Polyfolds, Cobordisms, and the strong Weinstein conjecture
aa r X i v : . [ m a t h . S G ] O c t POLYFOLDS, COBORDISMS, AND THE STRONG WEINSTEINCONJECTURE
STEFAN SUHR AND KAI ZEHMISCH
Abstract.
We prove the strong Weinstein conjecture for closed contact mani-folds that appear as the concave boundary of a symplectic cobordism admittingan essential local foliation by holomorphic spheres. Introduction
Given a closed (co-orientable) contact manifold (
M, ξ ) and a defining contactform α (i.e. ξ = ker α ) Weinstein conjectured in [39] that the Reeb vector field R ,which is uniquely defined by i R d α = 0 and α ( R ) = 1, admits a periodic solution.The Weinstein conjecture was proven by Taubes [36] for all closed contact manifoldsof dimension 3 and has been verified in higher dimensions in many situations mostrecently in the presence of contact connected sums in [16, 17, 18]. We refer thereader to [34] for the state of the art of the conjecture.A stronger conjecture was given in [1] that asks for a finite collection of periodicsolutions of R , a so-called null-homologous Reeb link , that oriented by R andeventually counted with a positive multiple of a period represents the trivial class inthe homology of M . We refer to the existence question of a null-homologous Reeblink as the strong Weinstein conjecture and remark that the stronger versionof the conjecture is not covered by Taubes result. The aim of the present work is toconfirm the strong Weinstein conjecture for closed contact manifolds ( M, ξ = ker α )that appear as the concave end of symplectic cobordisms with particular properties.This will generalize the results obtained in [14].The notion of a symplectic cobordism was introduced in [9, 23] in the contextof symplectic field theory. A symplectic cobordism is a compact connectedsymplectic manifold ( W, ω ) with boundary that admits a
Liouville vector field Y near ∂W , which by definition satisfies L Y ω = ω . According to the boundaryorientation induced by the orientation of the symplectic form the Liouville vectorfield Y points either in or out of W . This decomposes the boundary of W into the concave boundary M − (along which Y points in) and into the convex boundary M + (along which Y points out). The Liouville vector field defines contact forms α − and α + by restricting i Y ω to the tangent bundles of M − and M + , resp., so that( M − , α − ) and ( M + , α + ) are particular examples of contact type hypersurfaces in( W, ω ), cf. [31].
Date : 13.10.2016.2010
Mathematics Subject Classification.
A theorem of Hofer [20] relates the existence of closed Reeb orbits to the existenceof (punctured finite energy) holomorphic curves in symplectic cobordisms, cf. [2,14, 15]. In order to utilize this relation we will consider symplectic cobordisms thatadmit a so-called essential local foliation by holomorphic spheres . The definition isgiven in Section 2.3 below.Generalizing the notion of symplectic cobordisms we will consider compact con-nected symplectic manifolds (
W, ω ) that have boundary components that are eitherconcave, convex, or foliated by symplectic spheres. In the case the foliation is givenby an essential local foliation by holomorphic spheres in the sense of Section 2.3(
W, ω ) is called a symplectic cobordism as well.The symplectic area of the holomorphic spheres of the foliation as it will turnout induces an upper bound on the total action of a null-homologous Reeb link,which is by definition the sum of the actions of the link components counted withthe selected period multiplicities.
Theorem 1.1. If ( M, α ) is the concave boundary of a symplectic cobordism ( W, ω ) ,and ( W, ω ) admits an essential local foliation by holomorphic spheres of area π̺ ,then there exists a null-homologous Reeb link in ( M, α ) of total action smaller than π̺ . Additionally, if ( W, ω ) has no concave boundary it has no convex boundaryeither. Remark 1.2.
The only surface (
W, ω ) to which the theorem applies is C P .The qualitative content of the theorem is valid not only for one particular contactform on M . In fact, the construction from [14, Section 3.3] allows one to conclude forany contact form whose kernel is equal to ξ = ker α . Each contact form that defines( M, ξ ) appears as a graph over the zero section in the symplectisation of (
M, α ).After a shift in the negative R -direction the graph can be assumed to lie below thezero section. If ( M, α ) is the concave boundary of a symplectic cobordism (
W, ω ),then a positive constant multiple of any ξ -defining contact form can be realized asthe concave boundary of a slightly modified symplectic cobordism. The cobordismis obtained by gluing the symplectic cobordisms that is cut out by the shiftedgraph and the zero section to the cobordism ( W, ω ) along (
M, α ). To express thiscircumstance we will say that (
M, ξ ) is the concave boundary of (
W, ω ). Convexboundaries are handled similarly.
Corollary 1.3.
The strong Weinstein conjecture holds for all contact manifoldsthat are the concave boundary of a symplectic cobordism that is provided with anessential local foliation by holomorphic spheres.
