NON-SQUEEZING IN l. c. s. GEOMETRY AND HOLOMOPRHIC WEINSTEIN

CONJECTURE

YASHA SAVELYEV

Abstract. Based still on the theory of Kuranishi spaces of Fukaya-Oh-Ohta-Ono, we initiate herethe study of elements of Gromov-Witten theory of locally conformally symplectic manifolds or l. c. s.

manifolds, l. c. s.m.’s for short, which are a natural generalization of both contact and symplectic

manifolds. As a first application we show that in dimension 4 the classical Gromov non-squeezingtheorem has a certain C0 rigidity or persistence with respect to l. c. s. deformations, this is one version

of l. c. s. non-squeezing a first result of its kind. In a different direction we study Gromov-Wittentheory of the l. c. s.m. C × S1 induced by a contact manifold (C, λ), and show that the Gromov-

Witten invariant counting certain elliptic curves in C×S1 is identified with the classical Fuller index

of the Reeb vector field Rλ. Partly inspired by this we conjecture existence of certain elliptic curvesin some l. c. s.m.’s, which we call holomorphic Weinstein conjecture, and this is a direct extension of

the classical Weinstein conjecture. We show that either the holomorphic Weinstein conjecture holds

for any l. c. s. ω on S2k+1 × S1, with ω homotopic to the Hopf l. c. s. structure, or there exist skycatastrophes for families of holomorphic curves in l. c. s. manifold, in particular the conjecture holds

for a C0-neighborhood of the Hopf l. c. s. structure. Discovery of these holomorphic sky catastrophes

would be incredible, comparable to the discovery of sky catastrophes by Fuller for dynamical systems.

Contents

1. Introduction 21.1. Locally conformally symplectic manifolds and Gromov non-squeezing 31.2. Sky catastrophes 52. Introduction part II, holomorphic Weinstein conjecture 52.1. Holomorphic curves in the l. c. s.m. C × S1 52.2. Connection with the Fuller index 73. Elements of Gromov-Witten theory of an l. c. s. manifold 74. Rulling out sky catastrophes and non-squeezing 105. Connections of GW-theory of an l. c. s.m. C × S1 with the Fuller index 116. Preliminaries for the Proof of Theorem 2.3 146.1. Preliminaries on admissible homotopies 167. Proof of Theorem 2.3 177.1. Case I, X0 is finite type 177.2. Case II, X0 is infinite type 187.3. Case I, X0 is finite type and X1 is infinite type 187.4. Case II, Xi are infinite type 187.5. Case III, Xi are finite type 18A. Fuller index 19B. Virtual fundamental class 19References 20

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 of SEP, Mexico.

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2 YASHA SAVELYEV

1. 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 l. c. s. structurefor short. A locally conformally symplectic manifold or l. c. s.m. is a smooth 2n-fold M with an l. c. s.structure: which is a non-degenerate 2-form ω, which is locally diffeomorphic to f · ωst, for some(non-fixed) positive smooth function f , with ωst the standard symplectic form on R2n. 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 L2-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 Jt, t ∈ [0, 1], of ωt-compatible almost complex structures, we may have a continuous family ut of Jt-holomorphic curvess.t. energy(ut) 7→ ∞ as t 7→ a ∈ (0, 1) and s.t. there are no holomorphic curves for t ≥ a. These areanalogues of sky catastrophes discovered by Fuller [10].

We are able to tame these problems in certain situations, for example for some 4-d l. c. s.m.’s, andthis is how we arrive at a version of Gromov non-squeezing theorem for such l. c. s.m.’s. Even when itis impossible to tame these problems we show that there is still a potentially interesting theory which isanalogous to the theory of Fuller index in dynamical systems. For example we show that there is a directgeneralization of Seifert, Weinstein conjectures for certain l. c. s.m.’s, and which postulates existenceof certain elliptic curves in these l. c. s.m.’s. We call this the holomorphic Weinstein conjecture.

We prove this conjecture for l. c. s. structures C0 nearby to the Hopf l. c. s. structure on S2k+1×S1,and this uses a connection with the classical Fuller index. Note that the original Seifert conjecture[23] was likewise based on a nearby existence result. However in our case there is additional evidence,coming from the proof of original Weinstein conjecture. In fact we prove a stronger result that eitherholomorphic Weinstein conjecture holds for any l. c. s. structure homotopic to the Hopf l. c. s. structureor holomorphic sky catastrophes exist, as we shall discuss this would be fantastical.

We begin our more detailed discussion with the well known observation:

Theorem 1.1. [[18], [29]] Let (M,J) be a compact almost complex manifold, and u : (S2, j) → Ma J-holomorphic map. Given a Riemannian metric g on M , there is an ~ = ~(g, J) > 0 s.t. ifenergyg(u) < ~ then u is constant, where energyg is the L2-energy functional,

e(u) = energy(u) =

∫S2

|du|2dvol.

Using this we get the following extension of Gromov compactness to this setting. Let

Mg,n(J,A) =Mg,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 betweenu1 : Σ1 → M , and u2 : Σ2 → M is a biholomorphism of marked Riemann surfaces φ : Σ1 → Σ2 s.t.u2 φ = u1.

Theorem 1.2. Let (M,J) be an almost complex manifold. ThenMg,n(J,A) has a pre-compactification

Mg,n(J,A),

by Kontsevich stable maps, with respect to the natural metrizable Gromov topology see for instance[18], for genus 0 case. Moreover given E > 0, the subspace Mg,n(J,A)E ⊂ Mg,n(J,A) consisting ofelements u with e(u) ≤ E is compact. In other words energy 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 Mg,n(J,A), and we shall say that J is bounded (in class

A), later on we generalize this a bit in terms of what we call finite type . In this case Mg,n(J,A) is

NON-SQUEEZING IN l. c. s. GEOMETRY AND HOLOMOPRHIC WEINSTEIN CONJECTURE 3

compact, and has a virtual moduli cycle as in the original approach of Fukaya-Ono [9], or the morealgebraic approach [19]. So we may define functionals:

(1.3) GWg,n(ω,A, J) : H∗(Mg,n)⊗H∗(Mn)→ Q,

where Mg,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.

1.1. Locally conformally symplectic manifolds and Gromov non-squeezing. Let us give abit of background on l. c. s.m.’s. These were originally considered by Lee in [14], arising naturally aspart of an abstract study of “a kind of even dimensional Riemannian geometry”, and then furtherstudied by a number of authors see for instance, [1] and [27]. This is a fascinating object, an l. c. s.m.admits all the interesting classical notions of a symplectic manifold, like Lagrangian submanifoldsand Hamiltonian dynamics, while at the same time forming a much more flexible class. For exampleEliashberg and Murphy show that if a closed almost complex 2n-fold M has H1(M,R) 6= 0 then itadmits a l. c. s. 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 l. c. s. structure ω: the Lee class, α = αω ∈ H1(M,R). This has the property that on

the associated covering space M , ω is globally conformally symplectic. The class α may be defined asthe following Cech 1-cocycle. Let φa,b be the transition map for l. c. s. charts φa, φb of (M,ω). Thenφ∗a,bωst = ga,b · ωst for a positive real constant ga,b and ln ga,b gives our 1-cocycle. Thus an l. c. s.form is globally conformally symplectic iff its Lee class vanishes.

As we mentioned l. c. s.m.’s can also be understood to generalize contact manifolds. This worksas follows. First we have a natural class of explicit examples of l. c. s.m.’s, obtained by starting witha symplectic cobordism (see [5]) of a closed contact manifold C to itself, arranging for the contactforms at 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 1 (Banyaga). Let (C, ξ) be a contact manifold with a contact form λ and take M = C × S1

with 2-form ω = dαλ := dλ−α∧ λ, for α the pull-back of the volume form on S1 to C ×S1 under theprojection.

