice the area of the original disk and therefore you can embed a disk of radius 2RM . However we get much more, since we actually obtain RN #M N = ∞. This is because we can always formally contact embed M ×R2 into N #M N . Moreover, we can assume that the embedding restricted to a very small neighborhood U of the fiber M × {0} provides a honest fibered contact embedding into M × (−ε, ε) × S1 . Indeed, applying [2, Corollary 1.4] relative to the domain U we obtain a contact embedding of M × R2 thanks to the fact that N #M N is overtwisted. This just means that the contact embedding of the tubular neighborhood can be really sophisticated and its explicit construction is far from obvious. 1.2.2. Small loops of contactomorphisms. Theorem 1 allows to extend the result of non–existence of small positive loops of contactomorphisms in overtwisted 3–manifolds contained in [4] to arbitrary dimension. A loop of contactomorphisms or, more generally, a contact isotopy is said to be positive if it moves every point in a direction positively transverse to the contact distribution. The notion of positivity induces for certain manifolds, called orderable, a partial order on the universal cover of the contactomorphism group and it is related with non–squeezing and rigidity in contact geometry, see [6, 8]. As explained in [6], orderability is equivalent to the non–existence of a positive contractible loop of contactomorphisms. Any contact isotopy is generated by a contact Hamiltonian Ht : M → R that takes only positive values in case the isotopy is positive. The main result of [4] states that if (M, ker α) is an overtwisted 3–manifold there exists a constant C(α) such that any positive loop of contactomorphisms generated by a Hamiltonian H : M × S 1 → R+ satisfies ||H||C 0 ≥ C(α). The result has been recently extended to arbitrary hypertight or Liouville (exact symplectically) fillable 2 contact manifolds in [1]. As a consequence of Theorem 1, we can eliminate the restriction on the dimension in the overtwisted case: Theorem 4. Let (M, ker α) be an overtwisted contact manifold. There exists a constant C(α) such that the norm of a Hamiltonian H : M × S 1 → R+ that generates a positive loop {φθ } of contactomorphisms on M satisfies ||H||C 0 ≥ C(α) The strategy of the proof copies that of [4]. The first step is to prove that M × D2 (ε) is tight, this is provided by Theorem 1. The second step shows that a small positive loop provides a way to lift a plastikstufe in M (whose existence is equivalent to overtwistedness as discussed above [11]) to a plastikstufe in M × D2 (ε). This is exactly Proposition 9 in [4]. This provides a contradiction that forbids the existence of the small positive loop. It is worth mentioning that the argument forbids the existence of (possibly non–contractible) small positive loops. This is in contrast with [1] and the work in progress by S. Sandon [15] in which they need to add the contractibility hypothesis in order to conclude. Remark 5. The hypothesis in Theorem 4 can be changed by the probably weaker notion of GP S-overtwisted, see [13]. Indeed, assume that the manifold (M, ξ) is GP S-overtwisted. This means that there is an immersed GP S in the manifold. The positive loop produce a GP S in M × D2 (ε) by parallel transport of the GP S around a closed loop in the base D 2 (ε). In this case, we need to iterate the process k times to produce a GP S in M × P 2k (ε). Now, Theorem 10 concludes that this manifold embeds into a Stein fillable one providing a contradiction with the main result in [13]. Acknowledgements The authors express their gratitude to Roger Casals for the useful conversations around this article. The authors have been supported by the Spanish Research Projects SEV-20150554, MTM2015-63612-P, MTM2015-72876-EXP and MTM2016-79400-P. The second author was supported during the development of the article by a Master grant from ICMAT through the Severo Ochoa program. 