A contact manifold is called co-fillable if it is a boundary component of a sym-plectic cobordism that has more then one convex but no concave boundary compo-nents, cf. [40]. Examples of McDuff [30], Geiges [11, 12], and Massot-Niederkr¨uger-Wendl [29] show the existence of co-fillabe contact manifolds. Generalizing a resultof McDuff [30] we obtain (cf. Remark 6.1):
Corollary 1.4. If ( M, ξ ) is the concave boundary of a symplectic cobordism ( W, ω ) as in the theorem, then ( M, ξ ) is not co-fillable. In [14] the following capacity for symplectic manifolds (
V, ω ) that are not ofdimension 2 was introduced: c ( V, ω ) = sup ( M,α ) inf α . OLYFOLDS AND THE WEINSTEIN CONJECTURE 3
The supremum is taken over all closed contact type hypersurfaces (
M, α ) in (
V, ω )and inf α is the minimal total action of a null-homologous Reeb link in ( M, α ). Corollary 1.5. If ( W, ω ) as in the theorem has no concave (and hence no convex)boundary, then c ( W, ω ) ≤ π̺ . In particular, the Gromov radius of (
W, ω ) as introduced in [19] is bounded by π̺ from above. This can also be derived from the following uniruledness result.This is because no embedding of an open ball into W can intersect ∂B × C P ,which is the boundary (in case if non-empty) of the local foliation domain B × C P that we require to exist, cf. Section 2.3. Corollary 1.6. If ( W, ω ) as in the theorem has no contact type boundary compo-nents, then through each point of W there passes a (nodal) holomorphic sphere forany compatible almost complex structure that coincides with J B ⊕ i on U × C P ,where U is a collar neighborhood of ∂B in B . In order to prove the theorem we use holomorphic spheres corresponding tothe given local foliation. The associated moduli space is non-empty and regularin a neighborhood of the holomorphic spheres that come from the local foliation.The assumption of being essential results in uniqueness properties of the modulispace. In order to ensure global regularity properties of the moduli space, whichare obstructed by bubbling off of multiply covered holomorphic spheres of nega-tive first Chern class, semi-positivity of (
W, ω ) could be used. In order to get anunrestricted statement we will employ the regularity theory developed by Hofer-Wysocki-Zehnder instead. The space of (not necessarily holomorphic) stable curveshas a polyfold structure, see [28]. Using abstract perturbations (see [26]) the mod-uli space can be approximated by solution spaces of perturbed Fredholm problemsthat carry the structure of a smooth branched orbifold with weights. This enablesus to conclude as in [14].In Section 2 we formulate the definition of an essential local foliation by holomor-phic spheres. Examples and applications are presented in Section 3. The contentof Section 4-5 is a description of the moduli space of holomorphic curves relevantin our situation in view of the application of the theory of polyfolds and abstractperturbations. The proof of Theorem 1.1 is given in Section 6.2.
Definition
Completion.
A collar neighborhood of a concave boundary (
M, α ) of a sym-plectic cobordism (
W, ω ) is symplectomorphic to (cid:0) [0 , ε ) × M, d(e a α ) (cid:1) for some ε > T denote the set of all smooth strictly increasing functions ( −∞ , → (0 , R that restricted to [0 , ∞ ) coincides with a e a .A concave end is a symplectic manifold of the form (cid:0) ( −∞ , × M, d( τ α ) (cid:1) with τ ∈ T . Gluing the concave end to ( W, ω ) along (
M, α ) results in a symplecticmanifold ( W ′ , ω τ ) for τ ∈ T . In order to reflect the conformal nature we will call( W ′ , ω τ ) the completion of ( W, ω ).2.2.
Holomorphic spheres with nodes.
Consider a Riemann surface (
S, j ) withfinitely many connected components. A nodal pair is a subset of S that consists oftwo distinct points. Nodal pairs are required to be pairwise disjoint. Denote by D a finite set of nodal pairs such that the quotient space S/D is connected. A nodalRiemann surface is a triple (
S, j, D ). The nodal Riemann surface (
S, j, D ) is
STEFAN SUHR AND KAI ZEHMISCH said to be of genus zero if each connected component of S is diffeomorphic to the2-sphere and if the number of connected components of S is equal to the numberof nodal pairs D plus one; to phrase it differently, if S/D is simply connected.Let J denote an almost complex structure on the completion of W . We considera nodal Riemann surface ( S, j, D ) of genus zero. A nodal J -holomorphic sphere is a smooth map u : S → W ′ that solves the non-linear Cauchy-Riemann equation T u ◦ j = J ( u ) ◦ T u and that descends to a continuous map on the quotient
S/D .In case D is empty u is called un-noded .2.3. Essential holomorphic foliations.
Let (
B, ω B ) be an open symplectic mani-fold. If B has non-empty boundary we require that ∂B is closed. Denote by ω FS the Fubini-Study form on C P , which integrates to total area π . A local foliationby holomorphic spheres in ( W, ω ) is a symplectic embedding (cid:0) B × C P , ω B + ̺ ω FS (cid:1) −→ ( W, ω ) , where ̺ is a positive real number. If the boundary of B is non-empty we assumethat ∂B × C P is mapped diffeomorphically onto ∂W \ ( M − ∪ M + ) assuming thatbesides the concave and the convex boundary a further boundary component ( aposteriory equal to ∂B × C P ) exist.The local foliation B × C P is assumed to be equipped with a compatible al-most complex structure of the form J B ⊕ i. All almost complex structures on thecompletion W ′ are assumed to restrict to the split structure J B ⊕ i on B × C P and are called admissible .We remark that by a theorem of Moser [33] for any positive area form σ on S with σ ( S ) = π̺ there exists a diffeomorphism of S along which σ pulls back to ̺ ω FS . The area of a holomorphic sphere in the local foliation is π̺ . Definition 2.1.