Using above we may then faithfully embed the category of contact manifolds, and contactomorphisminto the category of l. c. s.m.’s, and loose l. c. s. morphisms. These can be defined as diffeomorphismsφ : (M1, ω1) → (M2, ω2) s.t. φ∗ω2 = fω1 for a smooth function f , see [1] for details on how thisembedding works.

Banyaga type l. c. s.m.’s give immediate examples of almost complex manifolds where the energyfunction is unbounded on the moduli spaces of fixed class pseudo-holomorphic curves, as well as wherenull-homologous J-holomorphic curves can be non-constant. We show that it is still possible to extracta variant of Gromov-Witten theory here. The story is closely analogous to that of the Fuller index indynamical systems, which is concerned with certain rational counts of periodic orbits. In that case skycatastrophes prevent us from obtaining a completely well defined invariant, but Fuller constructs certainpartial invariants which give dynamical information. In a very particular situation the relationshipwith the Fuller index becomes perfect as one of the results of this paper obtains the classical Fullerindex for Reeb vector fields on a contact manifold C as a certain genus 1 Gromov-Witten invariantof the l. c. s.m. C × S1. The latter also gives a conceptual interpretation for why the Fuller index isrational, as it is reinterpreted as an (virtual) orbifold Euler number.

1.1.1. 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 BR → D2(r) × R2n−2 forR > r, with BR the standard closed radius R ball in R2n centered at 0. Gromov’s non-squeezing is C0

persistent in the following sense.

4 YASHA SAVELYEV

Theorem 1.4. Given R > r, there is an ε > 0 s.t. for any symplectic form ω′ on S2×T 2n−2 C0-closeto a split symplectic form ω and satisfying

〈ω, (A = [S2]⊗ [pt])〉 = πr2,

there is no symplectic embedding φ : BR → (S2 × T 2n−2, ω′).

On the other hand it is natural to ask:

Question 1. Given R > r and every ε > 0 is there a (necessarily non-closed by above) 2-form ω′ onS2×T 2n−2 C0 or even C∞ ε-close to a split symplectic form ω, satisfying 〈ω,A〉 = πr2, and s.t. thereis an embedding φ : BR → S2 × T 2n−2, with φ∗ω′ = ωst?

The above theorem follows immediately by Gromov’s argument in [12], we shall give a certainextension of this theorem for l. c. s. structures. One may think that recent work of Muller [25] may berelated to the question above and our theorem below. But there seems to be no obvious such relationas pull-backs by diffeomorphisms of nearby forms may not be nearby. Hence there is no way to gofrom nearby embeddings that we work with to ε-symplectic embeddings of Muller.

Definition 1.5. Given a pair of l. c. s.m.’s (Mi, ωi), i = 0, 1, we say that f : M1 → M2 is a mor-phism, if f∗ω2 = ω1. A morphism is called an l. c. s. embedding if it is injective.

A pair (ω, J) for ω l. c. s. and J compatible will be called a compatible l. c. s. pair, or just a compatiblepair, where there is no confusion. Note that the pair of hypersurfaces Σ1 = S2×S1×pt ⊂ S2×T 2,Σ2 = S2 × pt × S1 ⊂ S2 × T 2 are naturally foliated by symplectic spheres, we denote by T folΣi thesub-bundle of the tangent bundle consisting of vectors tangent to the foliation. The following theoremsays that it is impossible to have certain “nearby” l. c. s. embeddings, which means that we have a firstrigidity phenomenon for l. c. s. structures. There is a small caveat here, that in what follows we takethe C0 topology on the space of l. c. s. structures that is coarser than the subspace topology of thenatural C0 topology (with respect to a metric) on the space of forms, cf. [1, Section 6].

Theorem 1.6. Let ω be a split symplectic form on M = S2 × T 2, and A as above with 〈ω,A〉 = πr2.Let R > r, then there is an ε > 0 s.t. if ωt is a continuous l. c. s. family on M C0 ε-close to ω, thenthere is no l. c. s. embedding

φ : (BR, ωst) → (M,ω1),

s.t φ∗j preserves the bundles T folΣi, for j the standard almost complex structure. In particular thereis no l. c. s. embedding

φ : (BR, ωst) → (M,ω1)− (Σ1 t Σ2),

the latter is a full-volume subspace diffeomorphic to S2 × R2.

We note that the image of the embedding φ would be of course a symplectic submanifold of (M,ω1).However it could be highly distorted, so that it might be impossible to complete φ∗ωst to a symplecticform on M nearby to ω. We also note that it is certainly possible to have a nearby volume preservingas opposed to l. c. s. embedding which satisfies all other conditions. Take ω = ω1, then if the symplecticform on T 2 has enough volume, we can find a volume preserving map φ : BR →M s.t. φ∗j preservesT folΣi. This is just the squeeze map, which as a map C2 → C2 is (z1, z2) 7→ ( z1a , a · z2). In fact we canjust take any volume preserving map φ, which doesn’t hit Σi, meanwhile there is no l. c. s. morphisminto → (M,ω1) − (Σ1 t Σ2). Note that ω1 must necessarily be globally conformally symplectic onM,ω1 − Σ1 t Σ2, as this is simply connected, however even in this special case the theorem above isnot an obvious extension of Gromov non-squeezing.

1.1.2. Toward non-squeezing for loose morphisms. In some ways loose morphisms of l. c. s.m.’s aremore natural, particularly when we think about l. c. s.m.’s from the contact angle. So what aboutnon-squeezing for loose morphisms as defined above? We can try a direct generalization of contactnon-squeezing of Eliashberg-Polterovich [4], and Fraser in [6]. Specifically let R2n × S1 be the pre-quantization space of R2n, or in other words the contact manifold with the contact form dθ − λ, forλ = 1

2 (ydx− xdy). Let BR now denote the open radius R ball in R2n.

NON-SQUEEZING IN l. c. s. GEOMETRY AND HOLOMOPRHIC WEINSTEIN CONJECTURE 5

Question 2. If R ≥ 1 is there a compactly supported, loose endomorphism of the l. c. s.m. R2n×S1×S1

which takes the closure of U := BR × S1 × S1 into U?

We expect the answer is no, but our methods here are not sufficiently developed for this conjecture,as we likely have to extend contact homology rather the Gromov-Witten theory as we do here.

1.2. Sky catastrophes. Given a continuous family Jt, t ∈ [0, 1] we denote by Mg(Jt, A) the

space of pairs (u, t), u ∈Mg(Jt, A).

Definition 1.7. We say that a continuous family Jt on a compact manifold M has a holomorphicsky catastrophe in class A if there is an element u ∈ Mg(Ji, A), i = 0, 1 which does not belong to

any open compact (equivalently energy bounded) subset of Mg(Jt, A).

Let us slightly expand this definition. If the connected components of Mg(Jt, A) are open, inother words if this space is locally connected, then we have a sky catastrophe in the sense above if andonly if there is a u ∈Mg(Ji, A) which has a non-compact connected component in Mg(Jt, A).

Proposition 1.8. Let M be a closed manifold, and suppose that Jt, t ∈ [0, 1] has no holomorphicsky catastrophes, then if Ji, i = 0, 1 are bounded:

GWg,n(A, J0) = GWg,n(A, J1),

if A 6= 0. If only J0 is bounded then there is at least one class A J1-holomorphic curve in M .

The assumption on A is for simplicity in this case. At this point in time there are no known examplesof families Jt with sky catastrophes, cf. [10].

Question 3. Do sky catastrophes exist?

Really what we are interested in is whether they exist generically. The author’s opinion is thatthey do appear even generically. However for locally conformally symplectic deformations (ωt, Jt) aspreviously defined, it might be possible that holomorphic sky catastrophes cannot exist generically, forexample it looks very unlikely that an example can be constructed with Reeb tori (see the followingsection), cf. [21].

In this direction we have the following result, that will be used in the proof of non-squeezing above.