2. M × P 2m (ε) admits a Stein fillable smooth compactification 2.1. Construction of a formal contact embedding M → ∂W with trivial normal bundle. Recall that (ξ, dα) defines a symplectic vector bundle over M , thus it is equipped with a complex bundle structure unique up to homotopy. Denote ξ ∗ the dual complex vector bundle of ξ. A standard result on the theory of vector bundles guarantees the existence of a complex vector bundle τ → M such that ξ ∗ ⊕ τ → M is trivial, that is, there is an isomorphism of complex vector bundles over M between ξ ∗ ⊕ τ and M × Ck = Ck , where k is a positive integer large enough. Denote π : T ∗ M → M the cotangent bundle projection and denote pr : π ∗ τ → T ∗ M the c = π ∗ τ as a smooth almost complex bundle projection. Define π̃ = π ◦ pr. Let us understand W manifold. Choosing a ξ–compatible contact form α, i.e ξ = ker α, it is clear that c∼ TW = π̃ ∗ τ ⊕ pr∗ T (T ∗ M ) ∼ = π̃ ∗ τ ⊕ π̃ ∗ T ∗ M ⊕ π̃ ∗ T M ∼ = π̃ ∗ τ ⊕ π̃ ∗ (ξ ∗ ⊕ hαi) ⊕ π̃ ∗ T M ∼ = π̃ ∗ (τ ⊕ ξ ∗ ) ⊕ π̃ ∗ hαi ⊕ π̃ ∗ T M ∼ = π̃ ∗ Ck ⊕ π̃ ∗ hαi ⊕ π̃ ∗ T M π̃ In particular, the vector bundle π ∗ τ → M is isomorphic to Ck ⊕ hαi. Fix a direct sum bundle metric h in π ∗ τ such that h(α, α) = 1. Now define c : h(v, v) ≤ 1}. W = {(v, p) ∈ W Given a complex structure j in ξ compatible with dα, we can extend it to a complex structure on T ∗ M and by a direct sum with a complex structure in τ we obtain a complex structure J 3 c . Then, (W, J) is an almost complex manifold with boundary ∂W that has a natural in T W formal contact structure ξ0 = T ∂W ∩ J(T ∂W ). Consider the embedding e0 : M → ∂W = S(Ck ⊕ hαi) : p 7→ (0, 1). We claim that its normal bundle is trivial because it is equal to π̃ ∗ Ck . The reason is that the normal bundle to a section of a vector bundle is the restriction of the vertical bundle to the section. In our case the restriction of the vertical bundle T (S(Ck ) ⊕ hαi)) to the image of e0 is clearly (π̃ ∗ Ck )|im(e0 ) . 2.2. W is Stein fillable and ∂W is contact. The distribution T ∂W ∩ J(T ∂W ) is not necessarily a contact structure in ∂W . However, we will deform this distribution to a genuine contact structure using the following result. Theorem 6 (Eliashberg [7]). Let (V 2n , J) be an almost complex manifold with boundary of dimension 2n > 4 and suppose that f : V → [0, 1] is a Morse function constant on ∂V such that indp (f ) ≤ n for every p ∈ Crit(f ). Then, there exists a homotopy of almost complex structures {Jt }1t=0 such that J0 = J, J1 is integrable and f is J1 –convex. We are clearly in the hypothesis since our manifold W is almost complex, has dimension 2k + 1 + dim M > 4 (because 2k ≥ dim ξ = dim M − 1) and deformation retracts to M . From Theorem 6 we obtain a homotopy of almost complex structures {Jt } in W such that J0 = J and, J1 is integrable. Moreover (W, J1 ) is a Stein domain and ∂W inherits a contact structure given by ξ1 = J1 (T ∂W ) ∩ T ∂W . In fact, there is a homotopy of formal contact structures between ξ0 and ξ1 provided by ξt = Jt (T ∂W ) ∩ T ∂W . 