A local foliation B × C P ⊂ W as readily defined is called es-sential if there exists a point ∗ ∈ B such that for all admissible compatible almostcomplex structures J on W ′ any nodal J -holomorphic sphere in W ′ that is • homologous to ∗ × C P , • non-constant restricted to any component of S , • and intersects B × C P non-triviallyis un-noded and up to a pre-composition with a M¨obius transformation equal to z ( b, z ) for some b ∈ B . Examples and Applications
In [14] the strong Weinstein conjecture was verified for contact manifolds thatappear as the concave boundary of a semipositive symplectic cobordism that canbe capped off along a convex boundary component in a particular way. The con-struction of the cap from [14, Section 5.1] generalizes to the present context asfollows.3.1.
Holomorphic foliations via caps.
We consider a non-empty closed contactmanifold (
N, α N ). For ε > V, ω V ) = (cid:0) ( − ε, × N, d(e a α N ) (cid:1) a cylindrical subset of the symplectization of ( N, α N ). We equip ( V, ω V ) with analmost complex structure J V that is compatible with the contact form α N ,i.e. J V is invariant under translation, restricts to a compatible complex structure OLYFOLDS AND THE WEINSTEIN CONJECTURE 5 on the symplectic bundle (ker α N , d α N ), and sends the Liouville vector field ∂ a tothe Reeb vector field of α N .Moreover, let ( Q, ω Q ) be a closed symplectic manifold and denote by J Q a com-patible almost complex structure on ( Q, ω Q ). The rational area spectrum of thesymplectic form ω Q is given by the image of the map H ( Q ; Q ) → R obtained byintegration against ω Q . The spectrum is countable and hence constitutes a residualsubset of R . Proposition 3.1.
Let ( W, ω ) be a symplectic manifold with boundary and consider ( V, ω V ) and ( Q, ω Q ) as stated above. We assume that ( W, ω ) admits a local foliation (cid:16) V × Q × C P , ω V + ω Q + ̺ ω FS (cid:17) with ∂W = N × Q × C P that is equipped with the almost complex structure J V ⊕ J Q ⊕ i . The local foliation is essential provided π̺ is not a rational spectral value of ω Q .Proof. Let u : S → W be a nodal holomorphic sphere that intersects the localfoliation V × Q × C P non-trivially. The restriction of u to u − (cid:0) V × Q × C P (cid:1) can be projected to the ( − ε, V . The composition of the resulting mapwith the exponential map is subharmonic and has an interior maximum. By themaximum principle and an open-closed argument applied to each component of S the image u ( S ) is contained in { a }× N × Q × C P for a suitable a ∈ ( − ε, u to the exact symplectic manifold ( − ε, × N must be constant(by compatibility) the image of u is in fact contained in { v } × Q × C P ≡ Q × C P for a suitable v ∈ V .In view of the Definition 2.1 we assume in addition that u is homologous to C P ≡ ∗ × C P in W for any choice of base point ∗ of V × Q and that u isnon-constant on each component of S . Choose an ordering on the componentsof S = S ⊔ . . . ⊔ S k and write u j for the restriction of u to the component S j .Further, denote by u Qj and ϕ j the projections of u j to Q and C P , resp. Accordingto K¨unneth’s formula with respect to Q × C P we get (cid:2) C P (cid:3) = k X j =1 (cid:2) u Qj (cid:3) + k X j =1 d j (cid:2) C P (cid:3) in H W , where d j is the degree of the holomorphic map ϕ j . Because π̺ is not inthe rational area spectrum of ω Q an application of ω W shows that k X j =1 ω Q (cid:16)(cid:2) u Qj (cid:3)(cid:17) = 0 and k X j =1 d j = 1 . Because the symplectic energy is non-negative by compatibility we get k = 1, d = 1, and u Q is constant. Therefore, there exists q ∈ Q and an automorphism ϕ of C P such that u ( z ) = (cid:0) v, q, ϕ ( z ) (cid:1) for all z ∈ C P . (cid:3) Remark 3.2.
The construction generalizes to a symplectic manifold (
V, ω V ) thathas a weakly convex contact type boundary, see Remark 6.1, or is the negative half-symplectisation of the stable Hamiltonian structure ( ω N , d θ ) that is induced by asymplectic fibration θ : N → S on the boundary N of V with respect to ω N := ω V | T N , see [5, p. 877]. In the second case one requires that π̺ is not a rational STEFAN SUHR AND KAI ZEHMISCH area spectral value of ω F + ω Q , where F is the typical fibre of θ , which is symplecticwith respect to ω F := ω V | T F . The compatible almost complex structures J V we arenow considering are translation invariant, turn the fibres of θ into a holomorphicsubmanifold in each level of ( − ε, × N , and send the Reeb vector field of the stableHamiltonian structure ( ω N , d θ ) to ∂ a , see [5, Section 5].3.2. Stabilized Weinstein conjecture.
