symplectic embeddings of polydisks

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Sel. Math. New Ser. DOI 10.1007/s00029-013-0146-2 Selecta Mathematica New Series Symplectic embeddings of polydisks R. Hind · S. Lisi © Springer Basel 2014 Abstract We show that the polydisk P (1, 2), the product of disks of areas 1 and 2, can be symplectically embedded in a ball B( R) of capacity R if and only if R 3. Hence, the inclusion map gives the optimal embedding and neither the Embedded Contact Homology nor Ekeland–Hofer capacities give sharp obstructions in this situation. Our proof applies the theory of pseudoholomorphic curves in manifolds with cylindrical ends, and in particular finite energy foliations. Keywords Symplectic polydisk · Symplectic embedding · Lagrangian submanifold · Pseudoholomorphic curve · Finite energy foliation Mathematical Subject Classification (2000) 53D05 · 53D12 · 53D42 1 Introduction and notation In this paper, we obtain new obstructions to symplectic embeddings of a product of disks (a polydisk) into a 4-dimensional ball. Let us equip R 4 with coordinates x 1 , y 1 , x 2 , y 2 and its standard symplectic form ω = dx 1 d y 1 + dx 2 d y 2 . The polydisk P (r, s ) is defined to be P (r, s ) = π(x 2 1 + y 2 1 ) r,π(x 2 2 + y 2 2 ) s with symplectic form induced from R 4 . The ball of capacity R (i.e., the ball of radius R π ) is R. Hind (B ) Department of Mathematics, University of Notre Dame, Notre Dame, IN, USA e-mail: [email protected] S. Lisi Department of Mathematics, Université de Nantes, Nantes, France e-mail: [email protected]

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Page 1: Symplectic embeddings of polydisks

Sel. Math. New Ser.DOI 10.1007/s00029-013-0146-2

Selecta MathematicaNew Series

Symplectic embeddings of polydisks

R. Hind · S. Lisi

© Springer Basel 2014

Abstract We show that the polydisk P(1, 2), the product of disks of areas 1 and 2, canbe symplectically embedded in a ball B(R) of capacity R if and only if R ≥ 3. Hence,the inclusion map gives the optimal embedding and neither the Embedded ContactHomology nor Ekeland–Hofer capacities give sharp obstructions in this situation. Ourproof applies the theory of pseudoholomorphic curves in manifolds with cylindricalends, and in particular finite energy foliations.

Keywords Symplectic polydisk · Symplectic embedding · Lagrangian submanifold ·Pseudoholomorphic curve · Finite energy foliation

Mathematical Subject Classification (2000) 53D05 · 53D12 · 53D42

1 Introduction and notation

In this paper, we obtain new obstructions to symplectic embeddings of a product ofdisks (a polydisk) into a 4-dimensional ball.

Let us equip R4 with coordinates x1, y1, x2, y2 and its standard symplectic form

ω = dx1 ∧ dy1 + dx2 ∧ dy2. The polydisk P(r, s) is defined to be

P(r, s) ={π(x2

1 + y21 ) ≤ r, π(x2

2 + y22 ) ≤ s

}

with symplectic form induced from R4. The ball of capacity R (i.e., the ball of radius√

) is

R. Hind (B)Department of Mathematics, University of Notre Dame, Notre Dame, IN, USAe-mail: [email protected]

S. LisiDepartment of Mathematics, Université de Nantes, Nantes, Francee-mail: [email protected]

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R. Hind, S. Lisi

B(R) ={π(x2

1 + y21 + x2

2 + y22 ) ≤ R

}

again with the induced symplectic form.Given r, s, the embedding problem for the polydisk P(r, s) into a ball is to find the

infimum of the set of R such that there exists a symplectic embedding P(r, s) → B(R).There are two well-known sequences of symplectic capacities giving obstructions

to symplectic embeddings in 4 dimensions, namely the Ekeland–Hofer capacities [4](which are actually defined in all dimensions) and the Embedded Contact Homology(ECH) capacities introduced by Hutchings [13]. In the case when (r, s) = (1, 2), boththe Ekeland–Hofer and ECH capacities give the obstruction R ≥ 2. As P(1, 2) andB(2) have the same volume, this restriction also follows since symplectic embeddingspreserve volume.

Our main theorem solves the embedding problem for polydisks in the case when(r, s) = (1, 2).

Theorem 1.1 There exists a symplectic embedding

P(1, 2):=D(1)× D(2) ↪→ B(R)

if and only if R ≥ 3.

The sufficiency of R ≥ 3 follows by the inclusion P(1, 2) ⊂ B(3). In general,we have P(r, s) ⊂ B(r + s). Hence, the theorem can be rephrased as saying thatfor P(1, 2); the inclusion map gives the optimal embedding. The necessity of R ≥ 3implies that for this particular embedding problem, neither the Ekeland–Hofer norECH capacities give a sharp obstruction. We contrast this with the case of ellipsoidembeddings into a ball when the ECH capacities give a complete list of obstructions,see [16]. Our obstruction does not come from a symplectic capacity, but instead frompseudoholomorphic foliations; thus, the techniques seem to be special to dimension 4.

The embedding problem is discussed at length by Schlenk in the book [18], andin particular the technique of symplectic folding. One consequence of this is thefollowing.

Theorem 1.2 ([18], Proposition 4.3.9) If s > 2, then there exists a symplectic embed-ding P(1, s) → B(R) for all R > 2 + s

2 .

When s > 2 we have 2 + s2 < 1 + s, and so this result implies that for P(1, s) with

s > 2 the inclusion map is never optimal.Finally, we remark that when s increases, even the embeddings in Theorem 1.2 are

far from optimal. Indeed, Theorem 3 in [18] implies that by taking s sufficiently large,we can find embeddings P(1, s) → B(R) whose image occupies an arbitrarily largeproportion of the volume.

1.1 Organization and outline of the proof

Define CP2(R) to be complex projective space, equipped with the Fubini-Studysymplectic form scaled so the area of the line is R, and denote by CP1(∞) the

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line at infinity. Then CP2(R) is the union of the open ball of capacity R andCP1(∞). Our goal is to show that for any R < 3, there does not exist an embed-ding P(1, 2) ↪→ CP2(R)\CP1(∞).

