Conformal symplectic Weinstein conjecture and non-squeezing
aa r X i v : . [ m a t h . S G ] F e b Direct link to author’s version
CONFORMAL SYMPLECTIC WEINSTEIN CONJECTURE ANDNON-SQUEEZING
YASHA SAVELYEV
Abstract.
We study here, from the Gromov-Witten theory point of view, some aspects of rigidity oflocally conformally symplectic manifolds, or lcs manifolds for short, which are a natural generalizationof both contact and symplectic manifolds. In particular, we give a first known analogue of the classicalGromov non-squeezing in lcs geometry. Another possible version of non-squeezing related to contactnon-squeezing is also discussed. In a different direction we study Gromov-Witten theory of the lcsmanifold C × S induced by a contact form λ on C , and show that the extended Gromov-Witteninvariant counting certain charged elliptic curves in C × S is identified with the extended classicalFuller index of the Reeb vector field R λ , by extended we mean that these invariants can be ±∞ -valued. Partly inspired by this, we conjecture existence of certain 1-d curves we call Reeb curves incertain lcs manifolds, which we call conformal symplectic Weinstein conjecture, and this is a directextension of the classical Weinstein conjecture. Also using Gromov-Witten theory, we show that theCSW conjecture holds for a C - neighborhood of the induced lcs form on C × S , for C a contactmanifold with contact form whose Reeb flow has non-zero extended Fuller index, e.g. S k +1 withstandard contact form, for which this index is ±∞ . We also show that in some cases the failureof this conjecture implies existence of sky catastrophes for families of holomorphic curves in a lcsmanifold. The latter phenomenon is not known to exist, but if it does, would be analogous to skycatastrophes in dynamical systems discovered by Fuller. Contents
1. Introduction 21.1. Locally conformally symplectic manifolds 21.2. Conformal symplectic Weinstein conjecture 31.3. Non-squeezing 71.4. Sky catastrophes 92. Elements of Gromov-Witten theory of an lcs manifold 103. Rulling out some sky catastrophes and non-squeezing 124. Genus 1 curves in the lcsm C × S and the Fuller index 145. Extended Gromov-Witten invariants and the extended Fuller index 195.1. Preliminaries on admissible homotopies 215.2. Invariance 225.3. Case I, X is finite type 225.4. Case II, X is infinite type 225.5. Case I, X is finite type and X is infinite type 235.6. Case II, X i are infinite type 235.7. Case III, X i are finite type 23A. Fuller index 23B. Remark on multiplicity 246. Acknowledgements 25References 25 Key words and phrases. locally conformally symplectic manifolds, conformal symplectic non-squeezing, Gromov-Witten theory, virtual fundamental class, Fuller index, Seifert conjecture, Weinstein conjecture.Partially supported by PRODEP grant. Introduction
The theory of pseudo-holomorphic curves in symplectic manifolds as initiated by Gromov and Floerhas revolutionized the study of symplectic and contact manifolds. What the symplectic form givesthat is missing for a general almost complex manifold is a priori energy bounds for pseudo-holomorphiccurves a fixed class. On the other hand there is a natural structure which directly generalizes bothsymplectic and contact manifolds, called locally conformally symplectic structure or lcs structure forshort. A locally conformally symplectic manifold or sometimes just lcsm is a smooth 2 n -fold M withan lcs structure: which is a non-degenerate 2-form ω , which is locally diffeomorphic to f · ω st , for some(non-fixed) positive smooth function f , with ω st the standard symplectic form on R n . It is naturalto try to do Gromov-Witten theory for such manifolds. The first problem that occurs is that a priorienergy bounds are gone, as since ω is not necessarily closed, the L -energy can now be unboundedon the moduli spaces of J -holomorphic curves in such a ( M, ω ). Strangely a more acute problem ispotential presence of holomorphic sky catastrophes - given a smooth family { J t } , t ∈ [0 , { ω t } -compatible almost complex structures, we may have a continuous family { u t } of J t -holomorphic curvess.t. energy( u t )
7→ ∞ as t a ∈ (0 ,
1) and s.t. there are no holomorphic curves for t ≥ a . These areanalogues of sky catastrophes discovered by Fuller [10] for closed orbits of dynamical systems.We can tame these problems in certain situations and this is how we arrive at a certain lcs extensionof Gromov non-squeezing. Even when it is impossible to tame these problems we show that there canstill be an extended Gromov-Witten type theory which is analogous to the theory of extended Fullerindex in dynamical systems, [25]. In a very particular situation the relationship with the Fuller indexbecomes perfect as one of the results of this paper obtains the (extended) Fuller index for Reeb vectorfields on a contact manifold C as a certain (extended) genus 1 Gromov-Witten invariant of the Banyagalcsm C × S , see Example 1. The latter also gives a conceptual interpretation for why the Fuller indexis rational, as it is reinterpreted as an (virtual) orbifold Euler number.Inspired by this, we conjecture that certain lcsm’s must poses certain curves that we call Reebcurves, and this is a direct generalization of the Weinstein conjecture, we may call this conformalsymplectic Weinstein conjecture. We prove this CSW conjecture for certain lcs structures C nearbyto Banyaga type lcs structures on C × S . This partly uses the above mentioned connection of Gromov-Witten theory of C × S with the classical Fuller index. Note that Seifert [26] was likewise initiallymotivated by a C neighborhood version of the Seifert conjecture for S k +1 , which he proved. We couldsay that in our case there is more evidence for globalizing, since the original Weinstein conjecture isalready proved, Taubes [29], for C a closed contact three-fold. In addition to the C neighborhoodversion, we also prove a stronger result that relates the CSW conjecture to existence of holomorphicsky catastrophes.Finally, we should exclaim that the Gromov-Witten theory in this story plays a local (in the space ofstructures) role, unless addition global geometric control is obtained. (As is the case for us sometimes.)This is analogous to what happens with Fuller index in dynamical systems. A global lcs invariant,which takes the form of a homology theory, is under development, but many ingredients for this arealready present here. (For example generators, and appropriate almost complex structures.)1.1. Locally conformally symplectic manifolds.
These were originally considered by Lee in [14],arising naturally as part of an abstract study of “a kind of even dimensional Riemannian geometry”,and then further studied by a number of authors see for instance, [1] and [30]. This is a fascinatingobject, a lcsm admits all the interesting classical notions of a symplectic manifold, like Lagrangiansubmanifolds and Hamiltonian dynamics, while at the same time forming a much more flexible class.For example Eliashberg and Murphy show that if a closed almost complex 2 n -fold M has H ( M, R ) = 0then it admits a lcs structure, [5], see also [2].To see the connection with the first cohomology group, let us point out right away the most basicinvariant of a lcs structure ω when M has dimension at least 4: the Lee class, α = α ω ∈ H ( M, R ).This has the property that on the associated α -covering space f M , the lift e ω is globally conformallysymplectic. The class α may be defined as the following Cech 1-cocycle. Let φ a,b be the transition map ONFORMAL SYMPLECTIC WEINSTEIN CONJECTURE AND NON-SQUEEZING 3 for lcs charts φ a , φ b of ( M, ω ). Then φ ∗ a,b ω st = g a,b · ω st for a positive real constant g a,b and { ln g a,b } gives our 1-cocycle. Thus an lcs form is globally conformally symplectic iff its Lee class vanishes.Again assuming M has dimension at least 4, the Lee class α has a natural differential form repre-sentative, called the Lee form and defined as follows. We take a cover of M by open sets U a in which ω = f a · ω a for ω a symplectic, and f a a positive smooth function. Then we have 1-forms d (ln f a ) in each U a which glue to a well defined closed 1-form on M , as shown by Lee. By slight abuse, we denote this1-form, its cohomology class and the Cech 1-cocycle from before all by α . It is moreover immediatethat for an lcs form ω dω = α ∧ ω, for α the Lee form as defined above.As we mentioned lcsm’s can also be understood to generalize contact manifolds. This works asfollows. First we have a natural class of explicit examples of lcsm’s, obtained by starting with asymplectic cobordism (see [5]) of a closed contact manifold C to itself, arranging for the contact formsat the two ends of the cobordism to be proportional (which can always be done) and then gluingtogether the boundary components. As a particular case of this we get Banyaga’s basic example. Example . Let (
C, λ ) be a contact (2 n + 1)-manifold where λ is a contact form, ∀ p ∈ C : λ ∧ λ n ( p ) = 0, and take M = C × S with 2-form ω = d α λ := dλ − α ∧ λ , for α := pr ∗ S dθpr S : C × S → S the projection, and λ likewise the pull-back of λ by the projection C × S → C .The operator d α : Ω k ( M ) → Ω k +1 ( M ) is called the Lichnerowicz differential with respect to a closed1-form α , and satisfies d α ◦ d α = 0 so that we have an associated Lichnerowicz complex.Using above we may then faithfully embed the category of contact manifolds, and contactomorphisminto the category of lcsm’s, and certain lcs morphisms as defined below. Definition 1.1.
A diffeomorphism φ : ( M , ω ) → ( M , ω ) is said to be an lcs map if φ ∗ ω ishomotopic through lcs forms { ω t } , in the same d α Lichnerowicz cohomology class, to ω , where α isthe Lee form of ω as before. In other words, for each t ∈ [0 , , d α ( ddt | t = t ω t ) = 0 . We also define, following Banyaga, conformal symplectomorphisms φ : ( M , ω ) → ( M , ω ) tobe diffeomorphisms satisfying φ ∗ ω = f ω for a smooth positive function f .1.2. Conformal symplectic Weinstein conjecture. An exact lcs structure on M is a pair ( λ, α )with α a closed 1-form, s.t. ω = d α λ is non-degenerate. This determines a generalized distribution V λ : V λ ( p ) = { v ∈ T p M | dλ ( v, · ) = 0 } , which we call the vanishing distribution . And we have a generalized distribution ξ λ , which isdefined to be the ω -orthogonal complement to V λ , which we call co-vanishing distribution . Foreach p ∈ M , V λ ( p ) has dimension at most 2 since dλ − α ∧ λ is non-degenerate. If M n is closed V λ cannot identically vanish since ( dλ ) n cannot be non-degenerate by Stokes theorem. Definition 1.2.
Let ( M, λ, α ) be an exact lcs structure. Then a smooth map o : S → M , s.t. o ∗ ( T S ) ⊂ V λ and s.t. ∀ s ∈ S : o ∗ λ ( s ) = 0 is called a Reeb curve for ( M, λ, α ) . We then have the following basic “conformal symplectic Weinstein conjecture”, later on we state astronger form of this conjecture.
Conjecture 1.
Let M be closed of dimension at least 4, and ( λ, α ) an exact lcs structure on M , thenthere is a Reeb curve for ( M, λ, α ) . The dimension 2 case is special but some version (possibly same version) of the conjecture shouldhold in this case. As one trivial example, given an exact lcs 2-manifold (
M, λ, α ), with dλ = 0 and with α rational, the conjecture holds automatically, just take the Reeb curve to parametrize a componentof a regular fiber of the map f : Σ → S classifying α , that is so that α = q · f ∗ dθ , for q ∈ Q . YASHA SAVELYEV
Lemma 1.3.