In the following we will construct a classof symplectic cobordisms with an essential local foliation as described. Let (
P, ω P )be a symplectic filling , i.e. a symplectic cobordism with all boundary componentsof contact type being convex. Consider a closed hypersurfaceΣ ⊂ P × Q × C that is of contact type with respect to ω P + ω Q + d x ∧ d y . The induced contact formon Σ is denoted by α Σ . Each component of Σ bounds a relatively compact opendomain the so-called bounded domain . We require that the bounded domainsare pairwise disjoint. We denote the union of the domains by D Σ .Let ̺ be a positive real number such that π̺ is not in the rational area spectrumof ( Q, ω Q ) and greater than the minimal area of a closed disc in C about the originthat contains the image of the projection map Σ ⊂ P × Q × C → C . We define the symplectic cap to be( C, ω C ) = (cid:16) P × Q × C P \ D Σ , ω P + ω Q + ̺ ω FS (cid:17) . Applying Theorem 1.1 to (
C, ω C ) with the components of D Σ glued back thatcorrespond to the concave boundary components we see that ( C, ω C ) cannot havea convex boundary. In other words (Σ , α Σ ) is the concave boundary of ( C, ω C ). Remark 3.3.
Let the dimension of P × Q × C be 2 n . If the ( n − ω = ω P + ω Q + d x ∧ d y has a primitive µ , as it is the case if ω P is exact, the helicity (see [32]) can be used as in [41] to show that Σ is the concaveboundary of ( C, ω C ). Indeed, by Stokes theorem the symplectic volume of ( D Σ , ω )equals R Σ µ ∧ ω , where Σ is equipped with the boundary orientation induced by thesymplectic orientation of ( D Σ , ω ). On the other hand, the restriction of ω to T Σequals d α Σ , where α Σ is the contact form induced by the local Liouville vector field Y , so that (cid:16) µ | T Σ − α Σ ∧ (cid:0) d α Σ (cid:1) n − (cid:17) ∧ d α Σ is an exact form on Σ. Consequently, R Σ µ ∧ ω equals the contact volume of (Σ , α Σ ),so that i Y ω n | T Σ = nα Σ ∧ (cid:0) d α Σ (cid:1) n − is a positive volume form on (Σ , α Σ ). Hence, the boundary orientation on Σ equalsthe orientation induced by α Σ , i.e. the local Liouville vector field Y points out.Consequently, if ( A, ω A ) is a symplectic cobordism such that (Σ , α Σ ) appears asconvex boundary gluing along (Σ , α Σ ) yields a symplectic manifold( W, ω W ) = ( A, ω A ) ∪ (Σ ,α Σ ) ( C, ω C )to which Theorem 1.1 applies. As an example we phrase. Corollary 3.4.
The Gromov radius of (cid:16) P × Q × D , ω P + ω Q + d x ∧ d y (cid:17) OLYFOLDS AND THE WEINSTEIN CONJECTURE 7 is lower or equal than π . The strong Weinstein conjecture holds true for any closedcontact type hypersurface in (cid:16) P × Q × C , ω P + ω Q + d x ∧ d y (cid:17) provided that P is not empty. In particular, we get the (very) stabilized strong Weinstein conjecture for hyper-surfaces of contact type in Q × C ℓ for ℓ ≥
2, cf. Floer-Hofer-Viterbo [10]. If H is aDonaldson hypersurface in ( Q, ω Q ) (see [6]) so that the complement of H has thestructure of a Stein manifold ( P, ω P ) then the strong Weinstein conjecture followsfor contact type hypersurfaces in ( Q \ H ) × C . By [3] it is possible to constructsymplectic hypersurfaces H that lie in the complement of a given compact isotropicsubmanifold in ( Q, ω Q ).3.3. Cotangent bundles.
We consider the unit sphere S m +1 in C m +1 . TheWeinstein conjecture holds true for any closed hypersurface Σ that is of contacttype in T ∗ S m +1 , cf. [37]. The contact structure on Σ is taken to be the one inducedfrom T ∗ S m +1 . Notice, that no assumption is made on the bounded component ofthe complement T ∗ S m +1 \ Σ in view of the zero section, cf. [21, 35]. We claim thatthe strong Weinstein conjecture holds for Σ as well.Indeed, S m +1 embeds into C m +1 × C P m as a Lagrangian submanifold L viathe map that sends z ∈ S m +1 ⊂ C m +1 to (cid:0) z, [¯ z ] (cid:1) . The map S m +1 ∋ w [ w ],where [ w ] denotes the complex line through w and the origin, is the so-called Hopffibration, along which the Fubini-Study form ω FS pulles back to d x ∧ d y . Becausethe complex conjugation z ¯ z is anti-symplectic the symplectic form d x ∧ d y + ω FS on C m +1 × C P m vanishes pulled back to S m +1 . Hence, the (2 m + 1)-dimensionalsubmanifold L ⊂ C m +1 × C P m is Lagrangian. Using the fibrewise radial Liouvilleflow of T ∗ S m +1 we can isotope Σ into a small neighborhood of the zero-section.Hence, with Weinstein’s tubular neighborhood theorem ([38]) Σ can be realized asa contact type hypersurface in C m +1 × C P m with the characteristic foliation to beconjugate to the one induced by the inclusion Σ ⊂ T ∗ S m +1 . The claim followswith Corollary 3.4.In fact, this shows the strong Weinstein conjecture for all cotangent bundles overclosed manifolds of the form X × S m +1 with m ≥