Our proof will be by contradiction. Assume such an embedding exists. Let L bethe Lagrangian torus at the ‘corner’ of the bidisk:

L = ∂D(1)× ∂D(2) ⊂ P(1, 2) ⊂ CP2(R)\CP1(∞) ⊂ CP2(R).

Note that the (closed) ball of capacity 1, centered at (0, 0), sits in the (closed) bidiskD(1)× D(2) and does not intersect L . Let X be the symplectic manifold obtained byblowing up this ball of capacity 1: X = CP2(R)�CP2(1). Let ωR be the symplecticform on X . Denote the exceptional divisor by E . Note that E is a symplectic sphereof area 1 and that X is the non-trivial CP1 bundle over CP1.

In Sect. 2, we describe two classes of holomorphic curves in X , namely curves inthe class of a fiber which foliate X , and sections S of high degree d = S • CP1(∞).Such curves exist for any tame almost complex structure on X .

Next, we will stretch the neck along the boundary of a small tubular neighborhoodof L as described in [2] and take limits of the closed curves described in Sect. 2. Thestretching is described in Sect. 3 together with the Fredholm theory for the limitingcurves.

We will also be interested in intersection properties of the limiting finite energycurves. Therefore, we briefly describe Siefring’s extended intersection number inSect. 4.

With all of this in place, the limits of curves in the fiber class are considered inSect. 5. They form a finite energy foliation, see [11].

Finally, in Sect. 6, we consider the limit of a high degree curve with pointwiseconstraints (the degree will depend on R). We are able to make deductions about thislimit based on the structure of the finite energy foliation, and we will see that for thelimit to have nonnegative area it is necessary that R ≥ 3, thus proving Theorem 1.1.

To fix notation, let (k, l) ∈ H1(L) denote the homology class k[∂D(1)]+l[∂D(2)].2 Closed curves in X

We study two classes of closed curves in X = CP2(R)�CP2(1), the blow-up ofa ball of capacity 1 in CP2 with its Fubini-Study form scaled such that lines havearea R. Throughout we work with tame almost complex structures J on X for whichthe exceptional divisor E and the line at infinity CP1(∞) are complex. Recall fromSect. 1.1 that the blown-up ball is located inside our polydisk and so disjoint fromCP1(∞).

The results are fairly standard, and so we merely outline the proofs. Throughout,by the degree of a holomorphic curve, we will mean its intersection with CP2(∞).

2.1 Fiber curves

The result here is the following, which does not use the assumption R < 3. Given theassumption above that both the line at infinity and the exceptional divisor are complex,

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R. Hind, S. Lisi

nor does it require genericity of J . (Our proof is very close in spirit to the analysis inMcDuff [15].)

Proposition 2.1 (see [5], Proposition 4.1) Let J be an almost complex structure asabove. The manifold X is foliated by J holomorphic spheres in the class [CP1(∞)]−[E].Outline of the proof

Let C be a holomorphic sphere in the class [CP1(∞)] − [E]. Then C • C = 0.The index formula, see for example [17, Theorem 3.1.5] gives the virtual dimensionindex(C) = 2. By positivity of intersection, such a curve C must intersect CP1(∞)

transversally in a single point and so be somewhere injective. Further, by the adjunctionformula [17, Section 2.6], C is embedded. Then by automatic regularity, see [17,Lemma 3.3.3], the corresponding moduli space has dimension 2.

We now claim that this homology class is not represented by any nodal curve. Sup-pose instead that we have such a nodal representative. It must have a single componentof degree 1, which we denote by C1. Let e denote its intersection number with theexceptional class E , with e ≥ 0 since E is represented by a holomorphic sphere. Thiscomponent C1 is somewhere injective by the fact it is of degree 1, and it represents theclass A:=[CP1(∞)]−e[E]. Observe now that 〈c1(T X), [CP1(∞)]−e[E]〉 = 3−eand A • A = 1 − e2. The adjunction inequality gives then

0 ≤ A • A − c1(A)+ χS2 = 1 − e2 − 3 + e + 2 = e − e2.

Thus, e = 1 or e = 0. If e = 1, the remaining components of the nodal curve representthe 0 homology class in X , which cannot be represented by a non-constant curve. Ife = 0, the remaining components of the nodal curve represent the class −[E], whichhas negative symplectic area and thus cannot be represented by a holomorphic curve.Thus, no such nodal curve can represent the fiber class.

It follows then that the moduli space of curves in the fiber class is compact and asC • C = 0 consists of curves with disjoint image. Hence, the J -holomorphic fibercurves foliate X whenever the moduli space is non-empty.

Now, we know that for a standard J , fiber curves exist. Indeed, for the integrableJ , they arise by blowing up lines passing through the origin. As curves in the class areembedded, automatic regularity again implies that the set of J for which the curvesexist is open. Since there are no nodal curves for any J , by compactness this set ofJ is also closed. Hence, J -holomorphic fiber curves exist for all J , and the proof iscomplete. ��

2.2 High degree curves

We now consider holomorphic spheres S in X of high degree d > 1. These come inhigh dimensional families, so to work with isolated curves we impose a collection ofpoint constraints. The following result is similar to [6, Proposition 2.2].

Proposition 2.2 Let J be a regular almost complex structure on X (so all somewhereinjective curves are regular).

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Then, there is a co-meager set P ⊂ X2d consisting of 2d constraint points so thatfor each tuple of constraints p1, . . . , p2d , there is a unique embedded holomorphicsphere S in the class d[CP1(∞)] − (d − 1)[E] passing through the points.

Outline of the proofLet J be a fixed, regular almost complex structure.Let S be a holomorphic sphere in the class d[CP1(∞)]− (d − 1)[E]. If the sphere

is smooth (i.e., not a nodal curve), it is somewhere injective because d and d − 1 arecoprime. By the adjunction formula as above, the curve is then embedded.