Theorem 1.9. Let M be a closed 4-fold and (ωt, Jt), t ∈ [0, 1], a continuous family of compatiblel. c. s. pairs on M , and αt its associated family of Lee classes. Let Σi ⊂ M , i = 0, . . . ,m be acollection of hypersurfaces s.t. PD(αt) =

∑i ai,t[Σi] for each t. (ai,t need not be continuous in t.) Let

Jt be such that for each t there is a foliation of Σi by Jt-holomorphic class B curves, then Jt hasno sky catastrophes in every class A s.t. A ·B ≤ 0.

2. Introduction part II, holomorphic Weinstein conjecture

2.1. Holomorphic curves in the l. c. s.m. C × S1. Let (C, λ) be a closed contact manifold witha contact form λ. Then T = S1 acts on C × S1 by rotation in the S1 coordinate. Let J be analmost complex structure on the contact distribution, compatible with dλ. There is an induced almostcomplex structure Jλ on C × S1, which is T -invariant, coincides with J on the contact distribution

ξ ⊂ TC ⊕ θ ⊂ T (C × S1),

for each θ and which maps the Reeb vector field

Rλ ∈ TC ⊕ 0 ⊂ T (C × S1)

tod

dθ∈ 0 ⊕ TS1 ⊂ T (C × S1),

for θ ∈ [0, 2π] the global angular coordinate on S1. This almost complex structure is compatible withdαλ.

6 YASHA SAVELYEV

We now consider the moduli space of marked holomorphic tori, (elliptic curves) in C × S1, ina certain class A. Our notation for this is M1,1(Jλ, A), where A is a class of the maps, (to beexplained). The elements are equivalence classes of pairs (u,Σ): u a Jλ-holomorphic map of a stablegenus 1, elliptic curve Σ into C × S1. So Σ is a nodal curve with principal component an ellipticcurve, and other components spherical. So the principal component determines an element of M1,1

the compactified moduli space of elliptic curves, which is understood as an orbifold. The equivalencerelation is (u,Σ) ∼ (u′,Σ′) if there is an isomorphism of marked elliptic curves φ : Σ → Σ′ s.t.u′ φ = u. When Σ is smooth, we may write [u, j] for an equivalence class where j is understood asa complex structure on the sole principal component of the domain, and u the map. Or we may justwrite [u], or even just u keeping track of j, and of the fact that we are dealing with equivalence classes,implicitly.

Let us explain what class A means. We need to be careful because it is now possible for non-constantholomorphic curves to be null-homologous. Here is a simple example take S3 × S1 with J determinedby the Hopf contact form as above, then all the Reeb tori are null-homologous. In many cases we canjust work with homology classes A 6= 0, and this will remove some headache, but in the above specificsituation this is inadequate.

Given u ∈M1,1(Jλ, A) we may compose Σu−→ C × S1 pr−→ S1, for Σ the nodal domain of u.

Definition 2.1. In the setting above we say that u is in class A, if (pr u)∗dθ can be completed to anintegral basis of H1(Σ,Z), and if the homology class of u is A, possibly zero.

It is easy to see that the above notion of class is preserved under Gromov convergence, and that aclass A J-holomorphic map cannot be constant for any A, in particular by Theorem 1.1 a class A maphas energy bounded from below by a positive constant, depending on (ω, J). And this holds for anyl. c. s. pair (ω, J) on C × S1.

2.1.1. Reeb tori. For the almost complex structure Jλ as above we have one natural class of holomor-phic tori in C × S1 that we call Reeb tori. Given a closed orbit o of Rλ, a Reeb torus uo for o, is themap

uo(θ1, θ2) = (o(θ1), θ2),

θ1, θ2 ∈ S1 A Reeb torus is Jλ-holomorphic for a uniquely determined holomorphic structure j on T 2.If

Dto(t) = c ·Rλ(o(t)),

then

j(∂

∂θ1) = c

∂

∂θ2.

Proposition 2.2. Let (C, λ) be as above. Let A be a class in the sense above, and Jλ be as above.Then the entire moduli space M1,1(Jλ, A) consists of Reeb tori.

Note that the formal dimension of M1,1(Jλ, A) is 0, for A as in the proposition above. It is givenby the Fredholm index of the operator (5.2) which is 2, minus the dimension of the reparametrizationgroup (for smooth curves) which is 2. In Theorem 2.6 we relate the count of these curves to theclassical Fuller index, which is reviewed in the Appendix A.

What follows is one non-classical application of the above theory.

Theorem 2.3. Let (S2k+1 × S1, dαλst) be the l. c. s.m. associated to a contact manifold (S2k+1, λst)for λst the standard contact form. Let A be as in the discussion above. Then for any l. c. s. pair (ω, J),homotopic through a path of l. c. s. pairs p = (ωt,, Jt) to (dαλ, Jλ), there exists an elliptic, class A,J-holomorphic curve in C × S1, provided p has no sky catastrophe in class A.

While not having sky catastrophes gives us a certain compactness control, the above theorem is notimmediate because we can still in principle have total cancellation of the infinitely many componentsof the moduli space M1,1(Jλ, A). In other words a virtual 0-dimension Kuranishi space M1,1(Jλ, A),with an infinite number of connected components, can certainly be null-cobordant, by a cobordism all

NON-SQUEEZING IN l. c. s. GEOMETRY AND HOLOMOPRHIC WEINSTEIN CONJECTURE 7

of whose components are compact. So for the theorem we need a certain additional geometric control,and a certain algebraic framework, to preclude such total cancellation.

Theorem 2.4. Let (S2k+1 × S1, dαλH) be the l. c. s.m. associated to a contact manifold (S2k+1, λH)for λH the standard contact form. There exists a δ > 0 s.t. for any l. c. s. pair (ω, J) C0 δ-close to(dαλH , J

λH ), there exists an elliptic, class A, J-holomorphic curve in C × S1. (Where A is as in thediscussion above.)

We shall call dαλH the Hopf l. c. s. structure . Note that Seifert [23] initially found a similarexistence phenomenon of orbits on S2k+1 for a vector field C0-nearby to the Hopf vector field. And heasked if the nearby condition can be removed, this was known as the Seifert conjecture. This turnedout not to be quite true [13]. Likewise it is natural for us to conjecture that the δ-nearby conditioncan be removed. This conjecture has more evidence however. For if ω = dαλ for λ the contact forminducing the standard contact structure on S2k+1, or any contact form on a threefold, and J = Jλ

then we know there are J-holomorphic class A tori, since we know there are λ-Reeb orbits, as theWeinstein conjecture is known to hold in these cases, [28], [26] and hence there are Reeb tori. We thenhave the following “holomorphic Seifert/Weinstein conjecture”.

Conjecture 1. For any l. c. s. pair (ω, J) on S2k+1 × S1 or on C × S1 for C a threefold, there is anelliptic, non-constant J-holomorphic curve.

We note that it implies the Weinstein conjecture for S2k+1 and for C a contact three-fold byProposition 2.2. The above conjecture is not just a curiosity. In contact geometry, rigidity is based onexistence phenomena of Reeb orbits, and l. c. s. manifolds should be understood as generalized contactmanifolds. To attack rigidity questions in l. c. s. geometry, like Question 2 we need an analogue of Reeborbits, we propose that this analogue is elliptic curves as above, from which point of view the aboveconjecture becomes very natural. From this point of view, we may reformulate Theorem 2.3 as follows.

Theorem 2.5. Either the holomorphic Weinstein conjecture holds for any l. c. s. pair (ω, J) on S2k+1×S1, with ω homotopic to the Hopf l. c. s. structure, or holomorphic sky catastrophes exist.

If holomorphic sky catastrophes are discovered, this will be a tremendous discovery. The originaldiscovery by Fuller [10] of sky catastrophes in dynamical systems is one of the most important indynamical systems, see also [24] for an overview. On the other hand pseudo-holomorphic curves inl. c. s. manifolds look to be much more rigid objects then periodic orbits of general smooth dynamicalsystems. So it is possible that holomorphic sky catastrophes do not exist.