Conjecture 1 implies the Weinstein conjecture.Proof.
Let o : S → C × S be a Reeb curve for the Banyaga lcs structure d α λ . Since o ∗ ( T S ) ⊂ V λ ,( pr C ) ∗ ◦ o ∗ ( T S ) ⊂ ker dλ ⊂ T C . Since in addition o ∗ ◦ u ∗ λ is non-vanishing on T S , pr C ◦ o isimmersed in C and is the image of a Reeb orbit. (cid:3) In what follows we use the following C metric on the space L ( M ) of exact lcs structures on M .For ( λ , α ) , ( λ , α ) ∈ L ( M ) define: d C (( λ , α ) , ( λ , α )) = d C ( λ , λ ) + d C ( α , α ) , where d C on the right side is the usual C metric. We say that an exact lcs structure ( M n , λ, α ) is regular if the set: V ( M, λ ) := { p ∈ M | ( dλ ) n ( p ) = 0 } , is a smooth submanifold of M . A regular C -neighborhood of an lcs structure is then a neighborhoodwith respect to d C intersected with the subset of all regular lcs structures.For λ H the standard contact structure on S k +1 , so that its Reeb flow is the Hopf flow, we will call ω H := d α λ H the Hopf lcs structure . Theorem 1.4.
Conjecture 1 holds for a regular C -neighborhood of the Hopf lcs structure ( λ H , α ) on S k +1 × S . This is proved in Section 4. Note that Seifert [26] initially found an analogous existence phenomenonof orbits on S k +1 for a non-singular vector field C -nearby to the Hopf vector field, . And he askedif the nearby condition can be removed, this became known as the Seifert conjecture. This turned outnot to be quite true [13]. Likewise it is natural for us to conjecture that the nearby condition can beremoved and this is the CSW conjecture. In our case this has some additional evidence that we discussin the next section.Directly extending Theorem 1.4 we have the following. Theorem 1.5.
Let C be a closed contact manifold with contact form λ . Let ( λ, α ) be the associatedBanyaga lcs structure on M = C × S , with i ( R λ , β ) = 0 , for some β , where the latter is the extendedFuller index, as described in Appendix A. Then either the conformal symplectic Weinstein conjectureholds for any regular exact lcs structure ( λ ′ , α ′ ) on M , so that ω = d α ′ λ ′ is homotopic through(general) lcs forms { ω t } to ω = d α λ or holomorphic sky catastrophes exist, (these are further discussedin Section 1.4).Example . Take C = S k +1 and λ = λ H , then i ( R λ ,
0) = ±∞ , (sign depends on k ), [25]. Or take C to be unit cotangent bundle of a hyperbolic manifold ( X, g ), λ the associated Louiville form, and( λ, α ) the associated Banyaga lcs structure, in this case i ( R λ , β ) = ± β = 0.To motivate the above conjecture we need pseudo-holomorphic curves in lcs manifolds.1.2.1. Pseudo-holomorphic curves in the lcsm C × S . Banyaga type lcsm’s give immediate examplesof almost complex manifolds where the L energy functional is unbounded on the moduli spaces offixed class J -holomorphic curves, as well as where null-homologous J -holomorphic curves can be non-constant. We are going to see this shortly.Let ( C, λ ) be a closed contact (2 n + 1)-fold with a contact form λ . The Reeb vector field R λ on C is a vector field satisfying dλ ( R λ , · ) = 0 and λ ( R λ ) = 1. We also denote by λ the pull-back of λ by theprojection C × S → C , and by ξ ⊂ T ( C × S ) the distribution ξ ( p ) = ker dλ ( p ).Identifying S = R / Z , S acts on C × S by s · ( x, θ ) = ( x, θ + s ). We take J to be an almostcomplex structure on ξ , which is S invariant, and compatible with dλ . The latter means that g J ( · , · ) := dλ | ξ ( · , J · ) is a J invariant Riemannian metric on the distribution ξ . With more careful analysis we can also likely relax C condition to C condition. ONFORMAL SYMPLECTIC WEINSTEIN CONJECTURE AND NON-SQUEEZING 5
There is an induced almost complex structure J λ on C × S , which is S -invariant, coincides with J on ξ and which satisfies: J λ ( R λ ⊕ { } ( p )) = ddθ ( p ) , where R λ ⊕{ } is the section of T ( C × S ) ≃ T C ⊕ R , corresponding to R λ , and where ddθ ⊂ { }⊕ T S ⊂ C × S denotes the vector field generating the action of S on C × S .We now consider a certain moduli space of holomorphic tori in C × S , which have a certain charge,this charge condition is also studied Oh-Wang [22], and I am grateful to Yong-Geun Oh for relateddiscussions.Partly the reason for introduction of “charge” is that it is now possible for non-constant holomorphiccurves to be null-homologous, so we need additional control. Here is a simple example take S × S with J = J λ , for the λ the standard contact form, then all the Reeb holomorphic tori (as definedfurther below) are null-homologous. In many cases we can just work with homology classes A = 0,but this is inadequate for our setup for conformal symplectic Weinstein conjecture.Let Σ be a complex torus with a chosen marked point z ∈ Σ. These are also known as ellipticcurves. An isomorphism φ : (Σ , z ) → (Σ , z ) is a biholomorphism s.t. φ ( z ) = z . The setof isomorphism classes forms a smooth orbifold M , , with a natural compactification, the Deligne-Mumford compactification M , , by adding a point at infinity corresponding to a nodal curve.Suppose then ( M, ω ) is an lcs manifold,
J ω -compatible almost complex structure, and α the Leeclass corresponding to ω . Assuming for simplicity, at the moment, (otherwise take stable maps) that( M, J ) does not admit non-constant J -holomorphic maps ( S , j ) → ( M, J ), we define: M , , ( J, A )as a set of equivalence classes of tuples ( u, S ), for S = (Σ , z ) ∈ M , , and u : Σ → M a J -holomorphicmap satisfying the charge (1,0) condition : there exists a pair of generators ρ, γ for H (Σ , Z ), suchthat h ρ, u ∗ α i = 1 h γ, u ∗ α i = 0 , and with [ u ] = A . The equivalence relation is ( u , S ) ∼ ( u , S ) if there is an isomorphism φ : S → S s.t. u ◦ φ = u .Note that the charge condition directly makes sense for nodal curves. And it is easy to see that thecharge condition is preserved under Gromov convergence, and obviously a charge (1,0) J -holomorphicmap cannot be constant for any A .By slight abuse we may just denote such an equivalence class above by u , so we may write u ∈M , , ( J, A ), with S implicit.1.2.2. Reeb holomorphic tori in ( C × S , J λ ) . For the almost complex structure J λ as above we haveone natural class of charge (1,0) holomorphic tori in C × S . Let o be a period c Reeb orbit o of R λ ,that is a map: o : S → C,D s o ( s ) = c · R λ ( o ( s )) , for c >
0, and ∀ s ∈ S := R / Z . A Reeb torus u o for o , is the map u o ( s, t ) = ( o ( s ) , t ) ,s, t ∈ S . A Reeb torus is J λ -holomorphic for a uniquely determined holomorphic structure j on T defined by: j ( ∂∂s ) = c ∂∂t . YASHA SAVELYEV
Let e S ( λ ) denote the space of general period λ -Reeb orbits. There is an S action on this space by θ · o ( s ) = o ( s + θ ). Let S ( λ ) := e S ( λ ) /S denote the quotient by this action. We have a map: R : S ( λ ) → M , , ( J λ , A ) , R ( o ) = u o . Proposition 1.6.
The map R is a bijection. So in the particular case of J λ , the domains of elliptic curves in C × S are “rectangular”, that isare quotients of the complex plane by a rectangular lattice, however for a more general almost complexstructure on C × S , tamed by more general lcs forms as we soon consider, the domain almost complexstructure on our curves can in principle be arbitrary, in particular we might have nodal degenerations.Also note that the expected dimension of M , , ( J λ , A ) is 0. It is given by the Fredholm index ofthe operator (4.2) which is 2, minus the dimension of the reparametrization group (for non-nodalcurves) which is 2. That is given an elliptic curve S = (Σ , z ), let G (Σ) be the 2-dimensional groupof biholomorphisms φ of Σ. And given a J -holomorphic map u : Σ → M , (Σ , z, u ) is equivalent to(Σ , φ ( z ) , u ◦ φ ) in M , , ( J λ , A ), for φ ∈ G (Σ).In Theorem 4.5 we relate the (extended) count (Gromov-Witten invariant) of these curves to the(extended) Fuller index, which is reviewed in the Appendix A. This will be one ingredient for thefollowing. Definition 1.7.
Let ( M, λ, α ) be an exact lcs structure, ω = d α λ . We say that an ω -compatible J is admissible if it preserves the vanishing distribution V λ , and the co-vanishing distribution ξ λ , that is J ( V λ ) ⊂ V λ and J ( ξ ( λ )) ⊂ ξ ( λ ) . We call ( M, λ, α, J ) as above a tamed exact lcs structure . The significance of an admissible almost complex structure is the following.
Lemma 1.8.
Let ( M, λ, α, J ) be a tamed exact lcs structure. Then given a smooth u : Σ → M , where Σ is a closed (nodal) Riemann surface, u is J -holomorphic only if image du ( z ) ⊂ V λ ( u ( z )) for all z ∈ Σ , in particular u ∗ dλ = 0 .Proof. We have I = Z Σ u ∗ dλ ≥ J preserves V λ . On the other hand I > I = 0. Since J also preserves ξ λ , this can happen only ifimage du ( z ) ⊂ V λ ( u ( z ))for all z ∈ Σ. (cid:3) Theorem 1.9.
Let M = S k +1 × S , d α λ H the Hopf lcs structure. Then there exists a δ > s.t. forany exact lcs structure ( λ ′ , α ′ ) on M C δ -close to ( λ H , α ) , and J compatible with ω ′ = d α ′ λ ′ and C δ -close to J λ H , there exists an elliptic, charge (1,0), J -holomorphic curve in S k +1 × S . Moreover,if k = 1 and J is admissible then this curve may be assumed to be non-nodal and embedded. The following is to be proved in Section 4.
Theorem 1.10.
Let ( M, λ, α, J ) be a tamed exact lcs structure, if α is rational then every non-constant J -holomorphic curve u : Σ → M contains a Reeb curve, meaning that there is a S ≃ S ⊂ Σ s.t. u | S is a Reeb curve. If moreover Σ is smooth, connected and immersed then Σ ≃ T . In a sense the above discussion tells us that J -holomorphic curves strictify Reeb curves, in the sensethat Reeb curves satisfy a partial differential relation while J -holomorphic curves satisfy a partialdifferential equation, but given a solution of the former we also the latter. Strictifying could be helpfulbecause the “strict” objects may possibly be counted in some way.It makes sense to try to partially strictify Reeb curves more directly. It is in fact an equivalence of the corresponding topological action groupoids, but we do not need this explicitly.