Next, an index calculation shows that nodal representatives of this homology classare of codimension at least 2. Thus, as in [6, Lemma 2.5], as index(S) = 4d and J isregular, the image of the nodal representatives of this class (equipped with 2d markedpoints) under the evaluation map to X2d has codimension 2. It follows that a nodalcurve will not intersect points p1, . . . , p2d if they are chosen generically, i.e., outsideof this codimension 2 set.

Also, as index(S) = 4d, the virtual dimension of curves in our class passing through2d constraint points is 0, and as S • S = 2d − 1, there can be at most one such smoothcurve.

Finally, we are left to check that there exists a smooth curve intersecting the con-straint points. We observe that there exists a smooth curve in the class for some J .Indeed, let us choose a complex structure such that the projection X → CP1(∞)

along the fibers of the above foliation is holomorphic, describing X as a non-trivialholomorphic CP1 bundle over CP1. Our curves are sections of the above bundle withd ‘zeros’ (intersections with CP1(∞)) and d − 1 ‘poles’ (intersections with E) andthese exist because the bundle has degree 1.

Given such a J -holomorphic curve, by choosing the constraint points to lie on thecurve in a generic position, by the uniqueness above we can deduce that the corre-sponding Gromov–Witten invariant counting curves in the class through 2d constraintsis ±1. Hence, we will be able to find a curve for all regular J and generic constraintpoints as required. ��

In the following, we will consider a sequence of almost complex structures JN , eachof which is regular. Since the countable intersection of co-meager sets is co-meager, ageneric choice of p1, . . . , p2d gives existence of an embedded smooth curve for eachJN .

3 Fredholm theory

In this section, we describe our neck stretching procedure and determine the Fredholmindex of the types of curve that can appear in the holomorphic buildings we obtain bystretching the neck.

3.1 Stretching the neck

To perform a neck stretch, we must choose a tubular neighborhood U of L in X . Recallthat L = ∂D(1)×∂D(2) ⊂ P(1, 2)�CP2(1) ⊂ X . By Weinstein’s theorem, a tubular

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R. Hind, S. Lisi

neighborhood can be identified with a neighborhood of the zero-section in T ∗L . Wefix a flat metric on L and let our neighborhood be a unit cotangent disk bundle withboundary � = T 3. The Liouville form on T ∗L restricts to a contact form λ on �and the associated Reeb vector field v generates the geodesic flow. We can identify aneighborhood of � in X with the symplectic manifold (� × (−ε, ε), d(etλ)), whereλ is pulled back from � to � × (−ε, ε) using the natural projection.

Now choose a sequence of almost complex structures J N on CP2(R). We arrangethat these all coincide outside of a neighborhood of � and also that both E andCP1(∞) are J N –holomorphic. Further, on the neighborhood�× (−ε, ε)we assumethat all J N preserve the planes ξ = {λ = 0} and coincide on these planes. However,we require J N (v) = − 1

N∂∂t . Then (CP2, J N ) admits a biholomorphic embedding of

� × (−N , N ) equipped with a translation invariant complex structure mapping theReeb vectors to the unit vectors in (−N , N ).

A sequence of J N holomorphic spheres of fixed degree has a limit in the senseof [2]. This is a holomorphic building whose components are finite energy curvesmapping into three almost complex manifolds with cylindrical ends, namely

(1) X\U ∪� × (−∞, 0];(2) � × R;(3) U ∪� × [0,∞).

The almost complex structures here coincide with the J N away from the ends andare translation invariant on the cylindrical ends. We note that given a compact set Kin any of these three cylindrical manifolds there exists a biholomorphic embeddingK → (X, J N ) for all N sufficiently large. For the theory of finite energy curves see[8–10]. We will often use diffeomorphisms to identify the first and third cylindricalmanifolds with X\L and T ∗L , respectively.

The finite energy curves constituting our holomorphic building have a level struc-ture. If the building has k + 1 levels, then we say that the curves mapping to T ∗L areat level 0, the curves mapping to X\L are at level k and the curves mapping to�× R

are at a level l with 0 < l < k. Finite energy curves of level 0 have only positiveends, curves of level k have only negative ends, and the punctures of curves mappingto � × R are positive if the t coordinate approaches +∞ as we approach the punc-ture and negative if the t coordinate approaches −∞. For a union of the finite energycurves to form a building, we require that the punctures be matched in pairs such thateach positive end of a curve of level l is matched with a negative end of a curve oflevel l + 1 asymptotic to the same geodesic. By identifying the underlying Riemannsurfaces (that is, the domains of the finite energy curves) at matching punctures, wecan think of the domain of a holomorphic building as a nodal Riemann surface.

3.2 Compactification and area

The finite energy curves in all three manifolds have punctures asymptotic to Reeborbits on � = T 3. Under our identification of the first manifold with X\L , thesecurves can be extended to maps from the oriented blow-up of the Riemann surface atits punctures, mapping the boundary circles to the asymptotic closed Reeb orbit onL , see Proposition 5.10 of [2]. Away from their singularities the finite energy curves

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are now symplectic immersions, that is, the pull back of the symplectic form on X isnon-degenerate. The integral of this symplectic form is defined to be the area of thefinite energy curve. As L is Lagrangian, it follows from the compactness theorem of[2] that for a converging sequence of J N holomorphic spheres in a fixed homologyclass A ∈ H2(X,Z), the sum of the areas of the components of the limiting buildingin X\L is equal to ωR(A).

3.3 Fredholm indices

By our construction, the Reeb dynamics on� = T 3 gives the geodesic flow on L = T 2

for the flat metric. The Reeb orbits on � = T 3 are then in bijective correspondencewith the geodesics on T 2. There is an S1 family of closed geodesics in each homologyclass (k, l) for (k, l) ∈ Z

2\{(0, 0)}. Here we recall from Sect. 1.1 that L lies inthe boundary of a polydisk P(1, 2) and (k, l) ∈ H1(L) denotes the homology classk[∂D(1)] + l[∂D(2)]

We recall our convention that a curve in X\L is of degree d if its intersection numberwith CP1(∞) is d. Denote its intersection number with E by e. Observe that sincethese are intersections with classes that have holomorphic representatives, d ≥ 0 ande ≥ 0, unless the holomorphic curve is a cover of E (in which case, e < 0 and d = 0).