2.3. Properties of the embedding e0 : (M, ξ) → (∂W, ξ1 ). Recall the following definition: Definition 7. An embedding e : (M0 , ξ0 , J0 ) → (M1 , ξ1 , J1 ) is called formal contact if there exists an homotopy of monomorphisms {Ψt : T M0 → T M1 }1t=0 such that Ψ0 = de, ξ0 = Ψ−1 1 (ξ1 ) and Ψ1 : (ξ0 , J0 ) → (ξ1 , J1 ) is complex. So far we have produced an embedding e0 : (M, ξ, j) → (∂W, ξ0 , J0 ) that is formal contact with the constant homotopy equal to de0 . Indeed, de−1 0 (ξ0 ) = ξ and de0 (ξ) is a complex subbundle of ξ0 . There is a family of complex isomorphisms Φt : ξ0 → ξt such that Φ0 = id. Fix a Reeb b0 = de0 (R). Build a family {Rt } of vector fields in vector field R associated to ξ and define R b T ∂W satisfying R0 |im e0 = R0 and hRt i ⊕ ξt = T ∂W . We take a family of metrics gt in ∂W defined in the following way: its restriction to ξt is hermitic for the complex bundle (ξt , Jt ) and Rt is unitary and orthogonal to ξt . Extend Φt to an isomorphism of T ∂W |im e0 in such a way that Φt (R0 ) = Rt . Define Et = Φt ◦ de0 : T M → T ∂W . The family {Et }1t=0 is composed of bundle monomorphisms and clearly satisfies that E1−1 (ξt ) = ξ and E1 (ξ) is a complex subbundle of ξ1 . Therefore, (e0 , Et ) is a formal contact embedding. Define Nt = Et (T M )⊥gt that is a bundle over im e0 which is complex by construction. N0 is isomorphic to Ck and therefore all the bundles Nt are trivial complex bundles. 2.4. Obtaining a contact embedding via h–principle. The only missing piece to complete the puzzle is to prove that the embedding e0 can be made contact. Using h–principle it is possible to deform (e0 , Et ) to a contact embedding thanks to the following theorem (cf. [5, Theorem 12.3.1]): Theorem 8. Let (e, Et ), e : (M0 , ξ0 = ker α0 ) → (M1 , ξ1 = ker α1 ), be a formal contact embedding between closed contact manifolds such that dim M0 + 2 < dim M1 . Then, there exists a family of embeddings eet : M0 → M1 such that: • ee0 = e and ee1 is contact, • de e1 is homotopic to E1 through monomorphisms Gt : T M0 → T M1 , lifting the embeddings eet , such that Gt (ξ0 ) ⊂ ξ1 and the restrictions Gt |ξ0 : (ξ0 , dα0 ) → (ξ1 , dα1 ) are symplectic. 4 Theorem 8 applied to (e0 , Et ) provides a family of embeddings {et } in which e1 : (M, ξ) → (∂W, ξ1 ) is a contact embedding and a family of monomorphisms Gt : T M → T ∂W that lift et such that G0 = E1 , G1 = de1 and Gt (ξ) ⊂ ξ1 is a complex subbundle. Lemma 9. The normal bundle of im(e1 ) in (∂W, ξ1 ) is trivial. Proof. Recall that N1 = E1 (T M )⊥g1 = G0 (T M )⊥g1 is a trivial complex vector bundle. Define, for t ∈ [1, 2], Nt = Gt−1 (T M )⊥g1 . Clearly, N2 is the normal bundle of the contact embedding e1 . Since N1 is a trivial vector bundle so is N2 .  Denote the 2m–dimensional polydisk by P 2m (r1 , . . . , rm ) = D2 (r1 ) × · · · × D2 (rm ) and abbreviate it as P 2m (r) when r1 = . . . = rm = r. The following result summarizes the work completed in this section and an important consequence (namely, the title of the section): M × P 2m (ε) admits a smooth compactification into a Stein fillable contact manifold. Theorem 10. Any closed contact manifold (M, ker α) contact embeds in the boundary of a Stein fillable manifold with trivial normal bundle. Furthermore, there exists k ≥ 1 such that for any m ≥ k !! m X ri2 dθi M × P 2m (ε), ker α + i=1 is tight with ε > 0 small enough depending only on α and