2.2. Connection with the Fuller index. If β is a free homotopy class of a loop in C denote by Aβthe induced homology class of a Reeb torus in C × S1. Then we have:

Theorem 2.6.GW1,1(N,Aβ , J

λ)([M1,1]⊗ [C × S1]) = i(N , Rλ, β),

where N ⊂ M1(Jλ, Aβ) is an open compact set, N the corresponding subset of periodic orbits of Rλ,

i(N , Rλ, 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 3, just below.

What about higher genus invariants of C×S1? Following Proposition 2.2, it is not hard to see thatall Jλ-holomorphic curves must be branched covers of Reeb tori. If one can show that these branchedcovers are regular when the underlying tori are regular, the calculation of invariants would be fairlyautomatic from this data, see [32], [30] where these kinds of regularity calculation are made.

3. Elements of Gromov-Witten theory of an l. c. s. manifold

Suppose (M,J) is a compact almost complex manifold, let N ⊂ Mg,k(J,A) be an open compactsubset with energy positive on N . The latter condition is only relevant when A = 0. We shall primarilyrefer in what follows to work of Pardon in [19], only because this is what is more familiar to the author,due to greater comfort with algebraic topology as opposed to analysis. But we should mention that

8 YASHA SAVELYEV

the latter is a follow up to a profound theory that is originally created by Fukaya-Ono [9], and laterexpanded with Oh-Ohta [8].

The construction in [19] of implicit atlas, on the moduli spaceM 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 [19], (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:

(3.1) GWg,n(N,A, J) : H∗(Mg,n)⊗H∗(Mn)→ Q.The first question is: how do these functionals depend on N, J?

Lemma 3.2. Let Jt, t ∈ [0, 1] be a continuous in the C∞ topology homotopy. Suppose that N isan open compact subset of the cobordism moduli space Mg,n(Jt, A) and that the energy function is

positive on N , (the latter only relevant when A = 0). Let

Ni = N ∩(Mg,n(Ji, A)

),

thenGWg,n(N0, A, J0) = GWg,n(N1, A, J1).

In particular if GWg,n(N0, A, J0) 6= 0, there is a class A J1-holomorphic stable map in M .

Proof of Lemma 3.2. We may construct exactly as in [19] a natural implicit atlas on N , with boundaryNop

0 tN1, (op denoting opposite orientation). And so

GWg,n(N0, A, J0) = GWg,n(N1, A, J1),

as functionals.

The most basic lemma in this setting is the following, and we shall use it in the following section.

Definition 3.3. 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.

Definition 3.4. We say that a pair of almost symplectic pairs (ωi, Ji) are δ-close, if ωi, and Jiare C0 δ-close, i = 0, 1.

Lemma 3.5. Given a compact M and an almost symplectic tuple (ω, J) on M , suppose that N ⊂Mg,n(J,A) is a compact and open component which is energy isolated meaning that

N ⊂(U = energy−1

ω (E0, E1))⊂(V = energy−1

ω (E0 − ε, E1 + ε)),

with ε > 0, E0 > 0 and with V ∩Mg,n(J,A) = N . Suppose also that GWg,n(N, J,A) 6= 0. Then thereis a δ > 0 s.t. whenever (ω′, J ′) is a compatible almost symplectic pair δ-close to (ω, J), there existsu ∈Mg,n(J ′, A) 6= ∅, with

E0 < energyω′(u) < E1.

Proof of Lemma 3.5.

Lemma 3.6. Given a Riemannian manifold (M, g), and J an almost complex structure, suppose thatN ⊂Mg,n(J,A) is a compact and open component which is energy isolated meaning that

N ⊂(U = energy−1

g (E0, E1))⊂(V = energy−1

g (E0 − ε, E1 + ε)),

with ε > 0, E0 > 0, and with V ∩Mg,n(J,A) = N . Then there is a δ > 0 s.t. whenever (g′, J ′) is C0

δ-close to (g, J) if u ∈Mg,n(J ′, A) and

E0 − ε < energyg′(u) < E1 + ε

thenE0 < energyg′(u) < E1.

NON-SQUEEZING IN l. c. s. GEOMETRY AND HOLOMOPRHIC WEINSTEIN CONJECTURE 9

Proof of Lemma 3.6. Suppose otherwise then there is a sequence (gk, Jk) C0 converging to (g, J),and a sequence uk of Jk-holomorphic stable maps satisfying

E0 − ε < energygk(uk) ≤ E0

or

E1 ≤ energygk(uk) < E1 + ε.

By Gromov compactness we may find a Gromov convergent subsequence ukj to a J-holomorphicstable map u, with

E0 − ε < energyg(u) ≤ E0

or

E1 ≤ energyg(u) < E1 + ε.

But by our assumptions such a u does not exist.

Lemma 3.7. Given a compact almost symplectic compatible triple (M,ω, J), so that N ⊂Mg,n(J,A)is exactly as in the lemma above. There is a δ′ > 0 s.t. the following is satisfied. Let (ω′, J ′) beδ′-close to (ω, J), then there is a smooth family of almost symplectic pairs (ωt, Jt), (ω0, J0) = (g, J),(ω1, J1) = (g′, J ′) s.t. there is open compact subset

N ⊂Mg,n(Jt, A),

and with

N ∩M(J,A) = N.

Moreover if (u, t) ∈ N then

E0 < energygt(u) < E1.

Proof. Given ε as in the hypothesis let δ be as in Lemma 3.6.

Lemma 3.8. Given a δ > 0 there is a δ′ > 0 s.t. if (ω′, J ′) is δ′-near (ω, J) there is an interpolatingfamily (ωt, Jt) with (ωt, Jt) δ-close to (ω, J) for each t.

Proof. Let gt be the family of metrics on M given by the convex linear combination of g = gωJ , g′ =

gω′,J′ . Clearly gt is δ′-close to g0 for each t. Likewise the family of 2 forms ωt given by the convexlinear combination of ω, ω′ is non-degenerate for each t if δ′ was chosen to be sufficiently small and isδ′-close to ω0 = ωg,J for each moment.

Let

ret : Met(M)× Ω(M)→ J (M)

be the “retraction map” (it can be understood as a retraction followed by projection) as defined in[17, Prop 2.50], where Met(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 structureret(g, ω) is compatible with ω. Then (ωt, ret(gt, ωt) is a compatible family. As ret is continuous inthe C∞-topology, δ′ can be chosen so that rett(gt, ωt are δ-nearby.

Let δ′ be chosen with respect to δ as in the above lemma and (ωt, Jt) be the corresponding family.

Let N consist of all elements (u, t) ∈M(Jt, A) s.t.

E0 − ε < energyωt(u) < E1 + ε.

Then by Lemma 3.6 for each (u, t) ∈ N , we have:

E0 < energyωt(u) < E1.

In particular N must be closed, it is also clearly open, and is compact as energy is a properfunction.

10 YASHA SAVELYEV

To finish the proof of the main lemma, let N be as in the hypothesis, δ′ as in Lemma 3.7, and N asin the conclusion to Lemma 3.7, then by Lemma 3.2

GWg,n(N1, J′, A) = GWg,n(N, J,A) 6= 0,

where N1 = N ∩Mg,n(J1, A). So the conclusion follows.

Proof of Proposition 1.8. For each u ∈ Mg,n(Ji, A), i = 0, 1 fix an open-compact subset Vu of

Mg,n(Jt, A) containing u. We can do this by the hypothesis that there are no sky catastrophes.

Since Mg,n(Ji, A) are compact we may find a finite subcover

Vui ∩ (Mg,n(J0, A) ∪Mg,n(J1, A))

of Mg,n(J0, A) ∪Mg,n(J1, A), considering Mg,n(J0, A) ∪Mg,n(J1, A) as a subset of Mg,n(Jt, A)

naturally. Then V =⋃i Vui is an open compact subset of Mg,n(Jt, A), s.t.