ONFORMAL SYMPLECTIC WEINSTEIN CONJECTURE AND NON-SQUEEZING 7
Definition 1.11.
Let ( M, λ, α ) be an exact lcs structure, Σ a closed possibly nodal Riemann surface.A smooth map u : Σ → M is called a Reeb 2-curve if u ∗ ( T Σ) ⊂ V ( λ ) and if there is a smooth map o : S → Σ s.t. ∀ s ∈ S : o ∗ u ∗ λ ( s ) = 0 . By Theorem 1.10 J -holomorphic curves give examples of Reeb 2-curves. More generally, for u satisfying the first condition, the second condition is satisfied for example if α is rational and u ∗ α ∧ u ∗ λ is symplectic except at finitely many points. The proofs of theorems 1.4, 1.5 actually produce Reeb 2-curves, through which we then deduce existence of Reeb curves. So it makes sense to further conjecturethe following. Conjecture 2.
Let M be closed, of dimension at least 4, and ω an exact lcs form on M whose Leeform α is rational, then there is a Reeb 2-curve in M . The above conjectures are not just a curiosity. In contact geometry, rigidity is based on existencephenomena of Reeb orbits, and lcs manifolds may be understood as generalized contact manifolds. Toattack rigidity questions in lcs geometry, like Question 2 further below, we need an analogue of Reeborbits, we propose that this analogue is Reeb curves.1.2.3.
Connection with the extended Fuller index.
One of the main ingredients for the above is aconnection of extended Fuller index with certain extended Gromov-Witten invariants. If β is a freehomotopy class of a loop in C set A β = [ β ] × [ S ] ∈ H ( C × S ) . Then we have:
Theorem 1.12.
Suppose that λ is a contact form on a closed manifold C , so that its Reeb flow isdefinite type, see Appendix A, then GW , ( A β , J λ )([ M , ] ⊗ [ C × S ]) = i ( R λ , β ) , where both sides are certain extended rational numbers Q ⊔{±∞} valued invariants, so that in particularif either side does not vanish then there are λ Reeb orbits in class β . What about higher genus invariants of C × S ? Following the proof of Proposition 1.6, it is nothard to see that all J λ -holomorphic curves must be branched covers of Reeb tori. If one can show thatthese branched covers are regular when the underlying tori are regular, the calculation of invariantswould be fairly automatic from this data, see [34], [32] where these kinds of regularity calculation aremade.1.3. Non-squeezing.
One of the most fascinating early results in symplectic geometry is the so calledGromov non-squeezing theorem appearing in the seminal paper of Gromov [12]. The most well knownformulation of this is that there does not exist a symplectic embedding B R → D ( r ) × R n − for R > r , with B R the standard closed radius R ball in R n centered at 0. Gromov’s non-squeezing is C persistent in the following sense.We say that a symplectic form ω on M × N is split if ω = ω ⊕ ω for symplectic forms ω , ω on M respectively N . Theorem 1.13.
Given
R > r , there is an ǫ > s.t. for any symplectic form ω ′ on S × T n − C -close to a split symplectic form ω and satisfying h ω, A i = πr , A = [ S ] ⊗ [ pt ] , there is no symplectic embedding φ : B R ֒ → ( S × T n − , ω ′ ) . On the other hand it is natural to ask:
Question . Given
R > r and every ǫ > ω ′ on S × T n − C or even C ∞ ǫ -close to a split symplectic form ω , satisfying h ω, A i = πr , and s.t. thereis an embedding φ : B R ֒ → S × T n − , with φ ∗ ω ′ = ω st ? YASHA SAVELYEV
The above theorem follows immediately by Gromov’s argument in [12], we shall give a certainextension of this theorem for lcs forms. One may think that recent work of M¨uller [28] may be relatedto the question above and our theorem below. But there seems to be no obvious such relation aspull-backs by diffeomorphisms of nearby forms may not be nearby. Hence there is no way to go fromnearby embeddings that we work with to ǫ -symplectic embeddings of M¨uller.We first give a ridid notion of a morphism of lcsm’s. Definition 1.14.
Given a pair of lcsm ’s ( M i , ω i ) , i = 0 , , we say that f : M → M is a sym-plectomorphism if f ∗ ω = ω . A symplectic embedding then as usual is an embedding by asymplectomorphism. Let now M = S × T n − , with ω a split symplectic form on M . The following theorem says that itis impossible to have certain symplectic embeddings into ( M, ω ′ ) with ω ′ C nearby to ω , even in theabsence of any volume obstruction. So that we have a first basic rigidity phenomenon for lcs structures.Note that in what follows we take a certain natural metric C topology T on the space of general lcsforms, defined in Section 3, which is finer than the standard C metric topology on the space of forms,cf. [1, Section 6]. The corresponding metric is denoted d .We have a real codimension 1 hypersurfacesΣ i = S × ( S × . . . × S × { pt } × S × . . . × S ) ⊂ M, where the singleton { pt } ⊂ S replaces the i ’th factor of T n − = S × . . . × S . These hypersurfaces arenaturally folliated by symplectic submanifolds diffeomorphic to S × T n − . We denote by T fol Σ i ⊂ T M , the distribution of all tangent vectors tangent to the leaves of the above mentioned folliation.
Theorem 1.15.
Let ω be a split symplectic form on M = S × T n − , and A as above with h ω, A i = πr . Let R > r , then there is an ǫ > s.t. if { ω t } is a T -continuous family of lcs forms on M , with d ( ω t , ω ) < ǫ for all t , then there is no symplectic embedding φ : ( B R , ω st ) ֒ → ( M, ω ) − ∪ i Σ i . The latter is a full-volume subspace diffeomorphic to S × R n − . More generally there is no symplecticembedding φ : ( B R , ω st ) ֒ → ( M, ω ) , s.t φ ∗ j preserves the bundles T fol Σ i , for j the standard almost complex structure on B R . We note that the image of the embedding φ would be of course a symplectic submanifold of ( M, ω ).However it could be highly distorted, so that it might be impossible to complete φ ∗ ω st to a symplecticform on M nearby to ω , so that it is impossible to deduce the above result directly from symplecticGromov non-squeezing. We also note that it is certainly possible to have a nearby volume preservingas opposed to lcs embedding which satisfies all other conditions, since as mentioned ( M, ω ) − ∪ i Σ i is a full ω -volume subspace diffeomorphic to S × R n − . This extension of Theorem 1.13 may beoptimal, since the ǫ condition cannot be removed from Theorem 1.13.1.3.1. Toward direct generalization of contact non-squeezing.
What about non-squeezing for lcs mapsas in Definition 1.1? We can try a direct generalization of contact non-squeezing of Eliashberg-Polterovich [4], and Fraser in [6]. Specifically let R n × S be the prequantization space of R n ,or in other words the contact manifold with the contact form dθ − λ , for λ = ( ydx − xdy ). Let B R now denote the open radius R ball in R n . Question . If R ≥ lcs embedding map φ : R n × S × S → R n × S × S , so that φ ( U ) ⊂ U , for U := B R × S × S and U the topological closure? ONFORMAL SYMPLECTIC WEINSTEIN CONJECTURE AND NON-SQUEEZING 9
Sky catastrophes.
This final introductory section will be of a more technical nature. Thefollowing is well known.
Theorem 1.16. [ [20] , [31] ] Let ( M, J ) be a compact almost complex manifold, and u : ( S , j ) → M a J -holomorphic map. Given a Riemannian metric g on M , there is an ~ = ~ ( g, J ) > s.t. if e g ( u ) < ~ then u is constant, where e g is the L -energy functional, e g ( u ) = energy g ( u ) = Z S | du | dvol. Using this we get the following (trivial) extension of Gromov compactness to lcs setting. Let M g,n ( J, A ) = M g,n ( M, J, A )denote the moduli space of isomorphism classes of class A , J -holomorphic curves in M , with domaina genus g closed Riemann surface, with n marked labeled points. Here an isomorphism between u : Σ → M , and u : Σ → M is a biholomorphism of marked Riemann surfaces φ : Σ → Σ s.t. u ◦ φ = u . Theorem 1.17.
Let ( M, J ) be an almost complex manifold. Then M g,n ( J, A ) has a pre-compactification M g,n ( J, A ) , by Kontsevich stable maps, with respect to the natural metrizable Gromov topology see for instance [20] , for genus 0 case. Moreover given E > , the subspace M g,n ( J, A ) E ⊂ M g,n ( J, A ) consisting ofelements u with e ( u ) ≤ E is compact, where e is the L energy with respect to an auxillary metric. Inother words e is a proper function. Thus the most basic situation where we can talk about Gromov-Witten “invariants” of (
M, J ) iswhen the energy function is bounded on M g,n ( J, A ), and we shall say that J is bounded (in class A ),later on we generalize this in terms of what we call finite type . In this case M g,n ( J, A ) is compact,and has a virtual moduli cycle as in the original approach of Fukaya-Ono [9], or the more algebraicapproach [23]. So we may define functionals:(1.18) GW g,n ( ω, A, J ) : H ∗ ( M g,n ) ⊗ H ∗ ( M n ) → Q , where M g,n denotes the compactified moduli space of Riemann surfaces. Of course symplectic man-ifolds with any tame almost complex structure is one class of examples, another class of examplescomes from some locally conformally symplectic manifolds.Given a continuous in the C ∞ topology family { J t } , t ∈ [0 ,
1] we denote by M g ( { J t } , A ) the spaceof pairs ( u, t ), u ∈ M g ( J t , A ). Definition 1.19.
We say that a continuous family { J t } on a compact manifold M has a holomorphicsky catastrophe in class A if there is an element u ∈ M g ( J i , A ) , i = 0 , which does not belong toany open compact (equivalently energy bounded) subset of M g ( { J t } , A ) . Let us slightly expand this definition. If M g ( { J t } , A ) is locally connected, so that the connectedcomponents are open, then we have a sky catastrophe in the sense above if and only if there is a u ∈ M g ( J i , A ) which has a non-compact connected component in M g ( { J t } , A ).At this point in time there are no known examples of families { J t } with sky catastrophes, cf. [10]. Question . Do sky catastrophes exist?Really what we are interested in is whether they exist generically. The author’s opinion is that theymay appear even generically. However, if we further constrain the geometry to exact lcs structures asin Section 1.2, then the question becomes much more subtle, see also [25] for a related discussion onpossible obstructions to sky catastrophes.Related to this we have the following technical result that will be used in the proof of non-squeezingdiscussed above.
Theorem 1.20.