0, which admit a Lagrangianembedding into T ∗ X × C m +1 × C P m . To get products with S observe that S embeds as a Lagrangian surface into the unit ball in C blown-up at two points, [5,Section 6 Example (3)]. To the blown-up ball the complement of the unit ball in C × ( C ∪ ∞ ) is glued on. With the construction of a symplectic cap (the cap being( C \ B ) × C P ) and Theorem 1.1 it follows that any closed hypersurface of contacttype in the cotangent bundle of X × S satisfies the strong Weinstein conjecture.Similarly, because for any closed orientable 3-manifold Y the connected sum L = Y S × S ) admits a Lagrangian embedding into C the strong Weinsteinconjecture holds true for T ∗ L , see [7]. This is of particular interest for contact typehypersurfaces in T ∗ Y that miss one fibre. Examples are given by energy surfaces ofclassical mechanical systems on Y with a sign changing potential function, cf. [35].4. Stable curves and the moduli space
We consider a symplectic cobordism (
W, ω ) that admits an essential local foli-ation by holomorphic spheres B × C P ⊂ W , see Section 2.3. Denote by ∗ the STEFAN SUHR AND KAI ZEHMISCH particular base point of B as required in Definition 2.1. We denote by ( M, α ) theconcave boundary of (
W, ω ) and associate the completion ( W ′ , ω τ ) attaching thenegative half-symplectisation of ( M, α ) as described in Section 2.1. Moreover, weassume the contact form α to be non-degenerate , that is along periodic solutionsof the Reeb vector field of α the linearised Poincar´e return map has no eigenvalueequal to 1. Let J be an admissible compatible almost complex structure on ( W ′ , ω τ )that is compatible with the contact forms α and α + on the concave end and in aneighborhood of M + , resp., as described at the beginning of Section 3. The aimof this section is to study stable holomorphic one-marked curves of genus zero in( W ′ , ω τ , J ).4.1. Stable maps.
Let (
S, j, D ) be a nodal Riemann surface of genus zero. The setof points, the so-called nodal points , that belong to a nodal pair is denoted by | D | .We provide ( S, j, D ) with a finite set M of pairwise distinct points in S \ | D | , theso-called marked points . The points in | D | ∪ M are called special . A connectedcomponent C of S is called stable if the number of special points in C is greateror equal than 3 − C ).A stable map ( S, j, D, M, u ) in W ′ is a continuous map u : S → W ′ defined ona marked nodal Riemann surface ( S, j, D, M ) that descends to a continuous mapon the quotient
S/D such that: • The map u is of Sobolev class H on S \ | D | and of weighted Sobolev class H ,δ near the nodal points | D | for some δ ∈ (0 , π ), see [28, Definition 1.1]. • The cohomological integral R C u ∗ ω τ is non-negative for any connected com-ponent C of S and for one τ ∈ T (and hence for all by Stokes theorem). • If a connected component C of S is not stable, then R C u ∗ ω τ > The moduli space.
Two stable maps (
S, j, D, M, u ) and ( S ′ , j ′ , D ′ , M ′ , u ′ )are said to be equivalent if there exists a diffeomorphism ϕ : S → S ′ such that ϕ ∗ j ′ = j , ϕ ∗ D = D ′ , ϕM = M ′ , and u ′ ◦ ϕ = u . Often we will write u for( S, j, D, M, u ). The equivalence class [
S, j, D, M, u ] of a stable map is called a stable curve and is denoted by u . We denote by Z the set of all one-marked genus zero stable curves u in W ′ such that the mapinduced by u on the quotient S/D is homologous to ∗ × C P , where ∗ ∈ B is thechosen base point of B .A stable curve u is called holomorphic if it can be represented by a stable map u that is holomorphic. The definition does not depend on the choice of u . Wedenote by M the moduli space of all holomorphic stable curves u ∈ Z .Observe that for all b ∈ B the class of (cid:0) C P , i , ∅ , ∞ , z ( b, z ) (cid:1) represents anelement of M that we will call a standard sphere . Due to the stability conditionand the local foliation B × C P being essential all non-standard holomorphic curvesin M do not intersect B × C P . We will identify the subset of standard spheresin M with B × C P , where the C P -factor corresponds to the marked point. Thecomplement is denoted by M cut = M \ B × C P . OLYFOLDS AND THE WEINSTEIN CONJECTURE 9
A priori uniform bounds.
The Dirichlet-energy of all nodal holomorphicspheres u induced by stable holomorphic curves u ∈ M equals Z S u ∗ ω τ = [ ω τ ] (cid:16)(cid:2) C P (cid:3)(cid:17) = π̺ for all τ ∈ T .The moduli space M admits upper bounds in the following sense. Because thealmost complex structure J is compatible with the contact form α + the maximumprinciple implies that no stable holomorphic curve can intersect a neighborhood( ε, × M + of M + , cf. the first part of the proof of Proposition 3.1. Similarly,because only standard curves u ∈ M intersect the foliation domain B × C P wecan bound M cut away from ∂B if the boundary of B is not empty.As it will turn out (see Section 5.5) there are no lower bounds for M along theconcave end. This will prove Theorem 1.1 as the following lemma shows: Lemma 4.1.