V ∩Mg,n(Ji, A) =Mg,n(Ji, A).

Now apply Lemma 3.2.Likewise if only J0 is bounded, for each u ∈ Mg,n(J0, A), fix an open-compact subset Vu of

Mg,n(Jt, A) containing u. Since Mg,n(J0, A) is compact we may find a finite subcover

Vui ∩ (Mg,n(J0, A)

of Mg,n(J0, A). Then V =⋃i Vui is an open compact subset of Mg,n(Jt, A), s.t.

V ∩Mg,n(Ji, A) =Mg,n(Ji, A).

Again apply Lemma 3.2.

Proof of Theorem 1.2. (Outline, as the argument is standard.) Suppose that we have a sequence uk

of J-holomorphic maps with L2-energy ≤ E. By [18, 4.1.1], a sequence uk of J-holomorphic curveshas a convergent subsequence if supk||duk||L∞ <∞. On the other hand when this condition does nothold rescaling argument tells us that a holomorphic sphere bubbles off. The quantization Theorem1.1, then tells us that these bubbles have some minimal energy, so if the total energy is capped byE, only finitely many bubbles may appear, so that a subsequence of uk must converge in the Gromovtopology to a Kontsevich stable map.

4. Rulling out sky catastrophes and non-squeezing

Proof of Theorem. 1.9 We shall actually prove a stronger statement that there is a universal energybound from above for class A, Jt-holomorphic curves. Suppose otherwise, then there is a sequenceuk of Jtk -holomorphic class A curves, with energyωtk

uk 7→ ∞ as k 7→ ∞. We may assume that tk

is convergent to t0 ∈ [0, 1]. Let uk be a lift of the curves to the covering space Mπ−→M determined

by the class α as described prior to the formulation of the theorem. If the image of uk is contained

(for a specific choice of lifts) in a compact K ⊂ M then we have:

energytk(uk) ' energyt0(uk) ≤ C〈ωsympt0 , A〉,

where ωt0 = fωsymp for ωsymp symplectic on M , f > 0 and C = supKf . Hence energy would beuniversally bounded for all uk.

Suppose there is no such K. Let uk be the corresponding sequence. We may suppose thatimageuk does not intersect Σi for all k and i, since otherwise uk must be a branched covering mapof a leaf of the Jtk -holomorphic foliation of Σi by the positivity of intersections, and consequently all

such uk have lifts contained in a specific compact subset of M .The Lee class α has a natural differential form representative, called the Lee form and defined as

follows. We take a cover of M by open sets Ua in which ω = fa ·ωa for ωa symplectic, and fa a positivesmooth function. Then we have 1-forms d(ln fa) in each Ua which glue to a well defined closed 1-formon M . We conflate the notation for this 1-form with its cohomology class, and the Cech 1-cocycle αdefined as before.

NON-SQUEEZING IN l. c. s. GEOMETRY AND HOLOMOPRHIC WEINSTEIN CONJECTURE 11

By our hypothesis that PD(α) =∑i ai[Σi] we have that π−1(M −

⋃i Σi) is a disjoint union tiKi

of bounded subsets, with respect to the proper function on M determined by the Lee 1-form α. Thenfor some k′ sufficiently large, the image of some lift uk′ intersects more then one of the Ki, and so uk′

intersects some Σi, a contradiction.

Proof of Theorem 1.6. We need to say what is our C0 topology on the space of l. c. s. forms. This isdefined via the following metric.

Definition 4.1. The metric d on on the space L(M) of l. c. s. 2-forms on M , is defined with respectto a fixed Riemannian metric g on M , and is given by

d(ω1, ω2) = dmass(ω1, ω2) + dmass(α1, α2),

for αi the Lee forms as above and dmass the metrics induced by the co-mass norms || · ||mass withrespect to g on differential k-forms. That is ||η||mass is the supremum over unit k-vectors v of |η(v)|.

Explicitly this means the following: a sequence ωk ⊂ L(M) converges to a symplectic form ω, iff

ωk, converges to ω, and iff given the lift sequence ωk = fkωsympk , on the universal cover M , with

ωsympk symplectic, the sequence fk is a sequence of positive functions converging to 1 on compactsets.

Fix an ε′ > 0 s.t. any 2-form ω1 on M , C0 ε′-close to ω, is non-degenerate, and is non-degenerateon the leaves of Σi. Suppose by contradiction that for every ε > 0 there exists an l. c. s. embedding

φ : BR → (M,ω1),

satisfying the conditions. Assume that ε < ε′, and let ωt be as in the hypothesis. By assumptionsωt is an l. c. s. form for each t, and is non-degenerate on the leaves of Σi. Extend φ∗j to an almostcomplex structure J1 on M , preserving T folΣi. We may then extend this to a family Jt of almostcomplex structures M , s.t. Jt is ωt-compatible for each t, and such that Jt preserves T folΣ for eachi, since the foliation of Σi is ωt-symplectic for each t. (For construction of Jt use for example themap ret from Lemma 3.8). When the image of φ does not intersect Σi this is vacuous. Then thefamily (ωt, Jt) satisfies the hypothesis of Theorem 1.9, and so has no sky catastrophes in class A.Consequently by Lemma 3.2 there is a class A J1-holomorphic curve u passing through φ(0). By the

proof of Theorem 1.9 we may choose a lift to M for each such curve u so that it is contained in a

compact set K ⊂ M , (independent of ε and all other choices). Now by definition of our d on L(M),for every δ we may find an ε so that if ω1 is ε-close to ω then ωsymp is δ-close to ωsymp1 on K. Since〈ωsymp, [u]〉 = πr2, if δ above is chosen to be sufficiently small then

|maxK

f1〈ωsymp1 , [u]〉 − π · r2| < πR2 − πr2,

since|〈ωsymp1 , [u]〉 − π · r2| ' |〈ωsymp, [u]〉 − π · r2| = 0,

for δ small enough, and maxK f1 ' 1 for δ small enough, where ' denotes approximate equality. Inparticular we get that ω1-area of u is less then πR2.

We may then proceed as in the classical proof Gromov [12] of the non-squeezing theorem to geta contradiction and finish the proof. More specifically φ−1(imageφ ∩ imageu) is a (nodal) minimalsurface in BR, with boundary on the boundary of BR, and passing through 0 ∈ BR. By constructionit has area strictly less then πR2 which is impossible by a classical result of differential geometry, (themonotonicity theorem.)

5. Connections of GW-theory of an l. c. s.m. C × S1 with the Fuller index

Proof of Proposition 2.2. Suppose we a have a curve without spherical nodal components u ∈M1,1(Jλ, A).We claim that (prC u)∗, has rank everywhere ≤ 1, for prC : C × S1 → C the projection. Supposeotherwise than it is immediate by construction of Jλ, that

∫Σu∗dλ > 0, for Σ domain of u, but dλ is

exact so that that this is impossible. It clearly follows from this that Σ must be smooth, (non-nodal).

12 YASHA SAVELYEV

Next observe that when the rank of (prC u)∗ is 1, its image is in the Reeb line sub-bundle of TC,for otherwise the image has a contact component, but this is Jλ invariant and so again we get that∫

Σu∗dλ > 0. We now show that the image of prC u is in fact the image of some Reeb orbit.

Pick an identification of the domain Σ of u with a marked Riemann surface (T 2, j), T 2 the standardtorus. We shall use throughout coordinates (θ1, θ2) on T 2 θ1, θ2 ∈ S1, with S1 unit complex numbers.Then by assumption on the class A (and WLOG) θ 7→ prS1 u(θ1

0× θ), is a degree 1 curve, whereprS1 : C × S1 → S1 is the projection. And so by the Sard theorem we have a regular value θ0, sothat S0 = u−1 pr−1

S1 (θ0) is an embedded circle in T 2. Now d(prS1 u) is surjective along T (T 2)|S0,

which means, since u is Jλ-holomorphic that prC u|S0has non-vanishing differential. From this and

the discussion above it follows that image of prC u is the image of some embedded Reeb orbit ou.Consequently the image of u is contained in the image of the Reeb torus of ou, and so (again by theassumption on A) u is a itself a Reeb torus map for some o covering ou.