Let M be closed and { ω t } , t ∈ [0 , , a continuous (with respect to the topology T )family of lcs forms on M . Let { J t } be a Frechet smooth family of almost complex structures, with J t compatible with ω t for each t . Let A ∈ H ( M ) be fixed, and let D ⊂ f M , with π : f M → M theuniversal cover of M , be a fundamental domain, and K := D its topological closure. Suppose thatfor each t , and for every x ∈ ∂K there is a e J t -holomorphic hyperplane H x through x , with H x ⊂ K ,such that π ( H x ) ⊂ M is a closed submanifold and such that A · π ∗ ([ H x ]) ≤ . Then { J t } has no skycatastrophes in class A . If holomorphic sky catastrophes are discovered, this would be a very interesting discovery. Theoriginal discovery by Fuller [10] of sky catastrophes in dynamical systems is one of the most importantin dynamical systems, see also [27] for an overview.2.
Elements of Gromov-Witten theory of an lcs manifold
Suppose (
M, J ) is a compact almost complex manifold, where the almost complex structures J areassumed throughout the paper to be C ∞ , and let N ⊂ M g,k ( J, A ) be an open compact subset withenergy positive on N . The latter condition is only relevant when A = 0. We shall primarily refer inwhat follows to work of Pardon in [23], only because this is what is more familiar to the author, dueto greater comfort with algebraic topology. But we should mention that the latter is a follow up to aprofound theory that is originally created by Fukaya-Ono [9], and later expanded with Oh-Ohta [8].The construction in [23] of implicit atlas, on the moduli space M of curves in a symplectic manifold,only needs a neighborhood of M in the space of all curves. So more generally if we have an almostcomplex manifold and an open compact component N as above, this will likewise have a naturalimplicit atlas, or a Kuranishi structure in the setup of [9]. And so such an N will have a virtualfundamental class in the sense of Pardon [23], (or in any other approach to virtual fundamental cycle,particularly the original approach of Fukaya-Oh-Ohta-Ono). This understanding will be used in otherparts of the paper, following Pardon for the explicit setup. We may thus define functionals:(2.1) GW g,n ( N, A, J ) : H ∗ ( M g,n ) ⊗ H ∗ ( M n ) → Q . How do these functionals depend on
N, J ? Lemma 2.2.
Let { J t } , t ∈ [0 , be a Frechet smooth family. Suppose that e N is an open compactsubset of the cobordism moduli space M g,n ( { J t } , A ) and that the energy function is positive on e N , (thelatter only relevant when A = 0 ). Let N i = e N ∩ (cid:0) M g,n ( J i , A ) (cid:1) , then GW g,n ( N , A, J ) = GW g,n ( N , A, J ) . In particular if GW g,n ( N , A, J ) = 0 , there is a class A J -holomorphic stable map in M .Proof of Lemma 2.2. We may construct exactly as in [23] a natural implicit atlas on e N , with boundary N op ⊔ N , ( op denoting opposite orientation). And so GW g,n ( N , A, J ) = GW g,n ( N , A, J ) , as functionals. (cid:3) The most basic lemma in this setting is the following, and we shall use it in the following section.
Definition 2.3. An almost symplectic pair on M is a tuple ( M, ω, J ) , where ω is a non-degenerate2-form on M , and J is ω -compatible, meaning that ω ( · , J · ) defines J -invariant Riemannian metric.When ω is lcs we call such a pair an lcs pair . Definition 2.4.
We say that a pair of almost symplectic pairs ( ω i , J i ) are δ -close , if { ω i } are C δ -close, and { J i } are C δ -close, i = 0 , . Define this similarly for a pair ( g i , J i ) for g a Riemannianmetric and J an almost complex structure. ONFORMAL SYMPLECTIC WEINSTEIN CONJECTURE AND NON-SQUEEZING 11
Definition 2.5.
For an almost symplectic pair ( ω, J ) on M , and a smooth map u : Σ → M define: e ω ( u ) = Z Σ u ∗ ω. By an elementary calculation this coincides with the L g J -energy of u , for g J ( · , · ) = ω ( · , J · ). Thatis e ω ( u ) = e g J ( u ). In what follows by f − ( a, b ), with f a function, we mean the preimage by f of theopen set ( a, b ). Lemma 2.6.
Given a compact M and an almost symplectic pair ( ω, J ) on M , suppose that N ⊂M g,n ( J, A ) is a compact and open component which is energy isolated meaning that N ⊂ (cid:0) U = e − ω ( E , E ) (cid:1) ⊂ (cid:0) V = e − ω ( E − ǫ, E + ǫ ) (cid:1) , with ǫ > , E > and with V ∩ M g,n ( J, A ) = N . Suppose also that GW g,n ( N, J, A ) = 0 . Then thereis a δ > s.t. whenever ( ω ′ , J ′ ) is a compatible almost symplectic pair δ -close to ( ω, J ) , there exists u ∈ M g,n ( J ′ , A ) = ∅ , with E < e ω ′ ( u ) < E . Proof of Lemma 2.6.
Lemma 2.7.
Given a Riemannian manifold ( M, g ) , and J an almost complex structure, suppose that N ⊂ M d,n ( J, A ) is a compact and open component which is energy isolated meaning that N ⊂ (cid:0) U = e − g ( E , E ) (cid:1) ⊂ (cid:0) V = e − g ( E − ǫ, E + ǫ ) (cid:1) , with ǫ > , E > , and with V ∩ M g,n ( J, A ) = N . Then there is a δ > s.t. whenever ( g ′ , J ′ ) is δ -close to ( g, J ) if u ∈ M g,n ( J ′ , A ) and E − ǫ < e g ′ ( u ) < E + ǫ then E < e g ′ ( u ) < E . Proof of Lemma 2.7.
Suppose otherwise then there is a sequence { ( g k , J k ) } converging to ( g, J ), anda sequence { u k } of J k -holomorphic stable maps satisfying E − ǫ < e g k ( u k ) ≤ E or E ≤ e g k ( u k ) < E + ǫ. By Gromov compactness, specifically theorems [20, B.41, B.42], we may find a Gromov convergentsubsequence { u k j } to a J -holomorphic stable map u , with E − ǫ ≤ e g ( u ) ≤ E or E ≤ e g ( u ) ≤ E + ǫ. But by our assumptions such a u does not exist. (cid:3) Lemma 2.8.
Let M be compact, and let ( M, ω, J ) be an almost symplectic triple, so that N ⊂M g,n ( J, A ) is exactly as in the lemma above with respect to some ǫ > . Then, there is a δ ′ > s.t. the following is satisfied. Let ( ω ′ , J ′ ) be δ ′ -close to ( ω, J ) , then there is a continuous in the C ∞ topology family of almost symplectic pairs { ( ω t , J t ) } , ( ω , J ) = ( g, J ) , ( ω , J ) = ( g ′ , J ′ ) s.t. there isopen compact subset e N ⊂ M g,n ( { J t } , A ) , and with e N ∩ M ( J, A ) = N. Moreover if ( u, t ) ∈ e N then E < e g t ( u ) < E . Proof.
For ǫ as in the hypothesis, let δ be as in Lemma 2.7. Lemma 2.9.
Given a δ > there is a δ ′ > s.t. if ( ω ′ , J ′ ) is δ ′ -near ( ω, J ) there is an interpolating,continuous in C ∞ topology family { ( ω t , J t ) } with ( ω t , J t ) δ -close to ( ω, J ) for each t .Proof. Let { g t } be the family of metrics on M given by the convex linear combination of g = g ω J , g ′ = g ω ′ ,J ′ . Clearly g t is δ ′ -close to g for each t . Likewise the family of 2 forms { ω t } given by the convexlinear combination of ω , ω ′ is non-degenerate for each t if δ ′ was chosen to be sufficiently small and is δ ′ -close to ω = ω g,J for each moment.Let ret : M et ( M ) × Ω( M ) → J ( M )be the “retraction map” (it can be understood as a retraction followed by projection) as defined in[19, Prop 2.50], where M et ( M ) is space of metrics on M , Ω( M ) the space of 2-forms on M , and J ( M )the space of almost complex structures. This map has the property that the almost complex structure ret ( g, ω ) is compatible with ω , and that ret ( g J , ω ) = J for g J = ω ( · , J · ). Then { ( ω t , ret ( g t , ω t ) } isa compatible family. As ret is continuous in C -topology, δ ′ can be chosen so that { ret t ( g t , ω t } are δ -nearby. (cid:3) Let δ ′ be chosen with respect to δ as in the above lemma and { ( ω t , J t ) } be the corresponding family.Let e N consist of all elements ( u, t ) ∈ M ( { J t } , A ) s.t. E − ǫ < e ω t ( u ) < E + ǫ. Then by Lemma 2.7 for each ( u, t ) ∈ e N , we have: E < e ω t ( u ) < E . In particular e N must be closed, it is also clearly open, and is compact as the energy e is a properfunction, as discussed. (cid:3) To finish the proof of the main lemma, let N be as in the hypothesis, δ ′ as in Lemma 2.8, and e N asin the conclusion to Lemma 2.8, then by Lemma 2.2 GW g,n ( N , J ′ , A ) = GW g,n ( N, J, A ) = 0 , where N = e N ∩ M g,n ( J , A ). So the conclusion follows. (cid:3) While not having sky catastrophes gives us a certain compactness control, the above is not immediatebecause we can still in principle have total cancellation of the infinitely many components of the modulispace M , ( J λ , A ). In other words a virtual 0-dimension Kuranishi space M , ( J λ , A ), with an infinitenumber of compact connected components, can certainly be null-cobordant, by a cobordism all of whosecomponents are compact. So we need a certain additional algebraic and geometric control to precludesuch a total cancellation. Proof of Theorem 1.17. (Outline, as the argument is standard.) Suppose that we have a sequence u k of J -holomorphic maps with L -energy ≤ E . By [20, 4.1.1], a sequence u k of J -holomorphic curveshas a convergent subsequence if sup k || du k || L ∞ < ∞ . On the other hand when this condition does nothold rescaling argument tells us that a holomorphic sphere bubbles off. The quantization Theorem1.16, then tells us that these bubbles have some minimal energy, so if the total energy is capped by E , only finitely many bubbles may appear, so that a subsequence of u k must converge in the Gromovtopology to a Kontsevich stable map. (cid:3) Rulling out some sky catastrophes and non-squeezing
Let M be a smooth manifold of dimension at least 4, which is an assumption as well for the rest ofthe paper, as dimension 2 case is special. The C k metric topology T k on the set LCS ( M ) of smoothlcs 2-forms on M is defined with respect to the following metric. ONFORMAL SYMPLECTIC WEINSTEIN CONJECTURE AND NON-SQUEEZING 13
Definition 3.1.
Fix a Riemannian metric g on M . For ω , ω ∈ LCS ( M ) define d k ( ω , ω ) = d C k ( ω , ω ) + d C k ( α , α ) , for α i the Lee forms of ω i and d C k the usual C k metrics induced by g . The following characterization of convergence will be helpful.
Lemma 3.2.