If there exists a sequence u k in M such that u k ( S k ) intersects ( −∞ , − k ] × M non-trivially, then ( M, α ) admits a null-homologous Reeb link oftotal action less than π̺ .Proof. If u k maps a component C of S k into the concave end u k | C is constant byStokes theorem. Therefore, we find sequences z k and w k of points on S k such that u k ( z k ) ∈ W and u k ( w k ) is contained in the concave end so that the projection of u k ( w k ) to the R -factor tends to −∞ . With respect to a metric on W ′ that equalsa product metric on the concave end (product with the Euclidean metric on the R -axis) the gradient of u k blows up. This follows with a mean value argument asin [15, 23]. The bubbling off analysis from [4] shows the existence of a holomor-phic building of height k − | k − ≥
1. The lowest level of the building isrepresented by a punctured finite energy surface in the symplectisation of (
M, α )that has Hofer-energy less than π̺ and positive punctures exclusively. Near thepunctures the finite energy surface converges to cylinders over closed Reeb orbits of α exponentially fast. Hence, the projection of a component to M along the R -axisdefines a 2-chain whose boundary is a Reeb link in ( M, α ) of total action less than π̺ , cf. [14, 20]. (cid:3) In other words, in order to prove Theorem 1.1 we have to exclude the casewhere there exists
K > u in M the image u ( S ) is contained in( − K, × M ∪ W .We remark that the proof of Lemma 4.1 uses the assumption α being non-degenerate. This is not a restriction as the arguments in Section 6 will show.5. Polyfold structure
Topology of the space of stable curves.
The space Z of stable curves hasa natural topology as described in [28, Section 2.1/3.4] that is second countable,paracompact, and Hausdorff, see [28, Theorem 1.6]. The topology is induced by the H -topology of maps on nodal Riemann surfaces that have exponential decay inholomorphic polar coordinates near the nodes. Part of the construction is a choiceof auxiliary marked points that stabilize all connected components of the domain ifnecessary. The additional points are fixed using local codimension-2 submanifoldsin W ′ that are transverse to the image of the curve intersecting in a single point. Itis required that the additional marked points are mapped to the intersection points. If a non-trivial automorphism group is acting on a stable curve the auxiliary markedpoints are supposed to be chosen equivariantely. The stabilization of the domainmakes it possible to use the topology of the corresponding Deligne-Mumford spacein terms of uniformizing families, see [28, Definition 2.12]. In order to describe thedesingularization of the nodes (i.e. the gluing) uniformizing families are used toobtain uniformizers for the space of stable maps Z , see [28, Section 3.1/3.2].We remark that the evaluation map ev : Z → W ′ that maps u to the value u ( z )at the marked point z is continuous, see [26, p. 2290] or [28, p. 7].5.2. The target space.
Let u be a stable map that represents a class u in Z . Wedenote by ξ a continuous section of Hom (cid:0) Λ T ∗ S, u ∗ T W ′ (cid:1) such that for each z ∈ S the map ξ ( z ) : T z S → T u ( z ) W ′ is complex anti-linear with respect to J (cid:0) u ( z ) (cid:1) .The section ξ is required to be of Sobolev class H on S \ | D | and of weightedSobolev class H ,δ near | D | for some δ ∈ (0 , π ), see [28, Section 1.2]. An equiva-lence ϕ of stable maps ( S, j, D, M, u ) and ( S ′ , j ′ , D ′ , M ′ , u ′ ) is an equivalence of( S, j, D, M, u, ξ ) and ( S ′ , j ′ , D ′ , M ′ , u ′ , ξ ′ ) if ξ ′ ◦ T ϕ = ξ . The equivalence class isdenoted by ξ and the space of all equivalence classes ξ by W . By [28, Theorem 1.9] W has a natural topology that is second countable, paracom-pact, and Hausdorff so that the projection p : W → Z that maps the class ξ to theclass u , if ξ is a section along u , is continuous. The Cauchy-Riemann operator ¯ ∂ J is a section of p whose value ¯ ∂ J u at a point u ∈ Z is the class represented by (cid:16) S, j, D, M, u, (cid:0) T u + J ( u ) ◦ T u ◦ j (cid:1)(cid:17) . Notice that the moduli space M equals the zero set { ¯ ∂ J u = } .5.3. Polyfold Fredholm section.