If the holomorphic curve in question is somewhere injective, then for a genericalmost complex structure, the moduli space of nearby finite energy curves will havedimension given by its Fredholm index, and in the remainder of this section, we listthe relevant indices.

Each of these index calculations follows immediately as an application of theappropriate version of Riemann–Roch and by the computation of the Conley–Zehnderindices of the Reeb orbits in T 3, where the computation is done with respect to thetrivialization induced by T ∗T 2 for Propositions 3.2 and 3.3, and with respect to thetrivialization from the bidisk D2 × D2 for Proposition 3.1 (see [3], Appendix A).

We start with curves in X\L .

Proposition 3.1 Let C be a curve in X\L of degree d, with e intersections with E,and with s negative ends asymptotic to geodesics in the classes (ki , li ), respectively,for 1 ≤ i ≤ s.

The index of C (as an unparametrized curve, allowing the asymptotic ends to movein the corresponding S1 family of Reeb orbits) is given by

index(C) = s − 2 + 6d − 2e + 2s∑

i=1

(ki + li ).

��For curves in R �, we have

Proposition 3.2 The index of a finite energy curve C in R × � of genus 0 with s+positive punctures and s− negative punctures is given by

index(C) = 2s+ + s− − 2.

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Finally, for curves in T ∗L we obtain,

Proposition 3.3 The index of a genus 0, finite energy curve C in T ∗T 2, with s (posi-tive) punctures, is current work of Siefring and Wendl will study Morse-Bott intersec-tion theory generally

index(C) = 2s − 2.

Note that we may consider a (sub)building consisting of a level in T ∗L and possiblymultiple levels in R × T 3 with the requirement of matching ends. If we now interprets to be the number of positive punctures of the uppermost level of the building, theindex formula of Proposition 3.3 also applies, giving the deformation index of thespace of buildings.

4 Intersections of finite energy curves

Siefring’s intersection theory for punctured pseudoholomorphic curves, extended bySiefring and Wendl to the Morse–Bott case [21–23, Section 4.1] is very useful to ourstudy and we briefly outline this here.

We require the existence of the following generalization of the intersection number,valid for curves with cylindrical ends. We say that a curve has cylindrical ends if it canbe parametrized by a punctured Riemann surface and is pseudoholomorphic of finiteenergy in a neighborhood of the punctures.

Theorem 4.1 ([21, Theorem 2.1; 23, Section 4.1]) Let (W, ω, J ) be an almost com-plex manifold with cylindrical ends that are symplectizations of contact manifolds,with almost complex structure adapted to Morse–Bott contact forms.

Then, for two curves with cylindrical ends, u, v in W , there exists an extendedintersection number u � v with the following properties:

(1) If u and v asymptote to disjoint sets of orbits, u � v is the intersection numberu • v.

(2) The extended intersection number u � v is invariant under homotopies of u and vthrough curves with cylindrical ends.

(3) If u and v are geometrically distinct (i.e., their images do not coincide on anyopen set), u � v ≥ 0.

Furthermore, if a sequence of curves representing classes C and C ′ in a compactmanifold X converge to holomorphic buildings with levels u1, . . . , uN and levelsu′

1, . . . , u′N , the following inequality holds: C • C ′ ≥ ∑

ui � u′i .

In our setting, the existence of this intersection number can be obtained by hand.Specifically, in X\L , the intersection number u � v of two curves u and v is obtainedby pushing u off of itself with a small perturbation that is tangent to the Morse–Bottfamily of orbits at each puncture to obtain a curve u′. Then take u � v = u′ • v.

A key fact necessary for the existence of this extended intersection number, but alsofor its computation, is the asymptotic convergence formula for pseudoholomorphiccurves. We will use this later, so we recall it here for the benefit of the reader. The

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statement we provide here is an immediate corollary of the more general results ofHofer, Wysocki and Zehnder and of Siefring.

First, we recall the definition of an asymptotic operator associated with a closedReeb orbit. Let (M, ker α) be a contact manifold with preferred contact form α, andlet R be the associated Reeb vector field. Let J be an almost complex structure onξ = ker α compatible with dα|ξ . Let γ : R/T Z → M be a closed Reeb orbit of periodT (not necessarily the minimal period). Choose a torsion-free connection ∇. Then,the asymptotic operator at the orbit γ is the unbounded self-adjoint operator:

Aγ,J : L2(γ ∗ξ) → L2(γ ∗ξ)η �→ −J (∇∂tη − ∇ηR).

Note that this differential operator does not depend on the choice of connection, butdoes depend on choice of almost complex structure.

We state the asymptotic convergence theorem for the case of a negative puncture,since this is the case we need. A positive puncture behaves analogously, with the signsmodified appropriately. By convention, we take S1 = R/Z.

Theorem 4.2 ([9], Theorem 1.3; [20], Theorems 2.2 and 2.3) Let (W, ω, J ) be analmost complex manifold with cylindrical ends that are symplectizations of contactmanifolds, with almost complex structure adapted to Morse–Bott contact forms.

Let u1 and u2 be finite energy punctured pseudoholomorphic half-cylindersu1, u2 : (−∞, 0] × S1 → X, asymptotic to the same (possibly multiply covered)orbit γ of period T > 0.

Then, for R << 0, there exist proper embeddings ψ1, ψ2 : (−∞, R] × S1 →(−∞, 0] × S1, asymptotic to the identity, maps Ui : (−∞, R] × S1 → (γ )∗ξ withUi (s, t) ∈ ξ(γ (T t)) so that

ui (ψi (s, t)) = (−T s, expγ (T t) Ui (s, t)).

Furthermore, either U1 − U2 vanishes identically, or there exists a positive eigen-value λ > 0 of the asymptotic operator Aγ,J and a corresponding eigenvector e(t) sothat

U1(s, t)− U2(s, t) = eTλs(e(T t)+ r(s, t)),

where r and all its derivatives decay exponentially fast.