Let M be compact and let { ω k } ⊂ LCS ( M ) be a sequence T converging to a symplecticform ω . Denote by { e ω k } the lift sequence on the universal cover f M . Then there is a sequence { ω sympk } of symplectic forms on f M , and a sequence { f k } of positive functions pointwise converging to , suchthat e ω k = f k ω sympk .Proof. We may assume that M is connected. Let α k be the Lee form of ω k , and g k functions on f M defined by g k ([ p ]) = R [0 , p ∗ α k , where the universal cover f M is understood as the set equivalenceclasses of paths p starting at x ∈ M , with a pair p , p equivalent if p (1) = p (1) and p − · p isnull-homotopic, where · is the path concatenation.Then we get: d e ω k = dg k ∧ e ω k , so that if we set f k := e g k then d ( f − k e ω k ) = 0 . Since by assumption | α k | C →
0, then pointwise g k → f k →
1, so that if we set e ω sympk := f − k e ω k then we are done. (cid:3) Proof of Theorem 1.20.
We shall actually prove a stronger statement that there is a universal (for all t ) energy bound from above for class A , J t -holomorphic curves. Lemma 3.3.
Let
M, K be as in the statement of the theorem, and A ∈ H ( M ) fixed. Let ( ω, J ) bea compatible lcs pair on M such that for every x ∈ ∂K there is a e J -holomorphic (real codimension2) hyperplane H x ⊂ K ⊂ f M through x , such that π ( H x ) ⊂ M is a closed submanifold and such that A · [ π ( H x )] ≤ . Then any genus , J -holomorphic class A curve u in M has a lift e u with image in K .Proof. For u as in the statement, let e u be a lift intersecting the fundamental domain D , (as in thestatement of main theorem). Suppose that e u intersects ∂K , otherwise we already have image e u ⊂ K ◦ ,for K ◦ the interior, since image e u is connected (any by elementary topology). Then e u intersects u x asin the statement, for some x . So u is a J -holomorphic map intersecting the closed hyperplane π ( H x )with A · [ π ( H x )] ≤
0. By positivity of intersections, [20], image u ⊂ π ( H x ), and so image e u ⊂ H x . Andso image e u ⊂ ∂K . (cid:3) Suppose otherwise, then there is a sequence { u k } ∞ k =1 , u k : S → M , of J t k -holomorphic class A curves, with Z S u ∗ k ω t k → ∞ , as k → ∞ . We may assume that t k is convergent to t ′ ∈ [0 , u t has a lift e u t contained in a compact K ⊂ f M . Then for every ǫ > N so that for k > N we have: Z S e u ∗ k ω t k ≤ C k h e ω sympt k , A i where e ω t k = f k e ω sympk , for e ω sympk symplectic on f M , and f k : f M → R positive functions constructed asin the proof of Lemma 3.2, and where C k = max K f k . Thenlim k →∞ Z S e u ∗ k ω t k ≤ C h e ω sympt ′ , A i , where e ω t ′ = f t ′ e ω sympt ′ , for e ω sympt ′ symplectic, and C = max K f t ′ . So we have obtained a contradiction. (cid:3) Proof of Theorem 1.15.
Fix an ǫ ′ > ω on M , C ǫ ′ -close to ω , is non-degenerateand is non-degenerate on the leaves of the folliation of each Σ i , discussed prior to the formulation ofthe theorem. Suppose by contradiction that for every ǫ > { ω t } of lcs forms,with ω = ω , such that ∀ t : d ( ω t , ω ) < ǫ and such that there exists a symplectic embedding φ : B R ֒ → ( M, ω ) , satisfying conditions of the statement of the theorem. Take ǫ < ǫ ′ , and let { ω t } be as in the hypothesisabove. In particular ω t is an lcs form for each t , and is non-degenerate on Σ i . Extend φ ∗ j to an ω -compatible almost complex structure J on M , preserving T fol Σ i . We may then extend this toa family { J t } of almost complex structures on M , s.t. J t is ω t -compatible for each t , with J is thestandard split complex structure on M and such that J t preserves T Σ i for each i . The latter conditioncan be satisfied since Σ i are ω t -symplectic for each t . (For construction of { J t } use for example themap ret from Lemma 2.9). When the image of φ does not intersect ∪ i Σ i these conditions can betrivially satisfied.Then the family { ( ω t , J t ) } satisfies the hypothesis of Theorem 1.20, and so has no sky catastrophesin class A . In addition if N = M , ( J , A ) (which is compact since J is tamed by the symplectic form ω ) then GW , ( N, A, J )([ pt ]) = 1 , as this is a classical, well known invariant, whose calculation already appears in [12]. Consequently byLemma 2.2 there is a class A J -holomorphic curve u passing through φ (0).By Lemma 3.3 we may choose a lift e u to f M , with homology class [ e u ] also denoted by A , of each u so that the image of e u is contained in a compact set K ⊂ f M , (independent of choice of ǫ, { J t } ). Let e ω sympt and f t be as in Lemma 3.2, then by this lemma for every δ > ǫ > d ( ω , ω ) < ǫ then d C ( e ω symp , e ω symp ) < δ on K .Since h e ω symp , A i = πr , if δ above is chosen to be sufficiently small then | Z S u ∗ ω − πr | ≤ | max K f h e ω symp , A i − π · r | < πR − πr , since lim ǫ → |h e ω symp , A i − π · r | = |h e ω symp , A i − π · r | = 0 , and since d ( ω , ω ) → ⇒ max K f → . In particular we get that R S u ∗ ω < πR .We may then proceed as in the now classical proof of Gromov [12] of the non-squeezing theorem toget a contradiction and finish the proof. More specifically φ − (image φ ∩ image u ) is a minimal surfacein B R , with boundary on the boundary of B R , and passing through 0 ∈ B R . By construction it hasarea strictly less then πR which is impossible by the classical monotonicity theorem of differentialgeometry. (cid:3) Genus 1 curves in the lcsm C × S and the Fuller index Proof of Proposition 1.6.
Suppose we a have a curve without spherical nodal components u ∈ M , , ( J λ , A ) , represented by u : Σ → M = C × S . Since by Lemma 1.8, u ∗ ( T Σ) ⊂ V λ , we get that( pr C ◦ u ) ∗ ( T Σ) ⊂ ker dλ ⊂ T C, where pr C : C × S → C is the projection. Note that this implies in particular that Σ is non-nodal.By charge (1,0) condition pr S ◦ u is surjective and so by the Sard theorem we have a regular value θ ∈ S , so that u − ◦ pr − S ( θ ) contains an embedded circle S ⊂ Σ, where pr S : C × S → S is theprojection. Now d ( pr S ◦ u ) is surjective along T (Σ) | S , which means, since u is J λ -holomorphic, that pr C ◦ u | S has non-vanishing differential. From this and the discussion above it follows that image of pr C ◦ u is the image of some Reeb orbit. Consequently, by assumption that u has charge (1 , u isequivalent to a Reeb torus for a uniquely determined Reeb orbit o u . ONFORMAL SYMPLECTIC WEINSTEIN CONJECTURE AND NON-SQUEEZING 15
The statement of the lemma follows when u has no spherical nodal components. On the other handnon-constant J λ -holomorphic spheres are impossible, which can be seen as follows. Any such a J λ -holomorphic sphere u lifts to the covering space f M = C × R of M , as a e J -holomorphic map e u , where e J is the lift of J λ , and is compatible with the lift e ω of ω = d α λ . On the other had e ω = dλ − dt ∧ λ isconformally symplectomorphic to the exact symplectic form d ( e t λ ), for t : C × R → R the projection.So that e u is constant by Stokes theorem. (cid:3) Proposition 4.1.
Let ( C, ξ ) be a general contact manifold. If λ is a non-degenerate contact 1-form for ξ then all the elements of M , , ( J λ , A ) are regular curves. Moreover, if λ is degenerate then for a period c Reeb orbit o the kernel of the associated real linear Cauchy-Riemann operator for the Reeb torus u o is naturally identified with the 1-eigenspace of φ λc, ∗ - the time c linearized return map ξ ( o (0)) → ξ ( o (0)) induced by the R λ Reeb flow.Proof.
We already known that all u ∈ M , , ( J λ , A ) are equivalent to Reeb tori. In particular haverepresentation by a J λ -holomorphic map u : ( T , j ) → ( Y = C × S , J λ ) . Since each u is immersed we may naturally get a splitting u ∗ T ( Y ) ≃ N × T ( T ), using the g J metric,where N → T denotes the pull-back, of the g J -normal bundle to image u , and which is identified withthe pullback of the distribution ξ λ on Y , (which we also call the co-vanishing distribution).The full associated real linear Cauchy-Riemann operator takes the form:(4.2) D Ju : Ω ( N ⊕ T ( T )) ⊕ T j M , → Ω , ( T ( T ) , N ⊕ T ( T )) . This is an index 2 Fredholm operator (after standard Sobolev completions), whose restriction toΩ ( N ⊕ T ( T )) preserves the splitting, that is the restricted operator splits as D ⊕ D ′ : Ω ( N ) ⊕ Ω ( T ( T )) → Ω , ( T ( T ) , N ) ⊕ Ω , ( T ( T ) , T ( T )) . On the other hand the restricted Fredholm index 2 operatorΩ ( T ( T )) ⊕ T j M , → Ω , ( T ( T )) , is surjective by classical Teichmuller theory, see also [33, Lemma 3.3] for a precise argument in thissetting. It follows that D Ju will be surjective if the restricted Fredholm index 0 operator D : Ω ( N ) → Ω , ( N ) , has no kernel.The bundle N is symplectic with symplectic form on the fibers given by restriction of u ∗ dλ , andtogether with J λ this gives a Hermitian structure on N . We have a linear symplectic connection A on N , which over the slices S × { t } ⊂ T is induced by the pullback by u of the linearized R λ Reeb flow. Specifically the A -transport map from the fiber N ( s ,t ) to the fiber N ( s ,t ) over the path[ s , s ] × { t } ⊂ T , is given by ( u ∗ | N ( s ,t ) ) − ◦ ( φ λc ( s − s ) ) ∗ ◦ u ∗ | N ( s ,t ) , where φ λc ( s − s ) is the time c · ( s − s ) map for the R λ Reeb flow, where c is the period of the Reeborbit o u , and where u ∗ : N → T Y denotes the natural map, (it is the universal map in the pull-backdiagram.)The connection A is defined to be trivial in the θ direction, where trivial means that the paralleltransport maps are the id maps over θ rays. In particular the curvature R A , understood as a liealgebra valued 2-form, of this connection vanishes. The connection A determines a real linear CRoperator on N in the standard way (take the complex anti-linear part of the vertical differential of asection). It is elementary to verify from the definitions that this operator is exactly D .We have a differential 2-form Ω on the total space of N defined as follows. On the fibers T vert N ,Ω = u ∗ ω , for ω = d α λ , and for T vert N ⊂ T N denoting the vertical tangent space, or subspace of vectors v with π ∗ v = 0, for π : N → T the projection. While on the A -horizontal distribution Ω isdefined to vanish. The 2-form Ω is closed, which we may check explicitly by using that R A vanishesto obtain local symplectic trivializations of N in which A is trivial. Clearly Ω must vanish on the0-section since it is a A -flat section. But any section is homotopic to the 0-section and so in particularif µ ∈ ker D then Ω vanishes on µ . But then since µ ∈ ker D , and so its vertical differential is complexlinear, it must follow that the vertical differential vanishes, since Ω( v, J λ v ) >
0, for 0 = v ∈ T vert N and so otherwise we would have R µ Ω >
0. So µ is A -flat, in particular the restriction of µ over allslices S × { t } is identified with a period c orbit of the linearized at o R λ Reeb flow, and which doesnot depend on t as A is trivial in the t variable. So the kernel of D is identified with the vector spaceof period c orbits of the linearized at o R λ Reeb flow, as needed. (cid:3)
Proposition 4.3.