The space of stable curves Z is a polyfold(see [26, Section 3]) so that the evaluation map ev : Z → W ′ is sc-smooth, see[28, Theorem 1.7/1.8], [26, Theorem 1.10]. The projection map p : W → Z is astrong polyfold bundle, see [28, Theorem 1.10]. By [28, Theorem 1.11] the Cauchy-Riemann operator ¯ ∂ J : Z → W is a sc-smooth, component proper Fredholm sectionwhich is naturally oriented. The Fredholm index of ¯ ∂ J equals the dimension of W because the first Chern class of ( T W ′ , J ) evaluates to 2 on (cid:2) ∗ × C P (cid:3) .On the level of objects the strong polyfold bundle structure of p : W → Z is obtained by gluing strong polyfold bundles in the sense of ep-groupoids. Tounderstand the Cauchy-Riemann section ¯ ∂ J : Z → W it is enough to consider alocal M-polyfold bundle chart. We would like to apply this principle for standardholomorphic spheres u ∈ M . Represent u by the map u : z ( b, z ) for some b ∈ B .Denote by X the sc-Hilbert manifold of pairs ( v, z ), where z ∈ C P is a markedpoint near ∞ and v a map C P → W ′ of class H that is close to u mapping 0 and1 into B × B ×
1, resp. Moreover, let E be the sc-Hilbert space bundle over X with fibre consisting of all H -maps from C P into the space of J ( v )-anti-linearmaps Λ T ∗ C P → v ∗ T W ′ for v ∈ X . A M-polyfold bundle chart of p : W → Z about u is then given by E → X because u is un-noded, so that the surroundingsplicing core is the full ambient space and the sc-retraction map equals the identity.The Cauchy-Riemann operator ¯ ∂ J : Z → W induces a local section f : X → E near x = ( u, ∞ ). The linearization f ′ ( x ) : T x X → E x is the vertical differential,which is the composition of the sc-differential T x f : T x X → T E and the projection OLYFOLDS AND THE WEINSTEIN CONJECTURE 11 T E = T x X ⊕ E x → E x , see [24, Section 4.4]. By [28, p. 55 and Definition 5.5] and[25, Section 3] the linearization f ′ ( x ) is a sc-Fredholm operator and coincides withthe linearized Cauchy-Riemann operator at ( u, ∞ ) in the C ∞ -sense (cf. [32]) on adense subset, see [24, Proposition 2.14/2.15]. Because a sc-Fredholm operator isregularizing f ′ ( x ) is surjective with kernel being of dimension equal to dim W thatconsists of smooth elements exclusively.5.4. Abstract Perturbations.
Let λ : W → Q ∩ [0 , ∞ ) denote a sc + -multisection.In a local representation P : E → X of p : W → Z we write Λ : E → Q ∩ [0 , ∞ )for the sc + -multisection that corresponds to λ . The local sc + -sections of P thatare attached to Λ are denoted by s i . A pair (cid:0) ¯ ∂ J , λ (cid:1) is called transverse if in localrepresentations f : X → E of the Cauchy-Riemann operator the linearizations f ′ ( u ) − s ′ i ( u ) : T u X −→ E u are surjective for all i and for all u ∈ Z that are contained in (cid:8) λ (cid:0) ¯ ∂ J (cid:1) > (cid:9) , see [26,Definition 4.7(1)]. Observe that in case f ′ ( u ) is onto for u ∈ M that has a simplerepresentation by an un-noded holomorphic sphere map u we can choose one localsection that is identically 0 in a neighborhood of u in X , i.e. λ (0 u ) = 1.5.5. Gromov-Witten integration.
In this section we give a proof of Theorem1.1 in the case the contact form α is non-degenerate. For K > W K the open domain in W ′ that is obtained from W ′ by removing ( −∞ , − K ] × M ,[ − /K, × M + , and B K × C P , where B K is a subset of B that is diffeomorphic toeither a ball of radius 1 /K or, if ∂B is not empty, a collar neighborhood [ − /K, × ∂B of ∂B . We have to exclude uniform lower bounds for M , see Lemma 4.1.Arguing by contradiction we assume that there exists K > u ∈ M cut the image u ( S ) is contained in W K . There exists aneighborhood U of M cut in Z such that for all u ∈ U the image u ( S ) is containedin W K , cf. Section 5.1. For example we can choose U = ev − ( W K ) because ofev( M cut ) ⊂ W K . Moreover, by [4] and [28, Proposition 4.9 and Remark 4.10] M is a compact subset of Z .By [26, Theorem 4.17] there exists a small sc + -multisection λ such that (cid:0) ¯ ∂ J , λ (cid:1) is transverse. Notice, that for all standard spheres u ∈ M the isotropy group of u is trivial and f ′ ( u ) is onto, where f is a local representation of ¯ ∂ J . The proof of [26,Theorem 4.17] implies that λ can be chosen to be trivial for all standard spheres u ∈ M , i.e. in a local representation Λ is identically 1 on the zero-section over theset of standard holomorphic spheres u , cf. [26, Definition 3.35]. The support of λ can be assumed to be contained in U . Therefore, the solution set S = n u ∈ Z (cid:12)(cid:12)(cid:12) λ (cid:0) ¯ ∂ J u (cid:1) > o of (cid:0) ¯ ∂ J , λ (cid:1) is an oriented compact branched suborbifold of dimension dim W withboundary ∂ S , see [26, Theorem 4.17] and [28, Section 1.4]. Moreover, S \ (cid:0) B × C P (cid:1) is contained in U and S is equipped with the weight function ϑ : Z −→ Q ∩ [0 , ∞ ) ; u λ (cid:0) ¯ ∂ J u (cid:1) . A neighborhood of ∂ S in S , which is equal to a neighborhood of ∂ M in M , can beidentified with a collar neighborhood of ∂B × C P in B × C P ⊂ W .In order to reach the desired contradiction let Ω be a top-dimensional differentialform on W ′ of total volume R W Ω = 1 that has support in a sufficiently small openball which we require to have closure contained in the interior of B K × C P . Notice, that the evaluation map ev : S → W ′ induces an embedding of B K × C P intothe solution space S , which coincides with the moduli space M along the image B K × C P . Therefore, 1 = Z W Ω = Z ( S ,ϑ ) ev ∗ Ω . Using a compactly supported diffeotopy in the interior of W ′ we can isotope thesupport of Ω into the complement of W K ∪ (cid:0) B K × C P (cid:1) . Denoting by Ω the imageof Ω under pull back along the diffeotopy the difference Ω − Ω has a primitive µ with compact support in the interior of W ′ . Because the support of Ω does notlie in the image of the evaluation map ev : S → W ′ the restriction of ev ∗ Ω to S vanishes. By Stokes theorem [27, Theorem 1.27] Z ( S ,ϑ ) ev ∗ Ω = Z ( ∂ S ,ϑ ) ev ∗ µ . But ev ∗ µ restricted to the boundary ∂ S must vanish because µ has compact sup-port in the complement of ∂W ′ . This is a contradiction that finishes the proof ofTheorem 1.1 for a non-degenerate contact form α .With the same argument one shows that if ( W, ω ) has no concave boundary(
W, ω ) cannot have a convex boundary, either.6.