The statement of the lemma follows when u has no spherical nodal components. On the other handnon-constant holomorphic spheres are impossible also by the previous argument. So there are no nodalelements in M1,1(Jλ, A) which completes the argument.

Proposition 5.1. Let (C, ξ) be a general contact manifold. If λ is non-degenerate contact 1-form forξ, then all the elements of M1,1(Jλ, A) are regular curves. Moreover if λ is degenerate then for aperiod P Reeb orbit o the kernel of the associated real linear Cauchy-Riemann operator for the Reebtorus of o is naturally identified with the 1-eigenspace of φλP,∗ - the time P linearized return map

ξ(o(0))→ ξ(o(0)) induced by the Rλ Reeb flow.

Proof. We have previously shown that all [u, j] ∈ M1,1(Jλ, A), are represented by smooth immersedcurves, (covering maps of Reeb tori). Since each u is immersed we may naturally get a splittingu∗T (C × S1) ' Nu × T (T 2), using gJ metric, where Nu denotes the pull-back normal bundle, whichis identified with the pullback along the projection of C × S1 → C of the distribution ξ.

The full associated real linear Cauchy-Riemann operator takes the form:

(5.2) DJu : Ω0(Nu ⊕ T (T 2))⊕ TjM1,1 → Ω0,1(T (T 2), Nu ⊕ T (T 2)).

This is an index 2 Fredholm operator (after standard Sobolev completions), whose restriction toΩ0(Nu ⊕ T (T 2)) preserves the splitting, that is the restricted operator splits as

D ⊕D′ : Ω0(Nu)⊕ Ω0(T (T 2))→ Ω0,1(T (T 2), Nu)⊕ Ω0,1(T (T 2), T (T 2)).

On the other hand the restricted Fredholm index 2 operator

Ω0(T (T 2))⊕ TjM1,1 → Ω0,1(T (T 2)),

is surjective by classical Teichmuller theory, see also [31, Lemma 3.3] for a precise argument in thissetting. It follows that DJ

u will be surjective if the restricted Fredholm index 0 operator

D : Ω0(Nu)→ Ω0,1(Nu),

has no kernel.The bundle Nu is symplectic with symplectic form on the fibers given by restriction of u∗dλ, and

together with Jλ this gives a Hermitian structure on Nu. We have a linear symplectic connection Aon Nu, which over the slices S1 × θ′2 ⊂ T 2 is induced by the pullback by u of the linearized Rλ

Reeb flow. Specifically the A-transport map from N |(θ′1,θ′2) to N |(θ′′1 ,θ′2) over [θ′1, θ′′2 ] × θ′2 ⊂ T 2,

0 ≤ θ′1 ≤ θ′′2 ≤ 2π is given by

(u∗|N |(θ′′1 ,θ′2))−1 φλmult·(θ′′1−θ′1) u∗|N |(θ′1,θ′2)

,

where mult is the multiplicity of o and where φλmult·(θ′′1−θ′1) is the time mult · (θ′′1 − θ′1) map for the Rλ

Reeb flow.The connection A is defined to be trivial in the θ2 direction, where trivial means that the parallel

transport maps are the id maps over θ2 rays. In particular the curvature RA of this connection vanishes.The connection A determines a real linear CR operator on Nu in the standard way (take the complex

NON-SQUEEZING IN l. c. s. GEOMETRY AND HOLOMOPRHIC WEINSTEIN CONJECTURE 13

anti-linear part of the vertical differential of a section). It is elementary to verify from the definitionsthat this operator is exactly D.

We have a differential 2-form Ω on the Nu which in the fibers of Nu is just the fiber symplecticform and which is defined to vanish on the horizontal distribution. The 2-form Ω is closed, whichwe may check explicitly by using that RA vanishes to obtain local symplectic trivializations of Nu inwhich A is trivial. Clearly Ω must vanish on the 0-section since it is a A-flat section. But any sectionis homotopic to the 0-section and so in particular if µ ∈ kerD then Ω vanishes on µ. But then sinceµ ∈ kerD, and so its vertical differential is complex linear, it must follow that the vertical differentialvanishes, since Ω(v, Jλv) > 0, for 0 6= v ∈ T vertNu and so otherwise we would have

∫µ

Ω > 0. So µ is

A-flat, in particular the restriction of µ over all slices S1 × θ′2 is identified with a period P orbit ofthe linearized at o Rλ Reeb flow, which does not depend on θ′2 as A is trivial in the θ2 variable. Sothe kernel of D is identified with the vector space of period P orbits of the linearized at o Rλ Reebflow, as needed.

Proposition 5.3. Let λ be a contact form on a 2n + 1-fold C, and o a non-degenerate, period P ,Rλ-Reeb orbit, then the orientation of [uo] induced by the determinant line bundle orientation ofM1,1(Jλ, A), is (−1)CZ(o)−n, which is

sign Det(Id |ξ(o(0)) − φλP,∗|ξ(o(0))).

Proof of Proposition 5.3. Abbreviate uo by u. Fix a trivialization φ of Nu induced by a trivializationof the contact distribution ξ along o in the obvious sense: Nu is the pullback of ξ along the composition

T 2 → S1 o−→ C.

Then the pullback A′ of A (as above) to T 2×R2n is a connection whose parallel transport path alongS1 × θ2, p : [0, 1] → Symp(R2n), starting at 1, is θ2 independent and so that the parallel transportpath of A′ along θ1 × S1 is constant, that is A′ is trivial in the θ2 variable. We shall call sucha connection A′ on T 2 × R2n induced by p. By non-degeneracy assumption on o, the map p(1) hasno 1-eigenvalues. Let p′′ : [0, 1] → Symp(R2n) be a path from p(1) to a unitary map p′′(1), withp′′(1) having no 1-eigenvalues, s.t. p′′ has only simple crossings with the Maslov cycle. Let p′ be theconcatenation of p and p′′. We then get

CZ(p′)− 1

2sign Γ(p′, 0) ≡ CZ(p′)− n ≡ 0 mod 2,

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 hby unitarity. Here sign Γ(p′, 0) is the index of the crossing form of the path p′ at time 0, in the notationof [20]. Consequently

(5.4) CZ(p′′) ≡ CZ(p)− n mod 2,

by additivity of the Conley-Zehnder index.Let us then define a free homotopy pt of p to p′, pt 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 pt. By the proof of Proposition 5.1, the CR operator Dt determined by each A′t issurjective except at some finite collection of times ti ∈ (0, 1), i ∈ N determined by the crossing timesof p′′ with the Maslov cycle, and the dimension of the kernel of Dti is the 1-eigenspace of p′′(ti), whichis 1 by the assumption that the crossings of p′′ are simple.

The operator D1 is not complex linear so we concatenate the homotopy Dt with the homotopy

Dt induced by the homotopy At of A′1 to a unitary connection A1, where the homotopy At,is through connections induced by paths pt, giving a homotopy relative end points of p′ = p0 to a

unitary path p1 (for example h above). Let us denote by D′t the concatenation of Dt with Dt.By construction in the second half of the homotopy D′t, D′t is surjective. And D′1 is induced by aunitary connection, since it is induced by unitary path p1. Consequently D′1 is complex linear. Bythe above construction, for the homotopy D′t, D′t is surjective except for N times in (0, 1), where

14 YASHA SAVELYEV

the kernel has dimension one. In particular the sign of [u] by the definition via the determinant linebundle is exactly

−1N = −1CZ(p)−n,

by (5.4), which was what to be proved.