Let λ be a contact form on a (2 n + 1) -fold C , and o a non-degenerate, period c , R λ -Reeb orbit, then the orientation of [ u o ] induced by the determinant line bundle orientation of M , , ( J λ , A ) , is ( − CZ ( o ) − n , which is sign Det(Id | ξ ( o (0)) − φ λc, ∗ | ξ ( o (0)) ) . Proof of Proposition 4.3.
Abbreviate u o by u . Let N → T be associated to u as in the proof ofProposition 4.1. Fix a trivialization φ of N induced by any trivialization of the contact distribution ξ along o in the obvious sense: N is the pullback of ξ along the composition T → S o −→ C. Let the symplectic connection A on N be defined as before. Then the pullback connection A ′ := φ ∗ A on T × R n is a connection whose parallel transport paths p t : [0 , → Symp( R n ), along the closedloops S × { t } , are paths starting at 1, and are t independent. And so the parallel transport path of A ′ along { s } × S is constant, that is A ′ is trivial in the t variable. We shall call such a connection A ′ on T × R n induced by p .By non-degeneracy assumption on o , the map p (1) has no 1-eigenvalues. Let p ′′ : [0 , → Symp( R n )be a path from p (1) to a unitary map p ′′ (1), with p ′′ (1) having no 1-eigenvalues, and s.t. p ′′ has onlysimple crossings with the Maslov cycle. Let p ′ be the concatenation of p and p ′′ . We then get CZ ( p ′ ) −
12 sign Γ( p ′ , ≡ CZ ( p ′ ) − n ≡ , since p ′ is homotopic relative end points to a unitary geodesic path h starting at id , having regularcrossings, and since the number of negative, positive eigenvalues is even at each regular crossing of h by unitarity. Here sign Γ( p ′ ,
0) is the index of the crossing form of the path p ′ at time 0, in the notationof [24]. Consequently(4.4) CZ ( p ′′ ) ≡ CZ ( p ) − n mod 2 , by additivity of the Conley-Zehnder index.Let us then define a free homotopy { p t } of p to p ′ , p t is the concatenation of p with p ′′ | [0 ,t ] ,reparametrized to have domain [0 ,
1] at each moment t . This determines a homotopy {A ′ t } of connec-tions induced by { p t } . By the proof of Proposition 4.1, the CR operator D t determined by each A ′ t issurjective except at some finite collection of times t i ∈ (0 , i ∈ N determined by the crossing timesof p ′′ with the Maslov cycle, and the dimension of the kernel of D t i is the 1-eigenspace of p ′′ ( t i ), whichis 1 by the assumption that the crossings of p ′′ are simple.The operator D is not complex linear. To fix this we concatenate the homotopy { D t } with thehomotopy { e D t } defined as follows. Let { e A t } be a homotopy of A ′ to a unitary connection e A , wherethe homotopy { e A t } is through connections induced by paths { e p t } , giving a path homotopy of p ′ = e p to h . Then { e D t } is defined to be induced by { e A t } .Let us denote by { D ′ t } the concatenation of { D t } with { e D t } . By construction in the second half ofthe homotopy { D ′ t } , D ′ t is surjective. And D ′ is induced by a unitary connection, since it is inducedby unitary path e p . Consequently D ′ is complex linear. By the above construction, for the homotopy ONFORMAL SYMPLECTIC WEINSTEIN CONJECTURE AND NON-SQUEEZING 17 { D ′ t } , D ′ t is surjective except for N times in (0 , u ] by the definition via the determinant line bundle is exactly − N = − CZ ( p ) − n , by (4.4), which was what to be proved. (cid:3) Theorem 4.5. GW , ( N, A β , J λ )([ M , ] ⊗ [ C × S ]) = i ( e N , R λ , β ) , where N ⊂ M , , ( J λ , A β ) is an open compact set, e N the corresponding subset of periodic orbits of R λ , i ( e N, R λ , β ) is the Fuller index as described in the appendix below, and where the left hand side of theequation is a certain Gromov-Witten invariant, that we discuss in Section 2.Proof. If N ⊂ M , , ( J λ , A β ) is open-compact and consists of isolated regular Reeb tori { u i } , corre-sponding to orbits { o i } we have: GW , ( N, A β , J λ )([ M , ] ⊗ [ C × S ]) = X i ( − CZ ( o i ) − n mult ( o i ) , where the denominator mult ( o i ) is there because our moduli space is understood as a non-effectiveorbifold, see Appendix B.The expression on the right is exactly the Fuller index i ( e N, R λ , β ). Thus the theorem follows for N as above. However in general if N is open and compact then perturbing slightly we obtain a smoothfamily { R λ t } , λ = λ , s.t. λ is non-degenerate, that is has non-degenerate orbits. And such thatthere is an open-compact subset e N of M , , ( { J λ t } , A β ) with ( e N ∩ M , , ( J λ , A β )) = N , cf. Lemma 2.8.Then by Lemma 2.2 if N = ( e N ∩ M , , ( J λ , A β ))we get GW , ( N, A β , J λ )([ M , ] ⊗ [ C × S ]) = GW , ( N , A β , J λ )([ M , ] ⊗ [ C × S ]) . By the previous discussion GW , ( N , A β , J λ )([ M , ] ⊗ [ C × S ]) = i ( N , R λ , β ) , but by the invariance of Fuller index (see Appendix A), i ( N , R λ , β ) = i ( N, R λ , β ) . (cid:3) Proof of Theorem 1.10.
Let u : Σ → M be a non-constant J -curve. We first show that [ u ∗ α ] = 0.Suppose otherwise. Let f M denote the α -covering space of M , that is the space of equivalence classesof paths p starting at x ∈ M , with a pair p , p equivalent if p (1) = p (1) and R [0 , p ∗ α = R [0 , p ∗ α .Then the lift of ω to f M is e ω = f d ( f λ ), where f = e g and where g is a primitive for the lift e α of α to f M , that is e α = dg . In particular e ω is conformally symplectomorphic to an exact symplectic form on f M . So if e J denotes the lift of J , any closed e J -curve is constant by Stokes theorem. Now [ u ∗ α ] = 0, so u has a lift to a e J -holomorphic map e u : Σ → f M . Since Σ is closed, it follows by the above that e u isconstant, which is a contradiction.Since α is rational we may construct a smooth p : M → S , so that α = c · p ∗ dθ for c ∈ Q . Let u : Σ → M be a non-constant J -curve. Let s ∈ S be a regular value of p ◦ u , and let S ⊂ Σ, S ≃ S be a component of ( p ◦ u ) − ( s ). Since the critical points of u are isolated we may suppose that u isnon-critical along S . In particular u ∗ ω is non-vanishing everywhere on T Σ | S , which together withLemma 1.8 implies that u ∗ λ ∧ u ∗ α is non-vanishing everywhere on T Σ | S . So if o : S → S is anyparametrization, u ◦ o is a Reeb curve.Now if u is an immersion then u ∗ ω is symplectic and by Lemma 1.8 u ∗ dλ = 0, so that ω = u ∗ α ∧ u ∗ λ is non-degenerate on Σ. Let e Σ be the u ∗ α -covering space of Σ so that ω = dH ∧ u ∗ λ for some proper H : e Σ → R . Since ω is non-degenerate, H has no critical points so that e Σ ≃ S × R by basic Morsetheory. It follows that Σ ≃ T . (cid:3) Lemma 4.6.
Let ( M, λ, α, J ) be a tamed exact lcs structure. Suppose that α is rational, then everynon-constant J -curve u : Σ → M , with Σ a closed possibly nodal Riemann surface, is smooth, that is Σ is a smooth Riemann surface.Proof. Since α is rational we may construct a smooth p : M → S , so that α = c · p ∗ dθ for c ∈ Q . Let u : Σ → M be a non-constant J -curve. Let s ∈ S be a regular value of p ◦ u , and let S ⊂ Σ, S ≃ S be a component of ( p ◦ u ) − ( s ). Since the critical points of u are isolated we may suppose that u isnon-critical along S . Suppose by contradiction that Σ is nodal. We may then find an embedded disk i : D → Σ with ∂i ( D ) = S .Since u ∗ dλ = 0 by Lemma 1.8, R S i ∗ u ∗ λ = 0 by Stokes theorem, and so u ∗ λ ( v ) = 0 for some v ∈ T S ( z ) ⊂ T z Σ, z ∈ S . And let w ∈ T z Σ be such that v, w form a basis for T z Σ. Now u ∗ ω issymplectic along S so that u ∗ ω ( v, w ) = 0 which implies that u ∗ α ∧ u ∗ λ ( v, w ) = 0 since u ∗ dλ ( v, w ) = 0,but u ∗ α ( v ) = 0 and u ∗ λ ( v ) = 0, so that we have a contradiction. (cid:3) Proof of Theorem 1.9.
Let N ⊂ M , , ( A, J λ ), be the subspace corresponding, (under the correspon-dence of Proposition 1.6) to the subspace e N of all period 2 π R λ -orbits. It is easy to compute, see forinstance [11], i ( e N , R λ ) = ± χ ( CP k ) = 0 . By Theorem 4.5 GW , ( N, J λ , A ) = 0. The first part of the theorem then follows by Lemma 2.6.We now verify the second part. Let U be a δ -neighborhood of ( d α λ H , J λ H ) guaranteed by thefirst part of the theorem. Let ( λ ′ , α ′ , J ) ∈ U and u ∈ M , , ( A, J ) guaranteed by the first part of thetheorem, with J admissible. Let u be a simple J -holomorphic curve covered by u , which is non-nodalby Lemma 4.6. Let us recall for convenience the adjunction inequality. Theorem 4.7 (McDuff-Micallef-White [21], [16]) . Let ( M, J ) be an almost complex 4-manifold and A ∈ H ( M ) be a homology class that is represented by a simple J-holomorphic curve u . Then δ ( u ) − χ (Σ) ≤ A · A − c ( A ) , with equality if and only if u is an immersion with only transverse self-intersections. In our case A = 0, χ (Σ) = 0, so that δ ( u ) = 0, and so u is an embedding. (cid:3) Proof of Theorem 1.4.