In particular, we will apply this theorem in the case where the two negative endscome from curves asymptotic to different covers of the same simple orbit. In thatcase, we can reparametrize the ends by factoring through a covering map so that bothconverge to the same (higher multiplicity) orbit.

5 A finite energy foliation

Recalling Proposition 2.1, we may assume that for each N < ∞ there exists a foliationof X by JN holomorphic spheres in the class of a fiber F = [CP1(∞)] − [E] of the

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bundle X → E . For this, JN is chosen so that both the line at infinity CP1(∞) andthe exceptional divisor E are complex. Since these are regular curves disjoint fromL , we may also assume that each JN is regular and induces regular almost complexstructures with cylindrical ends on the manifolds X\L , T ∗L and � × R as describedin Sect. 3.1.

We will follow the method of [11] to construct a foliation of X\L by finite energycurves. To this end, we fix a countable dense set of points {pi } ∈ X . For each N ,there exists a unique curve C N

i in the fiber class intersecting pi . Taking a diagonalsubsequence of N → ∞, we may assume that all C N

i converge as N → ∞ toholomorphic buildings having a component Ci passing through pi . By taking furtherlimits of the Ci , as in [11], we obtain a collection of curves passing through eachpoint of X\L . Denote this collection by F . Positivity of intersection implies that thesecurves are either disjoint or have identical image. It is useful to note that once a diagonalsubsequence of N has been fixed, in fact every curve in F has image equal to that ofa limit of JN holomorphic spheres in the fiber class, and moreover, for all limits, thecomponent in X\L lies in F . This also follows by positivity of intersection. Hence,we could also describe F as the components in X\L of all limits of JN holomorphicspheres in the fiber class.

In Sect. 5.1, we describe the curves that arise in F and show it is a finite energyfoliation in the sense of [11].

Similarly, the limits of the C Ni also have components in T ∗L and R×�. Proceeding

similarly, we obtain curves in T ∗L and in R×� passing through each point. In Sect. 5.2,we describe these curves and show they form a finite energy foliation.

5.1 Limit curves in X\L

Proposition 5.1 The curves in F in X\L are of three kinds:

(1) Embedded closed spheres in the fiber class.(2) Planes C0 of degree 0 asymptotic to (1, 0) geodesics.(3) Planes C1 of degree d = 1 with e = C1 • E = 1 and asymptotic to (−1, 0)

geodesics.

These curves form a foliation of X\L.

Proof We pick a point p ∈ X\L and look at a limit of J N -holomorphic spheres C N

which lie in the fiber class and intersect p. As remarked above, the component of thelimit through p is a curve of F through p. If C N converges to a closed curve then itis of the first type. Nodal curves in X\L are excluded as in Sect. 2.

Now suppose that the limit is a (non-trivial) holomorphic building, B. Observe thatthe symplectic area of each of the C N is R −1, and thus by our assumption that R < 3the sum of the areas of the components in X\L is also R − 1 < 2 (as symplecticarea is preserved in the limit, see the discussion in Sect. 3.2). Also, recall that for anycurve C in X\L , of degree d and intersection number e with the exceptional divisor,and negative ends asymptotic to geodesics in classes (ki , li ), we can compactify to asymplectic surface with boundary on the relevant geodesics (see again Sect. 3.2) andthen the area is given by

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area (C) = Rd − e +∑

(ki + 2li ). (1)

By positivity of intersections, exactly one curve of B in X\L has degree 1. Byformula (1), the degree 0 components have positive integral area, and so as the totalarea is R − 1 < 2, there can be at most one of these. Thus, for the building B to benon-trivial (i.e., not a closed curve), there must be exactly two components in X\L ,one of area 1 and the other of degree 1 and area R − 2. Also, by our area bounds, wesee that neither component can contain a closed curve. As B is a limit of curves ofgenus 0, after identifying matching ends the building itself must have genus 0. Thus,as there are no finite energy planes in T ∗L (as there are no contractible geodesics) weconclude that that two components must both be planes, each with a single negativeend.

We denote these two planes by C0 and C1, where C0 is of degree 0 and asymptotic toa (k, l) geodesic and C1 is of degree 1. As C0 and C1 are components of a holomorphicbuilding whose other components lie in T ∗L , we necessarily have that C1 is asymptoticto a (−k,−l) geodesic.

Note that by the area condition, C0 must be somewhere injective and that C1 mustbe somewhere injective since it is of degree 1.

By positivity of intersection, exactly one of C0 or C1 can intersect the exceptionaldivisor E and so we reduce to two cases.

Case 1: C0 • E = 0; C1 • E = 1.Since the curves are somewhere injective and our almost complex structure is

assumed regular, their Fredholm indices must be non-negative. Therefore,

index(C0) = −1 + 2(k + l) ≥ 0

area(C0) = k + 2l = 1.

index(C1) = 3 − 2(k + l) ≥ 0

area(C1) = R − 1 − (k + 2l) = R − 2.

Solving these inequalities gives k = 1, l = 0 and we get curves of types (2) and (3).

Case 2: C0 • E = 1; C1 • E = 0.Now we compute

index(C0) = −3 + 2(k + l) ≥ 0

area(C0) = k + 2l − 1 = 1.

index(C1) = 5 − 2(k + l) ≥ 0

area(C1) = R − (k + 2l) = R − 2.

Solving these, we get k = 2, l = 0. Recall that C0 is somewhere injective. If C0failed to be embedded, it would either have an isolated non-immersed point or anisolated self-intersection, the existence of either of which would imply the existenceof a non-immersed point or of a self-intersection of the embedded curves C N for Nsufficiently large. Thus, C0 is an embedded curve asymptotic to a (2, 0) geodesic.

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R. Hind, S. Lisi

We now claim no such plane exists. To this end, we will compute the self-intersectionnumber of C0 in two different ways, interpreting the self-intersection number as theintersection of C0 with C ′

0, where C ′0 is obtained by pushing C0 off of itself with a

vector field that is tangent to the Morse–Bott family of orbits at the puncture (as in theextended intersection number given by Theorem 4.1).