k. Proof. The map e1 proves the first part because by Lemma 9 the normal bundle of the contact embedding e1 : (M, ξ) → (∂W, ξ1 ) is trivial. Notice that the codimension of the embedding is equal to 2k = dim τ and by replacing τ with τ ′ = τ ⊕ Cm−k we obtain embeddings of arbitrary codimension 2m ≥ 2k. Suppose henceforth that m ≥ k. By an standard neighborhood theorem in contact geometry it follows that there is a contactomorphism between a neighborhood of im(e1 ) in (∂W, ξ1 ) and P a neighborhood of M × {0} in (M × R2m , ker(α + ki=1 ri2 dθi )). Therefore, for some ε0 > 0, the previous contactomorphism provides an embedding from M × P 2m (ε0 ) into ∂W . Finally, since (∂W, ξ1 ) is Stein filable, it is tight. Thus, any of its open subsets is also tight and the conclusion follows.  3. M × D2 (ε) is tight if ε is small The argument leading to Theorem 10 provided no bound on the first positive integer k such that M × P 2k (ε, . . . , ε) is tight. Indeed, k was fixed at the beginning of Section 2, depending on the rank of τ → M , the bundle constructed to make the sum ξ ∗ ⊕ τ trivial. The insight needed to prove Theorem 1 is supplied by the understanding of overtwisted contact manifolds briefly discussed in the introduction. To be more concrete, the precise statement we will use in this section, extracted from [3], is the following: Theorem 11. Suppose that (M, ker α) is an overtwisted contact manifold. Then, if R is large enough, (M × D2 (R), ker(α + r 2 dθ)) is also overtwisted. The idea is to embed ∂W × D2 (R) in the boundary ∂V of a Weinstein manifold. Using the embedding constructed in the previous section we obtain then an embedding M × D2 (R) → ∂V that has trivial normal bundle. This leads to the proof of a statement similar to Theorem 10 in which we replace (M, ker α) by (M × D2 (R), ker(α + r 2 dθ)). Note that it is key to make sure that R is arbitrarily large. A Weinstein manifold (W, ω, f, Y ) is a manifold with boundary W equipped with a symplectic structure ω, a Morse function f : W → R and a Liouville vector field Y that is a pseudo– gradient for f . Notice that the symplectic form is automatically exact, ω = LY ω = d iY ω, so the boundary of a Weinstein manifold is exact symplectically fillable. The product of Weinstein manifolds (W1 , ω1 , g1 , Y1 ) and (W2 , ω2 , g2 , Y2 ) can be equipped with a Weinstein structure. Indeed, define ω ′ = ω1 + ω2 and Y ′ = Y1 + Y2 . Clearly, Y ′ is Liouville 5 for ω ′ . Suppose for simplicity that g1 and g2 are strictly positive (a rescaling would make the argument work in general) and define a function on W = W1 × W2 by fq = (g1q + g2q )1/q for an arbitrary q > 1. It is easy to check that Crit(fq ) = Crit(g1 ) × Crit(g2 ), the function fq is Morse and Y ′ is pseudogradient for fq . The Stein fillable manifold W supplied by Theorem 10 is naturally equipped with a Weinstein structure (W = f −1 (0, 1], ω, f, Y ) that satisfies ξ1 = ker(iY ω|∂W ). By the preceeding discussion, ∂ and a Weinstein structure in W × R2 is given by ω + dx ∧ dy, X = Y + r ∂r q 1/q   2 x + y2 fq = f q + (2R)2 The critical points of fq have the form (p, 0, 0), where p ∈ Crit(f ). The Liouville vector field X is transverse to ∂W × R2 . Our aim is now to embed ∂W × D2 (R) into a level set of fq by following φt , the flow of X. We can easily show: Proposition 12. For any δ > 0, there exists q > 1 large enough and a function µ : ∂W × D2 (R) → R− such that ||µ||C 0 ≤ δ and φµ : ∂W × D2 (R) → W × D2 (R) satisfies φµ (∂W × D2 (R)) ⊂ fq−1 (1). Proof. For q → ∞, the level set fq−1 (1) gets C ∞ –close to the submanifold ∂W × D2 (R). Since X is transverse to both of them, the result follows.  @W × D2 (R) φµ (@W × D2 (R)) −X D2 (2R) W fq−1 (1) Figure 1. Contact embedding of ∂W × D2 (R) into fq−1 (1). Proposition 12 produces a contactomorphism as the next lemma states. Lemma 13. Let e : H ֒→ M be a hypersurface transverse to a nowhere vanishing Liouville vector field X in (M, ω), the 1–form e∗ iX ω defines a contact structure on H. Moreover, if φt denotes the Liouville flow starting at H and s : H → R is a fixed function, then φs ◦ e : H ֒→ M is contactomorphic to e provided the flow φs is well–defined. Notice that the level set fq−1 (1) is the boundary of the Weinstein manifold V = fq−1 (0, 1]. Denote α′ = iX (α+dx∧dy). A straightforward application of Lemma 13 concludes the following: Proposition 14. For any R > 0, the contact manifold (∂W × D2 (R), ker(α′ |∂W ×D2 (R) )) admits a contact embedding into the boundary of a Weinstein manifold. Combining the last proposition and the results from the previous section we obtain: Corollary 15. Given a contact manifold (M, α) there exists k ∈ N and ε0 > 0 such that for P 2 every R > 0 the contact manifold (M × P 2k+2 (ε0 , . . . , ε0 , R), ker(α + k+1 i=1 ri dθi )) is tight. 6 Let us emphasize that ε0 does not depend on R: for any R > 0, M × P 2k+2 (ε0 , . . . , ε0 , R) is tight. Proof. The integer k and the number ε0 P both come from Theorem 10. Denote by e′ the contact embedding from (M × P 2k (ε0 ), ker(α + ki=1 ri2 dθi )) into (∂W, ξ1 = ker(iY ω)) and let η be the conformal factor of e′ , (e′ )∗ iY ω = exp(η)α′ . If necessary, decrease the value of ε0 to guarantee that sup η is finite. Proposition 14 supplies a Weinstein manifold (V = fq−1 (0, 1], ω + dx ∧ dy, fq , X) and contact embedding ϕ : (∂W × D2 (exp(sup η/2)R), α′ ) ֒→ (∂V, α′ ). Therefore, the map ϕ e : M × P 2k+2 (ε0 , . . . , ε0 , R) → ∂V given by ϕ(p, e x, y, xk+1 , yk+1 ) = ϕ(e′ (p, x, y), exp(η/2)xk+1 , exp(η/2)yk+1 )) is a contact embedding. Since ∂V is exact symplectically fillable the conclusion follows.  We are ready now P to prove Theorem 1. To ease the notation, we shall understand the contact form is equal to α + ri2 dθi in case it is omitted. Let us proceed by contradiction. Suppose that M × D2 (ε) is overtwisted for ε smaller than ε0 . Applying Theorem 11 k times consecutively we obtain a radius Rε > 0 such that M × P 2k+2 (ε, Rε , . . . , Rε ) is overtwisted. As we will show below, this manifold contact embeds into M × P 2k+2 (ε0 , . . . , ε0 , R) provided R is large enough. From Corollary 15 we know that the latter manifold is tight so we reach a contradiction. Therefore, M × D2 (ε) is tight. The only missing ingredient is the announced contact embedding: (1) M × P 2k+2 (ε, Rε , . . . , Rε ) → M × P 2k+2 (ε0 , . . . , ε0 , R) Its existence, subject to the conditions ε < ε0 and R large enough, is a consequence of the following packing theorem in symplectic geometry proved by Guth [10, Theorem 1]. Theorem 16. For every m ∈ N there is a constant C(m) ≥ 1 such that for any pair of ordered ′ that satisfy m–tuples of positive numbers R1 ≤ . . . ≤ Rm and R1′ ≤ . . . ≤ Rm • C(m)R1 ≤ R1′ and ′ . • C(m)R1 · . . . · Rk ≤ R1′ · . . . · Rm there is a symplectic embedding ′ P 2m (R1 , . . . , Rm ) ֒→ P 2m (R1′ , . . . , Rm ) The symplectic embedding supplied by Theorem 16 is automatically