Define a pseudo-metric d measuring distance between subspaces W , W of aninner product space ( T, g ) as follows. If dim W = dim W then d ( W , W ) := | P W − P W | , for | · | the g -operator norm, and P W i g -projection operators onto W i . If W = T or W = T define d ( W , W ) := 0, in all other cases set d ( W , W ) := 1. We may generalize this to a C pseudo-metric d again in terms of these projection operators.Let U be a C metric ǫ -ball neighborhood of ( ω H , J H := J λ H ) as in the first part of Theorem 1.9.To prove the theorem we need to construct a tamed exact lcs structure ( λ, α, J ), with ( d α λ, J ) ∈ U asTheorem 1.9 then tells us that there is a class A , J -holomorphic elliptic curve u in M , and since J isadmissible, by Theorem 1.10 there is a Reeb curve for ( λ, α ).Suppose that ω = d α ′ λ ′ is δ -close to ω H for the C metric d as in the statement of the theorem.Then clearly for each p ∈ M , d ( V ω ( p ) , V ω H ( p )) < ǫ δ and d ( ξ ω ( p ) , ξ ω H ( p )) < ǫ δ where ǫ δ → δ →
0, and where d is the pseudo-metric as defined above for subspaces of the inner product space( T p M, g ).Then choosing δ to be suitably be small, for each p ∈ V := V ( M, λ ′ ) we have an isomorphism φ ( p ) : T p M → T p M , φ p := P ⊕ P , for P : V λ H ( p ) → V λ ′ ( p ), P : ξ ω H ( p ) → ξ ω ( p ) the g -projectionoperators. Define J ( p ) := φ ( p ) ∗ J H , and this defines J on the sub-bundle π − T M V , for π T M : T M → M ONFORMAL SYMPLECTIC WEINSTEIN CONJECTURE AND NON-SQUEEZING 19 the bundle projection. In addition, if δ was chosen to be sufficiently small ( ω, J ) is a compatible pairon π − T M V , and is ǫ -close to ( ω H , J H ) on π − T M V .Now take any extension of J to T M so that ( ω, J ) is a compatible pair ǫ -close to ( ω H , J H ) on π − T M V . This can be obtained by using a partition of unity. Explicitly, J defined on π − T M V and ω givea Riemannian metric g J ( · , · ) = ω ( · , J · ) on π − T M V . Use a partition of unity to extend this metric to T M , and then use the map: ret : M et ( M ) × Ω( M ) → J ( M ) , as in Lemma 2.9. (cid:3) Proof of Theorem 1.5.
Let { ω t } , t ∈ [0 , C ∞ topology homotopy of lcsforms on M = C × S , as in the hypothesis. Fix an almost complex structure J on M admissible withrespect to ( α ′ , λ ′ ). Extend to a Frechet smooth family { J t } of almost complex structures on M , sothat J t is ω t -compatible for each t . Then in the absence of holomorphic sky catastrophes, by Theorem5.11, there is a non-constant elliptic J -holomorphic curve in M , so that the result follows by Theorem1.10. (cid:3) Extended Gromov-Witten invariants and the extended Fuller index
In what follows M is a closed oriented 2 n -fold, n ≥
2, and J an almost complex structure on M . Much of the following discussion extends to general moduli spaces M g,n ( J, A, a , . . . , a n ) with a , . . . , a n homological constraints in M . We shall however restrict for simplicity to the case ( ω, J ) is acompatible lcs pair on M , g = 1 , n = 1, the homological constraint is [ M ], as this is the main interestin this paper. Moreover, we restrict our moduli space to consist of non-zero charge pair (for example(1 , α of ω as in Section 1.2.1, and this willbe implicit, so that we no longer specify this in notation.In what follows e ( u ) denotes the energy of a map u : Σ → M , with respect to the metric inducedby an lcs pair ( ω, J ). Definition 5.1.
Let h = { ( ω t , J t ) } be a homotopy of lcs pairs on M , so that { J t } is Frechet smooth,and { ω t } C continuous. We say that it is partially admissible for A if every element of M , ( M, J , A ) is contained in a compact open subset of M , ( M, { J t } , A ) . We say that h is admissible for A ifevery element of M , ( M, J i , A ) ,i = 0 , is contained in a compact open subset of M , ( M, { J t } , A ) . Thus in the above definition, a homotopy is partially admissible if there are sky catastrophes goingone way, and admissible if there are no sky catastrophes going either way.Partly to simplify notation, we denote by a capital X a compatible general lcs triple ( M, ω, J ), thenwe introduce the following simplified notation.(5.2) S ( X, A ) = { u ∈ M , ( X, A ) } S ( X, a, A ) = { u ∈ S ( X, A ) | e ( u ) ≤ a } S ( h, A ) = { u ∈ M , ( h, A ) } , for h = { ( ω t , J t ) } a homotopy as above S ( h, a, A ) = { u ∈ S ( h, A ) | e ( u ) ≤ a } Definition 5.3.
For an isolated element u of S ( X, A ) , which means that { u } is open as a subset, weset gw ( u ) ∈ Q to be the local Gromov-Witten invariant of u . This is defined as: gw ( u ) = GW , ( { u } , A, J )([ M , ] ⊗ [ M ]) , with the right hand side as in (2.1) . Denote by S ( M, A ) the set of equivalence classes of all smooth stable maps Σ → M , in class A , forΣ an (non-fixed) elliptic curve, and where equivalence has the same meaning as in Section 1.2.1. Definition 5.4.
Suppose that S ( X, A ) has open connected components. And suppose that we have acollection of lcs pairs { X a = ( M, ω a , J a ) } , a ∈ R + satisfying the following: • S ( X a , a, A ) consists of isolated curves for each a . • S ( X a , a, A ) = S ( X b , a, A ) , (equality of subsets of S ( M, A ) ) if b > a , • For b > a , and for each u ∈ S ( X a , a, A ) = S ( X b , a, A ) : GW , ( { u } , A, J a ) = GW , ( { u } , A, J b ) , thus we may just write gw ( u ) for the common number. • There is a prescribed homotopy h a = { X at } of each X a to X , called structure homotopy ,with the property that for every y ∈ S ( X a , A ) there is an open compact subset C y ⊂ S ( h a , A ) , y ∈ C y , which is non-branching which meansthat C y ∩ S ( X ai , A ) ,i = 0 , are connected. • S ( h a , a, A ) = S ( h b , a, A ) , (similarly equality of subsets) if b > a is sufficiently large.We will then say that P ( A ) = { ( X a , h a ) } is a perturbation system for X in the class A . We shall see shortly that, given a contact (
C, λ ), the associated Banyaga lcs structure on C × S always admits a perturbation system for the moduli spaces of charge (1,0) curves in any class, if λ isMorse-Bott. Definition 5.5.
Suppose that X admits a perturbation system P ( A ) so that there exists an E = E ( P ( A )) with the property that S ( X a , a, A ) = S ( X E , a, A ) for all a > E , where this as before is equality of subsets, and the local Gromov-Witten invariants ofthe identified elements are also identified. Then we say that X is finite type and set: GW ( X, A ) = X u ∈ S ( X E ,A ) gw ( u ) . Definition 5.6.
Suppose that X admits a perturbation system P ( A ) and there is an E = E ( P ( A )) > so that gw ( u ) > for all { u ∈ S ( X a , A ) | E ≤ e ( u ) ≤ a } respectively gw ( u ) < for all { u ∈ S ( X a , A ) | E ≤ e ( u ) ≤ a } , and every a > E . Suppose in addition that lim a X u ∈ S ( X,a,A ) gw ( u ) = ∞ , respectively lim a X u ∈ S ( X,a,β ) gw ( u ) = −∞ . Then we say that X is positive infinite type , respectively negative infinite type and set GW ( X, A ) = ∞ , ONFORMAL SYMPLECTIC WEINSTEIN CONJECTURE AND NON-SQUEEZING 21 respectively GW ( X, A ) = −∞ . These are meant to be interpreted as extended Gromov-Witten in-variants, counting elliptic curves in class A . We say that X is infinite type if it is one or theother. Definition 5.7.
We say that X is definite type if it admits a perturbation system and is infinite typeor finite type. With the above definitions GW ( X, A ) ∈ Q ⊔ ∞ ⊔ −∞ , when it is defined. Proof of Theorem 1.12.
Given the definitions above, and the definition of the extended Fuller index in[25], this follows by the same argument as the proof of Theorem 4.5. (cid:3)
Perturbation systems for Morse-Bott Reeb vector fields.
Definition 5.8.
A contact form λ on M , and its associated flow R λ are called Morse-Bott if the λ action spectrum σ ( λ ) - that is the space of critical values of o R S o ∗ λ , is discreet and if for every a ∈ σ ( λ ) , the space N a := { x ∈ M | F a ( x ) = x } ,F a the time a flow map for R λ - is a closed smooth manifold such that rank dλ | N a is locally constantand T x N a = ker( dF a − I ) x . Proposition 5.9.
Let λ be a contact form of Morse-Bott type, on a closed contact manifold C . Thenthe corresponding lcs pair X λ = ( C × S , d α λ, J λ ) admits a perturbation system P ( A ) , for modulispaces of charge (1,0) curves for every class A .Proof. This follows immediately by [25, Proposition 2.12 ], and by Proposition 1.6. (cid:3)
Lemma 5.10.
The Hopf lcs pair ( S k +1 × S , d α λ H , J λ H ) , for λ H the standard contact structure on S k +1 is infinite type.Proof. This follows immediately by [25, Lemma 2.13], and by Proposition 1.6. (cid:3)
Theorem 5.11.
Let ( C, λ ) be a closed contact manifold so that R λ has definite type, and supposethat i ( R λ , β ) = 0 . Let ω = d α λ be the Banyaga structure, and suppose we have a partially admissiblehomotopy h = { ( ω t, , J t ) } , for class A β , then there in an element u ∈ M , , ( J , A β ) . The proof of this will follow.5.1.
Preliminaries on admissible homotopies.Definition 5.12.
Let h = { X t } be a smooth homotopy of lcs pairs. For b > a > we say that h is partially a, b - admissible , respectively a, b - admissible (in class A ) if for each y ∈ S ( X , a, A ) there is a compact open subset C y ⊂ S ( h, A ) , y ∈ C y with e ( u ) < b , for all u ∈ C y . Respectively, if foreach y ∈ S ( X i , a, A ) ,i = 0 , there is a compact open subset C y ∋ y of S ( h, A ) with e ( u ) < b , for all u ∈ C y . Lemma 5.13.
Suppose that X has a perturbation system P ( A ) , and { X t } is partially admissible, thenfor every a there is a b > a so that { e X bt } = { X t } · { X bt } is partially a, b -admissible, where { X t } · { X bt } is the (reparametrized to have t domain [0 , ) concatenation of the homotopies { X t } , { X bt } , and where { X bt } is the structure homotopy from X b to X .Proof. This is a matter of pure topology, and the proof is completely analogous to the proof of [25,Lemma 3.8]. (cid:3)
The analogue of Lemma 5.13 in the admissible case is the following:
Lemma 5.14.