First, we claim that C0 • C ′0 ≥ 0. Applying automatic transversality [23] to the

problem of finding nearby planes with asymptotic limits in the Morse–Bott family,as index(C0) = 1 we see that C0 comes in a local 1-parameter family with movingasymptotic end. Thus, for sufficiently small perturbation, C ′

0 can be realized by apseudoholomorphic curve and thus C0 •C ′

0 ≥ 0. (We note that by a relative adjunctionformula, for instance in Hutchings [12, Proposition 3.1], we actually have the strongerstatement C0 • C ′

0 ≥ 1, but we will not need this.)The self-intersection number C0 • C ′

0 can also be computed by interpreting C0 andC ′

0 as smooth disks in X with boundary on L (after a slight change of smooth structurenear the boundary). The boundaries of C0 and of C ′

0 are then doubly covered geodesicson L in the class (2, 0) and thus are parallel to each other. Taking a double cover ofdisks D(1)× {p} and D(1)× {p′} in the boundary of the polydisk (this part remainsafter our blow-up) for a suitable choices of p, p′ gives disks in C

2 with the sameboundaries as C0 and C ′

0. The disks do not intersect the exceptional divisor and thusmay be seen as disjoint disks in X . Since these two disks do not intersect each other,we may glue them to C0 and C ′

0 along their common boundaries to obtain spheres Sand S′ with S • S′ = C0 •C ′

0. But now, we have that S •[CP1(∞)] = 0 and S • E = 1,so [S] = [S′] = −E , giving S • S′ = −1 and hence C0 • C ′

0 = −1, thus providingthe contradiction and showing that planes C0 cannot exist.

We have now excluded Case 2, establishing the types of curves that can appear aslimits. We have already observed that there is a curve through each point, and the curvesare either disjoint or have identical images. Observe now that each curve obtained hasSiefring–Wendl self-intersection number 0 and is Fredholm regular by the automatictransversality criteria. From this, we obtain that curves in F give a foliation of X\Las claimed (see also [24, Theorem 7]).

The following lemma gives further structure of F .

Lemma 5.2 The foliation F contains a unique leaf of degree 1 asymptotic to each(−1, 0) geodesic and a unique leaf of degree 0 asymptotic to each (1, 0) geodesic.

Proof Note that the curves C0 and C1 in Proposition 5.1 both satisfy automatictransversality [23, Theorem 1] and have Fredholm index 1 when considered in themoduli space of curves with the asymptotic limit free to move in the Morse–Bott fam-ily of orbits. Their respective moduli spaces are compact since no further breaking canoccur for energy reasons, and thus, these moduli spaces consist of a collection of cir-cles. If we constrain the asymptotic limit of the curve, the corresponding constrainedproblem has Fredholm index 0 and also satisfies the automatic transversality criterion.As discussed in more detail in Appendix A, the linearized Cauchy–Riemann opera-tors corresponding to the constrained and unconstrained problem may be identified asdifferential operators, acting in the constrained case on sections of the normal bundleto the curve C0 (respectively, C1) decaying at the puncture, and in the unconstrained

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case acting on sections of the normal bundle asymptotic to a vector field along theorbit pointing in the direction of the Morse–Bott family of orbits. Since both operatorsare transverse, we see that the dimension of the kernel of the linearized operator jumpsby the dimension of the Morse–Bott family of orbits, and thus, the linearization of theevaluation map from the unconstrained moduli space to the Morse–Bott family mustbe surjective. Combining with compactness, this argument shows that the evaluationmap from our unconstrained moduli space to the Morse–Bott family of orbits (takinga curve to its asymptotic limit) is a submersion, and in particular then is surjective.

It remains to show that no two leaves are asymptotic to the same orbit in class(1, 0) (respectively, (−1, 0)). We consider the case of the (1, 0) geodesic—the sameargument applies for the (−1, 0) geodesic.

Suppose that both u and v are distinct degree 0 planes asymptotic to the same orbitin the class (1, 0). These curves do not intersect as they are leaves in the foliation.By the asymptotic convergence formula in Theorem 4.2, the two curves can eachbe represented near their ends as a graph over the asymptotic limit. The differencebetween them is then of the form

U (s, t)− V (s, t) = eTλs(e(T t)+ r(s, t)),

where λ is a positive eigenvalue of the asymptotic operator associated with the closedReeb orbit, e(t) is the corresponding eigenvector, and r(s, t) decays exponentially withs. As computed in Appendix A, the eigenvectors corresponding to positive eigenvalueshave winding number at least one with respect to the framing given by the S1 family oforbits. Pushing u off in the constant direction then introduces an additional intersectionpoint, implying that u � v ≥ 1. The homotopy invariance, however, gives u � v = 0.Thus, no two planes in the family can be asymptotic to the same orbit. ��

5.2 Limiting curves in T ∗L and R ×�.

Proposition 5.3 Limits of the C Ni in T ∗L and R × � are all cylinders which either

have positive ends asymptotic to Reeb orbits of type (1, 0) and (−1, 0), or lie in R×�and are trivial cylinders over an orbit of type (1, 0) or (−1, 0).

These curves foliate T ∗L. Furthermore, each geodesic in the S1 families of orbits inthe (1, 0) and (−1, 0) classes arises as the asymptotic limit of a curve in the foliation.

Proof As our limiting building has genus 0 and, by Proposition 5.1 (in the case whenthe limit is not a closed curve), contains exactly two curves in X\L asymptotic toorbits of type (1, 0) and (−1, 0), the limiting components in T ∗L and R ×� must fittogether along their matching ends to form a cylinder with ends asymptotic to (1, 0)and (−1, 0) orbits.

As there are no contractible orbits, this implies that all curves in T ∗L and R × �

are cylinders and every asymptotic limit is of the type described.Now observe again that these curves have Siefring–Wendl intersection number 0

and satisfy automatic regularity, establishing that these form a finite energy foliation(following [24, Theorem 7]). ��

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R. Hind, S. Lisi

Note that the nature of the curves in R × � and T ∗L does not require genericityof the almost complex structure J . Therefore, we can choose a J as in [14] so as toconclude the following.