extended to our desired contact embedding (1) thanks to the following lemma: Lemma 17. Let Ψ : (D1 , dλ1 ) → (D2 , dλ2 ) be an exact symplectic embedding. For any contact manifold (M, ker α) with a choice of contact form α that makes the associated Reeb flow complete, Ψ induces a (strict) contact embedding (M × D1 , α + λ1 ) → (M × D2 , α + λ2 ). Proof. Since Ψ is exact, there exists a smooth function H : D1 → R such that dH = Ψ∗ λ2 − λ1 . If we denote the Reeb flow in M by Φ, ϕ : (M × D1 , α + λ1 ) → (M × D2 , α + λ2 ), is a contact embedding. ϕ(p, x) = (Φ−H(x) (p), Ψ(x))  7 4. Extension to contact submanifolds The results from the previous sections can be extended to a more general setting: contact submanifolds with arbitrary normal bundle. In the presence of a nowhere vanishing section of the normal bundle we will prove that the contact submanifold has a tight neighborhood. This is the content of Theorem 3. Let π : E → M be a complex vector bundle over a contact manifold equipped with an hermitian metric and a unitary connection ∇. The associated vertical bundle is denoted by V = ker(dπ). The standard Liouville form in R2n is U (n)–invariant and induces a global 1– e This real 1–form can be extended to T E by the expression form in V that will be denoted λ. e ◦ πV after we choose a projection onto the vertical direction πV : T E → V. The map πV λ=λ is determined by the choice of unitary connection so it is not canonical. The 1–form in T E associated to the connection ∇ is α e = π ∗ α + λ. Even though α e can be seen as the lift of the contact form α to E, it is not a globally defined contact form in general. However, it defines a contact form around the zero section E0 of the vector bundle. Lemma 18. α e is a contact form in a neighborhood of E0 . The restriction (E0 , ker(e α|E0 )) is contactomorphic to (M, ξ = ker α). Moreover, given any other contact structure ker β that coincides with ker(e α) in E0 and with the same complex structure in the normal bundle, there exist neighborhoods U, V of E0 such that (U, ker(β|U )) and (V, ker(e α|V )) are contactomorphic. Suppose henceforth that π has a global nowhere vanishing section s : M → E. The section s creates a complex line subbundle π|L : L → M . Then, the bundle E splits as E = F ⊕ L and L is trivial, i.e. there is an isomorphism φ : L → C that sends s(p) to 1p ∈ C in the fiber above every point p ∈ M . A suitable choice of unitary connection on π : E → M ensures that the associated contact form can be written as α e = α′ + λ, where α′ is a contact form in F and λ is the radial Liouville 2 form in R . Proposition 19. There exists U , a neighborhood of the zero section F0 of F , and ε > 0 such that (U × D2 (ε), ker(α′ + λ)) is tight. Note that this statement is exactly Theorem 1 except from the fact that F is not closed. The proof of Proposition 19 follows by embedding (U, ker α′ ) in a closed contact manifold (Fe, ker α e′ ) 2 ′ and then applying Theorem 1 to this manifold to deduce that (Fe × D (ε), ker(e α + λ)) is tight 2 ′ if ε > 0 is small. This result evidently implies that (U × D (ε), ker(α + λ)) is also tight. The aforementioned embedding is defined by the natural inclusion of F in the projectivization of F ⊕ C: F ֒→ Q = P(F ⊕ C) The complex bundle πQ : Q → M carries a natural formal contact structure ξ ′ = (dπQ )−1 (ξ) Indeed, an almost complex structure in ξ ′ is obtained as the sum of the pullback of a complex structure in ξ compatible