Thus if N ⊂ M1,1(Jλ, Aβ) is open-compact and consists of isolated regular Reeb tori ui, corre-sponding to orbits oi we have:

GW1,1(N,Aβ , Jλ)([M1,1]⊗ [C × S1]) =

∑i

(−1)CZ(oi)−n

mult(oi),

where the denominator mult(oi) 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(N , Rλ, β). Thus the theorem follows for Nas above. However in general if N is open and compact then perturbing slightly we obtain a smoothfamily Rλt, λ0 = λ, s.t. λ1 is non-degenerate, that is has non-degenerate orbits. And such that

there is an open-compact subset N ofM1,1(Jλt, Aβ) with (N ∩M1,1(Jλ, Aβ)) = N , cf. Lemma 3.7.Then by Lemma 3.2 if

N1 = (N ∩M1,1(Jλ1 , Aβ))

we get

GW1,1(N,Aβ , Jλ)([M1,1]⊗ [C × S1]) = GW1,1(N1, Aβ , J

λ1)([M1,1]⊗ [C × S1]).

By previous discussion

GW1,1(N1, Aβ , Jλ1)([M1,1]⊗ [C × S1]) = i(N1, R

λ1 , β),

but by the invariance of Fuller index (see Appendix A),

i(N1, Rλ1 , β) = i(N,Rλ, β).

This finishes the proof of Theorem 2.6

Proof of Theorem 2.4. Let N ⊂M1,1(A, Jλ), be the subspace corresponding to the subspace N of allperiod 2π Rλ-orbits. (Under the Reeb tori correspondence.) It is easy to compute see for instance [11]

i(N , Rλ) = ±χ(CPk) 6= 0.

By Theorem 2.6 GW1,1(N, Jλ, A) 6= 0. The theorem then follows by Lemma 3.5.

6. Preliminaries for the Proof of Theorem 2.3

Much of the following discussion holds verbatim for general moduli spaces Mg,n(J,A, a1, . . . , an)with a1, . . . , an homological constraints. We shall however restrict for simplicity to the case g = 1, n = 1with trivial constraint [M ], as this is the interest of Theorem 2.4.

Definition 6.1. For a smooth homotopy h = (ωt, Jt) of l. c. s. pairs on M , we say that it is partiallyadmissible for A, if every element of

M1,1(M,J0, A)

is contained in a compact open subset of M1,1(M, Jt, A). We say that h is admissible for A ifevery element of

M1,1(M,Ji, A),

i = 0, 1 is contained in a compact open subset of M1,1(M, Jt, A).

NON-SQUEEZING IN l. c. s. GEOMETRY AND HOLOMOPRHIC WEINSTEIN CONJECTURE 15

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 capitalX a general l. c. s. pair (M,ω, J), then we introducethe following simplified notation.

(6.2)

S(X,A) = u ∈M1,1(X,A)S(X, a,A) = u ∈ S(X,A) | e(u) ≤ a.S(h,A) = u ∈M1,1(h,A)

S(h, a,A) = u ∈ S(h,A) | e(u) ≤ a

Definition 6.3. For an isolated element u of S(X,A), which means that u is open as a subset, weset gw(u, p) ∈ Q to be the local Gromov-Witten invariant of u. This is defined as:

gw(u, p) = GW1,1(u, A, J)([M1,1]⊗ [M ]),

with the right hand side as in (3.1).

Definition 6.4. Suppose that S(X,A) has open connected components. And suppose that we have acollection of l. c. s. pairs ⋃

a>0

(Xa = (M,ωa, Ja)),

satisfying the following:

• S(Xa, a, A) consists of isolated curves for each a.•

S(Xa, a, A) = S(Xb, a, A),

(equality of subsets of M1,1(X,A) × R+) if b > a, and the local Gromov-Witten invariantscorresponding to the identified elements of these sets coincide.• There is a prescribed homotopy ha = Xa

t of each Xa to X, called structure homotopy,with the property that for every

y ∈ S(Xa0 , A)

there is an open compact subset Cy 3 y of S(ha, A) which is non-branching which means that

Cy ∩ S(Xai , A),

i = 0, 1 are connected.•

S(ha, a, A) = S(hb, a, A),

(equality of subsets) if b > a is sufficiently large.

We will then say that

P(A) = (Xaa, ha)

is a perturbation system for X in the class A.

We shall see shortly that given a contact (C, λ), the associated l. c. s. structure on C × S1 alwaysadmits a perturbation system as above, if λ is Morse-Bott.

Definition 6.5. Suppose that X admits a perturbation system P(A) so that there exists an E =E(P(A)) with the property that

S(Xa, a, A) = S(XE , 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) =∑

u∈S(XE ,A)

gw(u).

16 YASHA SAVELYEV

Definition 6.6. Suppose that X admits a perturbation system P(A) and there is an E = E(P(A)) > 0so that gw(u) > 0 for all

u ∈ S(Xa, A) |E ≤ e(u) ≤ arespectively gw(u) < 0 for all

u ∈ S(Xa, A) |E ≤ e(u) ≤ a,and every a > E. Suppose in addition that

lima7→∞

∑u∈S(X,a,A)

gw(u) =∞, respectively lima7→∞

∑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) =∞,

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 6.7. We say that X is definite type if it is infinite type or finite type.

With the above definitions

GW (X,A) ∈ Q t∞t−∞,when it is defined.

Definition 6.8. An l. c. s. pair X is admissible if it admits a perturbation system, and if it is definitetype.

6.0.1. Perturbation systems for Morse-Bott Reeb vector fields.

Definition 6.9. A contact form λ on M , and its associated flow Rλ are called Morse-Bott if the λaction spectrum σ(λ) - that is the space of critical values of o 7→

∫S1 o

∗λ, is discreet and if for everya ∈ σ(λ), the space

Na := x ∈M |Fa(x) = x,Fa the time a flow map for Rλ - is a closed smooth manifold such that rank dλ|Na is locally constantand TxNa = ker(dFa − I)x.

Proposition 6.10. Let λ be a contact form of Morse-Bott type, on a closed contact manifold C. Thenthe corresponding l. c. s. pair Xλ = (C × S1, dαλ, Jλ) admits a perturbation system P(A), for everyclass A.

Proof. This follows immediately by [22, Proposition 2.12 ], and by Proposition 2.2.

Lemma 6.11. The Hopf l. c. s. pair (S2k+1 × S1, dαλH , JλH ), for λH the standard contact structure

on S2k+1 is infinite type.

Proof. This follows immediately by [21, Lemma 2. 13], and by Proposition 2.2.

6.1. Preliminaries on admissible homotopies.

Definition 6.12. Let h = Xt be a smooth homotopy of l. c. s. pairs. For b > a > 0 we say that h ispartially a, b-admissible, respectively a, b-admissible (in class A) if for each

y ∈ S(X0, a, A)

there is a compact open subset Cy 3 y of S(h,A), with e(u) < b, for all u ∈ Cy. Respectively, if for each

y ∈ S(Xi, a, A),

i = 0, 1 there is a compact open subset Cy 3 y of S(h,A) with e(u) < b, for all u ∈ Cy.

NON-SQUEEZING IN l. c. s. GEOMETRY AND HOLOMOPRHIC WEINSTEIN CONJECTURE 17

Lemma 6.13. Suppose that X0 has a perturbation system P(A), and Xt is partially admissible, then

for every a there is a b > a so that Xbt = Xt · Xb

t is partially a, b-admissible, where Xt · Xbt

is the (reparametrized to have t domain [0, 1]) concatenation of the homotopies Xt, Xbt , and where

Xbt is the structure homotopy from Xb to X0.

Proof. This is a matter of pure topology, and the proof is completely analogous to the proof of [21,Lemma 3.8].

The analogue of Lemma 6.13 in the admissible case is the following:

Lemma 6.14. Suppose that X0, X1 and Xt are admissible, then for every a there is a b > a so that

(6.15) Xbt = Xb

1,t−1 · Xt · Xb0,t

is a, b-admissible, where Xbi,t are the structure homotopies from Xb

i to Xi.