Corollary 5.4 ([14, section 10]) The almost complex structure on T ∗L may be chosenso that the limiting cylinders in T ∗L and R ×� have both ends asymptotic to a lift ofthe same geodesic in T 2 = L.

Remark 5.5 The construction of this foliation depended quite heavily on our assump-tion that the embedding of P(1, 2) is in C P2(R) for R < 3. If we considered R > 3,many other buildings could arise in the foliation. In particular, we would expect to seean S1 family of buildings with levels in X\L consisting of a degree 1 plane asymptoticto a (−1,−1) geodesic and a degree 0 plane intersecting E , asymptotic to a (1, 1)geodesic. The key idea of our argument is that the nonexistence of these curves is whatprovides the contradiction later.

6 Degree d curves with point constraints

In this section, to complete a proof by contradiction of Theorem 1.1, we will considerthe SFT limits of curves of degree d as we stretch the neck under the assumption thatR < 3.

Let us assume that all JN are regular. By Proposition 2.2, to any set of 2d pointsin sufficiently general position {qi } ∈ X , for each N there exists a unique embeddedJN -holomorphic curve SN of degree d and with e = SN • E = d −1 intersecting eachof the 2d points {qi }. We also observe that SN • F = 1. Let us fix the 2d points in asmall Weinstein neighborhood U of L . As the cylinders described by Proposition 5.3have deformation index at most 2, we may assume that each such cylinder intersectsat most one of the points.

We now take the limit as N → ∞. As described in Sect. 3 the limiting holomorphicbuilding will have components in X\L , in R ×� and in T ∗L . Denote the building byF and let F0 denote the components of the building in X\L .

We will study F0 using the foliation F described by Proposition 5.1. A generic curvein the foliation is a closed curve in the fiber class, but such curves can converge tobroken configurations, that is, the union of a curve of type (2) and (3) from Proposition5.1 asymptotic to the same geodesic with opposite orientations, see Corollary 5.4.

Lemma 6.1 All components of F0 except for one are (possibly branched) covers ofcurves in the foliation F . Denote by f0 ⊂ F0 ⊂ F the component that is not a cover.

For each closed curve C in F , the intersection number f0 • C = 1. For the brokenconfigurations in F , we have f0 � C0 + f0 � C1 = 1.

Furthermore, the component f0 is of some degree d ′ with 1 ≤ d ′ ≤ d and hasintersection number with E given by e′ = f0 • E = d ′ − 1.

Proof Fix a component f of F0. For any closed curve C of F , the intersection numberC • f is well defined and independent of C . Furthermore, this is also the intersectionnumber of f with a broken configuration in F , as long as the asymptotic limits of

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the broken configuration are geometrically distinct from the asymptotic limits of f .Hence, for C0 and C1 with limits different from those of f , we have

f • C0 + f • C1 = f • C. (2)

Broken configurations have codimension 1 in F . Most of these broken configura-tions have asymptotic limits on Reeb orbits that are geometrically distinct from theasymptotic limits of f . However, there are finitely many broken configurations whichhave at least one asymptotic limit in common with the underlying simple orbit of anasymptotic limit of f . These exceptional broken configurations are of codimension 2in the 2-dimensional space of leaves of F .

Let now C0 ∪ C1 be one of these broken configurations with asymptotic endscoinciding with an end of the curve f . We then have f • C0 + f • C1 ≤ f • C ,where C is a nearby closed curve in F . However, we now claim the following for theSiefring–Wendl extended intersection number (as introduced in Theorem 4.1).

Claimf � C0 + f � C1 = f � C = f • C. (3)

Proof of claim This identity essentially follows from the homotopy invariance of theSiefring–Wendl extended intersection number, where homotopies are taken throughpunctured curves, which are asymptotically holomorphic and converge to the sameMorse–Bott family of orbits [21–23, Section 4.1]. However, in order to keep the expo-sition self-contained, we briefly explain why the identity holds for the extended inter-section number in our specific setting. Recall from the discussion following Theorem4.1, that in our case with orbits in T

3, the extended intersection numbers f � C0 andf �C1 are obtained by perturbing C0 and C1 off of their asymptotic limits in the direc-tion of the S1 family of closed Reeb orbits. By Lemma 5.2, these perturbations maybe chosen to be holomorphic and both asymptotic to the same orbit, which is differentfrom the limit of f . The result now follows from the case of broken configurationswith asymptotics disjoint from f . ��

Given these preliminaries, we proceed to prove Lemma 6.1. Denote the componentsof the building F0 by f0, f1, . . . , fK .

As F is a limit of curves SN having intersection number 1 with fiber curves, if Cis a closed curve in F , we have the intersection number

C • f0 + C • f1 + · · · + C • fK = 1.

By positivity of intersection, each of the terms on the left here is a nonnegativeinteger, and so we conclude that exactly one component, which we call f0, has C• f0 =1 and all other components have C • fi = 0.

By the analysis above, for i ≥ 1 we have C • fi = 0 for all curves C in F ,including broken configurations. As F is a foliation, fi cannot be disjoint from allcurves and so we conclude by positivity of intersection that it must cover one of them.The intersection numbers for f0 also follow from the above analysis.

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R. Hind, S. Lisi

Finally, by Proposition 5.1, we note that each curve in F has the same intersectionnumber with E as its degree. This then holds for any cover of a curve in F , sofor each i = 1, . . . , K , fi • CP1(∞) = fi • E . The high degree curves SN haveSN • CP1(∞) = SN • E + 1, and thus, as intersection numbers are preserved in thelimit, we must have:

f0 • CP1(∞) = f0 • E + 1

d = f0 • CP1(∞)+ f1 • CP1(∞)+ · · · + fK • CP1(∞).