Suppose that X , X and { X t } are admissible, then for every a there is a b > a so that (5.15) { e X bt } = { X b ,t } − · { X t } · { X b ,t } is a, b -admissible, where { X bi,t } are the structure homotopies from X bi to X i . Invariance.Theorem 5.16.
Suppose we have a definite type lcs pair X , with GW ( X , A ) = 0 , which is joined to X by a partially admissible homotopy { X t } , then X has non-constant elliptic class A curves.Proof of Theorem 5.11. This follows by Theorem 5.16 and by Theorem 1.12. (cid:3)
We also have a more a more precise result.
Theorem 5.17. If X , X are definite type lcs pairs and { X t } is admissible then GW ( X , A ) = GW ( X , A ) .Proof of Theorem 5.16. Suppose that X is definite type with GW ( X , A ) = 0, { X t } is partiallyadmissible and M , ( X , A ) = ∅ . Let a be given and b determined so that e h b = { e X bt } is a partially( a, b )-admissible homotopy. We set S a = [ y C y ⊂ S ( e h b , A ) , for y ∈ S ( X b , a, A ). Here we use a natural identification of S ( X b , a, A ) = S ( e X b , a, A ) as a subset of S ( e h b , A ) by its construction. Then S a is an open-compact subset of S ( h, A ) and so admits an implicitatlas (Kuranishi structure) with boundary, (with virtual dimension 1) s.t: ∂S a = S ( X b , a, A ) + Q a , where Q a as a set is some subset (possibly empty), of elements u ∈ S ( X b , b, A ) with e ( u ) ≥ a . So wehave for all a :(5.18) X u ∈ Q a gw ( u ) + X u ∈ S ( X b ,a,A ) gw ( u ) = 0 . Case I, X is finite type. Let E = E ( P ) be the corresponding cutoff value in the definition offinite type, and take any a > E . Then Q a = ∅ and by definition of E we have that the left side is X u ∈ S ( X b ,E,A ) gw ( u ) = 0 . Clearly this gives a contradiction to (5.18).5.4.
Case II, X is infinite type. We may assume that GW ( X , A ) = ∞ , and take a > E , where E = E ( P ( A )) is the corresponding cutoff value in the definition of infinite type. Then X u ∈ Q a gw ( u ) ≥ , as a > E ( P ( A )). While lim a X u ∈ S ( X b ,a,A ) gw ( u ) = ∞ , as GW ( X , A ) = ∞ . This also contradicts (5.18). (cid:3) ONFORMAL SYMPLECTIC WEINSTEIN CONJECTURE AND NON-SQUEEZING 23
Proof of Theorem 5.17.
This is somewhat analogous to the proof of Theorem 5.16. Suppose that X i , { X t } are definite type as in the hypothesis. Let a be given and b determined so that e h b = { e X bt } ,see (5.15) is an ( a, b )-admissible homotopy. We set S a = [ y C y ⊂ S ( e h b , A )for y ∈ S ( X bi , a, A ). Then S a is an open-compact subset of S ( h, A ) and so has admits an implicit atlas(Kuranishi structure) with boundary, (with virtual dimension 1) s.t: ∂S a = ( S ( X b , a, A ) + Q a, ) op + S ( X b , a, A ) + Q a, , with op denoting opositite orientation and where Q a,i as sets are some subsets (possibly empty), ofelements u ∈ S ( X bi , b, A ) with e ( u ) ≥ a . So we have for all a :(5.19) X u ∈ Q a, gw ( u ) + X u ∈ S ( X b ,a,A ) gw ( u ) = X u ∈ Q a, gw ( u ) + X u ∈ S ( X b ,a,A ) gw ( u )5.5. Case I, X is finite type and X is infinite type. Suppose in addition GW ( X , A ) = ∞ andlet E = max( E ( P ( A )) , E ( P ( A ))), for P i , the perturbation systems of X i . Take any a > E . Then Q a, = ∅ and the left hand side of (5.19) is X u ∈ S ( X b ,E,A ) gw ( u ) . While the right hand side tends to ∞ as a tends to infinity since, X u ∈ Q a, gw ( u ) ≥ , as a > E ( P ( A )), and lim a X u ∈ S ( X b ,a,A ) gw ( u ) = ∞ , Clearly this gives a contradiction to (5.19).5.6.
Case II, X i are infinite type. Suppose in addition GW ( X , A ) = −∞ , GW ( X , A ) = ∞ andlet E = max( E ( P ( A )) , E ( P ( A ))), for P i , the perturbation systems of X i . Take any a > E . Then P u ∈ Q a, gw ( u ) ≤
0, and P u ∈ Q a, gw ( u ) ≥
0. So by definition of GW ( X i , A ) the left hand side of (5.18)tends to −∞ as a tends to ∞ , and the right hand side tends to ∞ . Clearly this gives a contradictionto (5.19).5.7. Case III, X i are finite type. The argument is analogous. (cid:3) A. Fuller index
Let X be a vector field on M . Set S ( X ) = S ( X, β ) = { ( o, p ) ∈ L β M × (0 , ∞ ) | o : R / Z → M is a periodic orbit of pX } , where L β M denotes the free homotopy class β component of the free loop space. Elements of S ( X ) willbe called orbits. There is a natural S reparametrization action on S ( X ), and elements of S ( X ) /S will be called unparametrized orbits , or just orbits. Slightly abusing notation we write ( o, p ) for theequivalence class of ( o, p ). The multiplicity m ( o, p ) of a periodic orbit is the ratio p/l for l > o . We want a kind of fixed point index which counts orbits ( o, p ) with certain weights- however in general to get invariance we must have period bounds. This is due to potential existenceof sky catastrophes as described in the introduction. Let N ⊂ S ( X ) be a compact open set. Assume for simplicity that elements ( o, p ) ∈ N are isolated.(Otherwise we need to perturb.) Then to such an ( N, X, β ) Fuller associates an index: i ( N, X, β ) = X ( o,p ) ∈ N/S m ( o, p ) i ( o, p ) , where i ( o, p ) is the fixed point index of the time p return map of the flow of X with respect to alocal surface of section in M transverse to the image of o . Fuller then shows that i ( N, X, β ) has thefollowing invariance property. Given a continuous homotopy { X t } , t ∈ [0 ,
1] let S ( { X t } , β ) = { ( o, p, t ) ∈ L β M × (0 , ∞ ) × [0 , | o : R / Z → M is a periodic orbit of pX t } . Given a continuous homotopy { X t } , X = X , t ∈ [0 , e N is an open compact subsetof S ( { X t } ), such that e N ∩ ( LM × R + × { } ) = N. Then if N = e N ∩ ( LM × R + × { } )we have i ( N, X, β ) = i ( N , X , β ) . In the case where X is the R λ -Reeb vector field on a contact manifold ( C n +1 , ξ ), and if ( o, p ) isnon-degenerate, we have:(A.1) i ( o, p ) = sign Det(Id | ξ ( x ) − F λp, ∗ | ξ ( x ) ) = ( − CZ ( o ) − n , where F λp, ∗ is the differential at x of the time p flow map of R λ , and where CZ ( o ) is the Conley-Zehnderindex, (which is a special kind of Maslov index) see [24].There is also an extended Fuller index i ( X, β ) ∈ Q ⊔ {±∞} , for certain X having definite type.This is constructed in [25], and is conceptually completely analogous to the extended Gromov-Witteninvariant constructed in this paper.B. Remark on multiplicity
This is a small note on how one deals with curves having non-trivial isotropy groups, in the virtualfundamental class technology. We primarily need this for the proof of Theorem 4.5. Given a closedoriented orbifold X , with an orbibundle E over X Fukaya-Ono [9] show how to construct using multi-sections its rational homology Euler class, which when X represents the moduli space of some stablecurves, is the virtual moduli cycle [ X ] vir . When this is in degree 0, the corresponding Gromov-Witteninvariant is R [ X ] vir . However they assume that their orbifolds are effective. This assumption is notreally necessary for the purpose of construction of the Euler class but is convenient for other technicalreasons. A different approach to the virtual fundamental class which emphasizes branched manifoldsis used by McDuff-Wehrheim, see for example McDuff [15], [18] which does not have the effectivityassumption, a similar use of branched manifolds appears in [3]. In the case of a non-effective orbibundle E → X McDuff [17], constructs a homological Euler class e ( E ) using multi-sections, which extends theconstruction [9]. McDuff shows that this class e ( E ) is Poincare dual to the completely formally naturalcohomological Euler class of E , constructed by other authors. In other words there is a natural notionof a homological Euler class of a possibly non-effective orbibundle. We shall assume the following blackbox property of the virtual fundamental class technology. Axiom B.1.
Suppose that the moduli space of stable maps is cleanly cut out, which means that it isrepresented by a (non-effective) orbifold X with an orbifold obstruction bundle E , that is the bundleover X of cokernel spaces of the linearized CR operators. Then the virtual fundamental class [ X ] vir coincides with e ( E ) . ONFORMAL SYMPLECTIC WEINSTEIN CONJECTURE AND NON-SQUEEZING 25
Given this axiom it does not matter to us which virtual moduli cycle technique we use. It issatisfied automatically by the construction of McDuff-Wehrheim, (at the moment in genus 0, butsurely extending). It can be shown to be satisfied in the approach of John Pardon [23]. And it issatisfied by the construction of Fukaya-Oh-Ono-Ohta [7], the latter is communicated to me by KaoruOno. When X is 0-dimensional this does follow immediately by the construction in [9], taking anyeffective Kuranishi neighborhood at the isolated points of X , (this actually suffices for our paper.)As a special case most relevant to us here, suppose we have a moduli space of elliptic curves in X , which is regular with expected dimension 0. Then its underlying space is a collection of orientedpoints. However as some curves are multiply covered, and so have isotropy groups, we must treat thisis a non-effective 0 dimensional oriented orbifold. The contribution of each curve [ u ] to the Gromov-Witten invariant R [ X ] vir ± u ])] , where [Γ([ u ])] is the order of the isotropy group Γ([ u ]) of [ u ], in theMcDuff-Wehrheim setup this is explained in [15, Section 5]. In the setup of Fukaya-Ono [9] we mayreadily calculate to get the same thing taking any effective Kuranishi neighborhood at the isolatedpoints of X . 6. Acknowledgements
I thank Yong-Geun Oh for a number of discussions on related topics, and for an invitation to IBS-CGP, Korea. Thanks also to John Pardon for receiving me during a visit in Princeton. I also thankDusa McDuff for comments on earlier versions and Kaoru Ono, Emmy Murphy, Viktor Ginzburg, YaelKarshon, Helmut Hofer and Richard Hind, for helpful discussions on related topics.
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