The proof of this is completely analogous to the proof of Lemma 6.13.

7. Proof of Theorem 2.3

Let us state a more general claim.

Theorem 7.1. Suppose we have an admissible l. c. s. pair X0, with GW (X0, A) 6= 0 which is joinedto X1 by a partially admissible homotopy Xt, then X1 has non-constant elliptic class A curves.

Theorem 2.3 clearly follows by the above and by Lemma 6.11. We also have a more a more preciseresult.

Theorem 7.2. If X0, X1 are admissible l. c. s. pairs and Xt is admissible then GW (X0, A) =GW (X1, A).

Proof of Theorem 7.1. Suppose that X0 is admissible with GW (X0, A) 6= 0, Xt is partially ad-

missible and M1,1(X1, A) = ∅. Let a be given and b determined so that hb = Xbt is a partially

(a, b)-admissible homotopy. We set

Sa =⋃y

Cy ⊂ S(hb, A),

for y ∈ S(Xb0, a, A). Here we use a natural identification of S(Xb, a, A) = S(Xb

0, a, A) as a subset of

S(hb, A) by its construction. Then Sa is an open-compact subset of S(h,A) and so has admits animplicit atlas (Kuranishi structure) with boundary, (with virtual dimension 1) s.t:

∂Sa = S(Xb, a, A) +Qa,

where Qa as a set is some subset (possibly empty), of elements u ∈ S(Xb, b, A) with e(u) ≥ a. So wehave for all a:

(7.3)∑u∈Qa

gw(u) +∑

u∈S(Xb,a,A)

gw(u) = 0.

7.1. Case I, X0 is finite type. Let E = E(P) be the corresponding cutoff value in the definition offinite type, and take any a > E. Then Qa = ∅ and by definition of E we have that the left side is∑

u∈S(Xb,E,A)

gw(u) 6= 0.

Clearly this gives a contradiction to (7.4).

18 YASHA SAVELYEV

7.2. Case II, X0 is infinite type. We may assume that GW (X0, A) = ∞, and take a > E, whereE = E(P(A)) is the corresponding cutoff value in the definition of infinite type. Then∑

u∈Qa

gw(u) ≥ 0,

as a > E(P(A)). While

lima 7→∞

∑u∈S(Xb,a,A)

gw(u) =∞,

as GW (X0, A) =∞. This also contradicts (7.4).

Proof of Theorem 7.2. This is somewhat analogous to the proof of Theorem 7.1. Suppose thatXi, Xtare admissible as in the hypothesis. Let a be given and b determined so that hb = Xb

t , see (6.15) isan (a, b)-admissible homotopy. We set

Sa =⋃y

Cy ⊂ S(hb, A)

for y ∈ S(Xbi , a, A). Then Sa 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:

∂Sa = (S(Xb0, a, A) +Qa,0)op + S(Xb

1, a, A) +Qa,1,

with op denoting opositite orientation and where Qa,i as sets are some subsets (possibly empty), ofelements u ∈ S(Xb

i , b, A) with e(u) ≥ a. So we have for all a:

(7.4)∑

u∈Qa,0

gw(u) +∑

u∈S(Xb0 ,a,A)

gw(u) =∑

u∈Qa,1

gw(u) +∑

u∈S(Xb1 ,a,A)

gw(u)

7.3. Case I, X0 is finite type and X1 is infinite type. Suppose in addition GW (X1, A) =∞ andlet E = max(E(P0(A)), E(P1(A))), for Pi, the perturbation systems of Xi. Take any a > E. ThenQa,0 = ∅ and the left hand side of (7.4) is ∑

u∈S(Xb0 ,E,A)

gw(u).

While the right hand side tends to ∞ as a tends to infinity since,∑u∈Qa,1

gw(u) ≥ 0,

as a > E(P1(A)), and

lima 7→∞

∑u∈S(Xb1 ,a,A)

gw(u) =∞,

Clearly this gives a contradiction to (7.4).

7.4. Case II, Xi are infinite type. Suppose in addition GW (X0, A) = −∞, GW (X1, A) =∞ andlet E = max(E(P0(A)), E(P1(A))), for Pi, the perturbation systems of Xi. Take any a > E. Then∑u∈Qa,0 gw(u) ≤ 0, and

∑u∈Qa,1 gw(u) ≥ 0. So by definition of GW (Xi, A) the left hand side of (7.4)

tends to −∞ as a tends to ∞, and the right hand side tends to ∞. Clearly this gives a contradictionto (7.4).

7.5. Case III, Xi are finite type. The argument is analogous.

NON-SQUEEZING IN l. c. s. GEOMETRY AND HOLOMOPRHIC WEINSTEIN CONJECTURE 19

A. Fuller index

Let X be a vector field on M . Set

S(X) = S(X,β) = (o, p) ∈ LβM × (0,∞) | o : R/Z→M is a periodic orbit of pX,

where LβM denotes the free homotopy class β component of the free loop space. Elements of S(X) willbe called orbits. There is a natural S1 reparametrization action on S(X), and elements of S(X)/S1

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 > 0 theleast period of 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) =∑

(o,p)∈N/S1

1

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 Xt, t ∈ [0, 1] let

S(Xt, β) = (o, p, t) ∈ LβM × (0,∞)× [0, 1] | o : R/Z→M is a periodic orbit of pXt.

Given a continuous homotopy Xt, X0 = X, t ∈ [0, 1], suppose that N is an open compact subsetof S(Xt), such that

N ∩ (LM × R+ × 0) = N.

Then if

N1 = N ∩ (LM × R+ × 1)we have

i(N,X, β) = i(N1, X1, β).

In the case where X is the Rλ-Reeb vector field on a contact manifold (C2n+1, ξ), and if (o, p) isnon-degenerate, we have:

(A.1) i(o, p) = sign Det(Id |ξ(x) − Fλp,∗|ξ(x)) = (−1)CZ(o)−n,

where Fλp,∗ is the differential at x of the time p flow map of Rλ, and where CZ(o) is the Conley-Zehnderindex, (which is a special kind of Maslov index) see [20].

B. Virtual fundamental class

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 2.6. 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. (Note that the story of the Euler class is older than thework of Fukaya-Ono, and there is possibly prior work in this direction.) When this is in degree 0,the corresponding Gromov-Witten invariant is

∫[X]vir

1. However they assume that their orbifolds are

effective. This assumption is not really necessary for the purpose of construction of the Euler class butis convenient for other technical reasons. A different approach to the virtual fundamental class whichemphasizes branched manifolds is used by McDuff-Wehrheim, see for example McDuff [15], which doesnot have the effectivity assumption, a similar use of branched manifolds appears in [3]. In the caseof a non-effective orbibundle E → X McDuff [16], constructs a homological Euler class e(E) usingmulti-sections, which extends the construction [9]. McDuff shows that this class e(E) is Poincare dualto the completely formally natural cohomological Euler class of E, constructed by other authors. In

20 YASHA SAVELYEV

other words there is a natural notion of a homological Euler class of a possibly non-effective orbibundle.We shall assume the following black box 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).

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 [19]. And it issatisfied by the construction of Fukaya-Oh-Ono-Ohta [7], although not quiet immediately. This isalso communicated to me by Kaoru Ono. When X is 0-dimensional this does follow immediately bythe construction in [9], taking any effective Kuranishi neighborhood at the isolated points of X, (thisactually suffices for our paper.)

As a special case most relevant to us here, suppose we have a moduli space of elliptic curves inX, 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

∫[X]vir

1 is ±1[Γ([u])] , where [Γ([u])] is the order of the isotropy group Γ([u]) of [u], in the

McDuff-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. I thank John Pardon, Viktor Ginzburg and Dusa McDuff, for related discussions.Thanks also to Yael Karshon, Yasha Eliashberg, Peter Topping, Kaoru Ono and Kenji Fukaya, forhelpful comments.

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E-mail address: yasha.savelyev@gmail.com

University of Colima, CUICBAS