with dα and a complex structure on the fibers of πQ . This formal contact structure is genuine (i.e., it is a true contact structure) in a neighborhood U of F0 by Lemma 18. The h–principle for closed manifolds proved in [2, Theorem 1.1] provides a homotopy from any formal contact structure to a contact structure. Furthermore, the homotopy can be made relative to a closed set in which the formal contact structure is already genuine. Applying this theorem we obtain a contact structure ξe′ on Q that agrees with ker α′ in U . We can reformulate Proposition 19 in the following way: Theorem 20. Let π : E → M be a complex vector bundle over a closed contact manifold (M, ξ). Suppose that π has a global nowhere vanishing section. Then, there exists a neighborhood U of e is tight for any contact structure ξe extending ξ the zero section of the bundle such that (U, ξ) and preserving the complex structure of E. An immediate application of Theorem 20 to the case in which M is a contact submanifold and π is its normal bundle yields Theorem 3. 8 References [1] Peter Albers, Urs Fuchs, and Will J. Merry. Positive loops and L∞ –contact systolic inequalities. Selecta Math. (N.S.), 23(4):2491–2521, 2017. [2] Matthew Strom Borman, Yakov Eliashberg, and Emmy Murphy. Existence and classification of overtwisted contact structures in all dimensions. Acta Math., 215(2):281–361, 2015. [3] R. Casals, E. Murphy, and F. Presas. Geometric criteria for overtwistedness. ArXiv e-prints, March 2015. [4] Roger Casals, Francisco Presas, and Sheila Sandon. Small positive loops on overtwisted manifolds. J. Symplectic Geom., 14(4):1013–1031, 2016. [5] Y. Eliashberg and N. Mishachev. Introduction to the h-principle, volume 48 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2002. [6] Y. Eliashberg and L. Polterovich. Partially ordered groups and geometry of contact transformations. Geom. Funct. Anal., 10(6):1448–1476, 2000. [7] Yakov Eliashberg. Topological characterization of Stein manifolds of dimension > 2. Internat. J. Math., 1(1):29–46, 1990. [8] Yakov Eliashberg, Sang Seon Kim, and Leonid Polterovich. Geometry of contact transformations and domains: orderability versus squeezing. Geom. Topol., 10:1635–1747, 2006. [9] F. Gironella. On some examples and constructions of contact manifolds. ArXiv e-prints, November 2017. [10] Larry Guth. Symplectic embeddings of polydisks. Invent. Math., 172(3):477–489, 2008. [11] Yang Huang. On plastikstufe, bordered Legendrian open book and overtwisted contact structures. J. Topol., 10(3): 720–743, 2017. [12] Klaus Niederkrüger. The plastikstufe—a generalization of the overtwisted disk to higher dimensions. Algebr. Geom. Topol., 6:2473–2508, 2006. [13] Klaus Niederkrüger and Francisco Presas. Some remarks on the size of tubular neighborhoods in contact topology and fillability. Geom. Topol., 14(2):719–754, 2010. [14] Francisco Presas et al. A class of non–fillable contact structures. Geom. Topol., 11(4): 2203–2225, 2007. [15] Sheila Sandon. Floer homology for translated points. 2016. Instituto de Ciencias Matematicas CSIC–UAM–UCM–UC3M, C. Nicolás Cabrera, 13–15, 28049, Madrid, Spain E-mail address: [email protected] Departamento de Álgebra, Geometrı́a y Topologı́a, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040 Madrid, Spain E-mail address: [email protected] Instituto de Ciencias Matematicas CSIC–UAM–UCM–UC3M, C. Nicolás Cabrera, 13–15, 28049, Madrid, Spain E-mail address: [email protected] 9