Proof of Theorem 1.1
We claim that there exist N ∈ N and periodic solutions x , . . . , x N of the Reebvector field R of α that are of (not necessary prim) period T , . . . , T N , resp., suchthat T + . . . + T N < π̺ and [ x ] + . . . + [ x N ] = 0 in the homology of M . WithSection 5.5 and Lemma 4.1 the claim follows for a non-degenerate contact form α . If α is degenerate we find a sequence of contact forms α k on M that are non-degenerate such that α k tends to α in C ∞ , see [22, Proposition 6.1]. Observe, thatthe Reeb vector field R k of α k tends to R in the C ∞ -topology too.First of all we consider a sequence of periodic solutions x k of R k that are ofperiod T k < π̺ . To the reparametrised sequence y k ( t ) = x k ( T k t ) the Arzel`a-Ascoli theorem applies so that a subsequence of y k converges in C ∞ ( R / Z ) to a loop y in M that is tangent to R R and has action T := Z y ∗ α = lim k →∞ Z y ∗ k α k ≤ π̺ , because R y ∗ k α k = T k . The loop x ( t ) := y ( t/T ) is a T -periodic solution of R . Toexpress this circumstance we will say that a subsequence of x k converges to x afterreparametrisation.With this preliminaries we apply the established existence result to the non-degenerate contact form α k for each k . This results in a sequence of periodicsolutions x k , . . . , x kN k of R k that are of period T k , . . . , T kN k , resp., such that theperiods sum up to total acton less than π̺ and the loops represent the zero classin the homology of M . By the flow-box theorem applied to R we get A k > A/ k sufficiently large, where A and A k denote the minimal action of a periodicsolution of R and R k , resp. Hence, N k is bounded from above by 2 π̺ /A so thatwe can assume the number of link components to be constantly equal to K ∈ N .Therefore, we find a subsequence x k , . . . , x kK that converges after reparametrisationto periodic solutions x , . . . , x K of R with period T + . . . + T K ≤ π̺ such that OLYFOLDS AND THE WEINSTEIN CONJECTURE 13 [ x k ] = [ x ] , . . . , [ x kK ] = [ x K ] for k sufficiently large. In particular, the sum [ x ] + . . . + [ x K ] must vanish. The desired claim follows with the exception of a non-strict inequality for the total action of the null-homologous Reeb link. In order toobtain the strict inequality observe that we can realize (cid:0) M, (1 + δ ) α (cid:1) as the concaveboundary of ( W, ω ) for some small δ >
0. Q.E.D.
Remark 6.1. (Weak contact type boundary condition)
In Theorem 1.1 theconvex boundary can be replaced by a J -convex boundary in the sense of [8] or[30], cf. [14, Section 3.2 (C4)] or [29]. This means the boundary components M + oriented as the boundary of ( W, ω ) admit a positive contact form α + such that ina neighborhood of M + there exists an almost complex structure J satisfying thefollowing properties: • J is tamed by ω , • J leaves the kernel ξ + of α + invariant, i.e. ξ + = T M ∩ JT M , and • J restricted to the contact structure ξ + is tamed by d α + .The reason is that there exists a collar neighborhood U of M such that any J -holomorphic sphere that intersects U must be constant. Indeed, U can be chosento be equal to ( − ε, × M for ε > ∂ a to the boundary M equals − JR + , where R + is the Reeb vector field of α + .Shrinking ε > K = 1 /ε the symplectic form d(e Ka α + )tames J on U and − d(de Ka ◦ J ) is positive on J -complex lines, cf. [13, Remark 4.3]. Proof of Corollary 1.6.
In the case (
W, ω ) has no contact type boundary theproof of Theorem 1.1 shows that the evaluation map restricted to the solutionspace S is surjective. Sending the abstract perturbations to zero one obtains afamily of corresponding nodal solutions through each point of W that admit con-verging subsequences in the topology of Z , cf. [26, Theorem 4.17]. The limits areholomorphic nodal spheres. (cid:3) Acknowledgement.
The research of this article was carried out while the au-thors stay at the
Institut for Advanced Studies
Princeton. We would like to thankHelmut Hofer for the invitation and for introducing us to the beauty of polyfolds.We thank Peter Albers, Matthew Strom Borman, Joel Fish, and Hansj¨org Geigesfor enlightening discussions. We thank Chris Wendl and the referees for valuablecomments on the first manuscript.
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