The result now follows from the nonnegativity of each of these intersections. ��As in Lemma 6.1, f1, f2, . . . , fK denote the components of F0 which cover leaves

of F . Then their asymptotic limits must be closed Reeb orbits corresponding to geo-desics in class (a, 0) for 0 �= a ∈ Z. As our building has genus 0 and there are nonull-homologous Reeb orbits, the homology classes of the asymptotic limits of f0must match with the asymptotic limits of the other fi up to orientation and so mustall be in classes (a, 0) also.

Lemma 6.2 Either all punctures of f0 are asymptotic to geodesics in classes (k, 0)for k > 0, or all punctures are asymptotic to geodesics in classes (k, 0) for k < 0.

Proof Suppose that f0 has a puncture asymptotic to a closed Reeb orbit in class(k, 0), k > 0. Denote this orbit by γ k . Then by Lemma 5.2 there exists a plane in thefamily C0 asymptotic to the underlying simple cover γ of the same orbit. We will nowestimate f0 �C0 using this specific plane. Note that f0 is somewhere injective since itsintersection numbers with CP1(∞) and with E are coprime, see Lemma 6.1. The planeC0 is also somewhere injective, and the two curves have geometrically distinct images.Therefore, f0 • C0 is well defined and nonnegative. By the asymptotic convergenceformula for holomorphic curves in Theorem 4.2, the two curves do not have anyintersection near the puncture. Furthermore, near their asymptotic limits, the curvesf0 and the k-fold cover of the restriction of C0 to a neighborhood of γ can be written asgraphs over γ k . In other words, there exist maps Ui : (−∞, R]×S1 → (γ )∗ξ, i = 1, 2with Ui (s, t) ∈ ξ(γ k(T t)) so that the images of f0 and of C0 restricted to a sufficientlysmall neighborhood of their punctures, are described, respectively, by

(−T s, expγ k (T t) Ui (s, t)), i = 1, 2.

Since the curves are geometrically distinct, U1−U2 does not vanish, and so there existsa positive eigenvalue λ > 0 of the asymptotic operator Aγ k ,J and a correspondingeigenvector e(t) so that

U1(s, t)− U2(s, t) = ekTλs(e(kT t)+ r(s, t)),

where r decays exponentially fast. As computed in Appendix A, the eigenvectorscorresponding to positive eigenvalues have winding number at least one with respectto the framing given by the S1 family of orbits. Pushing C0 off in the constant direction

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then introduces at least 1 additional intersection point, giving f0 �C0 ≥ 1. By Lemma6.1 this implies that f0 � C0 = 1 and f0 � C1 = 0.

By applying the same argument, if f0 has a puncture asymptotic to a closed Reeborbit in class (−k, 0), k > 0, it follows that f0 � C1 = 1 and f0 � C0 = 0.

Finally, by the homotopy invariance of the extended intersection number, eitherall punctures of f0 asymptote to Reeb orbits in classes (k, 0) for k ≥ 1, or they allasymptote to Reeb orbits in classes (−k, 0) for k ≥ 1 as required. ��

For components in T ∗L and � × R, we have the following.

Lemma 6.3 The components of F in T ∗L and � × R are covers (perhaps but notnecessarily multiple covers) of the cylinders asymptotic to orbits of type (1, 0) or(−1, 0) described in Proposition 5.3.

Proof We argue by contradiction and suppose that a component F1 of F does notcover such a cylinder. Then as the cylinders foliate T ∗L and � × R, we can find acylinder C which intersects F1 non-trivially.

Let f0 denote the component of F in X\L that does not cover a leaf of F , as givenby Lemma 6.1.

By perturbing C as necessary, we may assume that the asymptotic limits of C aredisjoint from the asymptotic limits of f0. Now fix a point on C and take a limit ofthe J N holomorphic spheres QN in the fiber class passing through p. Denote thelimiting building by G. By Proposition 5.3, one component of the limit in T ∗L or� × R coincides with C , and other components form a broken configuration in X\Lwith asymptotic limits distinct from f0. By Lemma 6.1 the curve f0 has intersectionnumber 1 with this broken configuration.

The building F now has two isolated intersections with G, one on F1 and the otheron f0. By positivity of intersection, there are then at least two isolated intersectionsbetween SN and QN for large N , and this is a contradiction as SN has intersectionnumber 1 with fiber curves. ��

6.1 Completion of the proof of Theorem 1.1

By Lemma 6.3, the components of F in T ∗L and � × R consist only of covers ofcylinders and by the choice of our point constraints {qi } at the start of this section,there exist at least 2d such cylinders. Suppose that such a cylinder has ends whichare not asymptotic to any end of f0. Then they are asymptotic to the ends of fi fori ≥ 1, that is, asymptotic to curves of F . In particular, as one end of the cylinder isasymptotic to a geodesic in the class (−1, 0), this end must match with a componentof F which is a cover of a curve of type C1 from Proposition 5.1. Suppose there are msuch cylinders not asymptotic to f0. Then f0 must have at least 2d − m negative endsand degree d ′ ≤ d − m (since the curves of type C1 asymptotic to the m cylindershave total degree at least m).

Now, if the asymptotic limits of f0 are all of the form (k, 0) for k ≥ 1, then sincethe building F has genus 0 there must be at least 2d − m planes in F that are coversof degree 1 planes in F and asymptotic to limits of f0 with the opposite orientation.

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R. Hind, S. Lisi

Combined with the m curves asymptotic to other orbits, we find 2d covers of curves oftype C1 in F . The degree of the building F is only d, however, so this is a contradiction.Hence, by Lemma 6.2 all negative ends of f0 must be in classes (−k, 0), and we musthave f0 � C0 = 0 and f0 � C1 = 1.

Let s ≥ 2d − m be the total number of negative ends of f0. Let 1 ≤ d ′ ≤ d − mbe the intersection number of f0 with the line at infinity (i.e., its degree). Recall thatthe intersection of f0 with the exceptional divisor is f0 • E = d ′ − 1. The area of f0then satisfies

0 < Area( f0) ≤ d ′ R − (d ′ − 1)− s

= d ′(R − 2)+ 1 + d ′ − s

≤ d(R − 2)+ 1 + (d − m)− (2d − m)

where the final inequality uses d ′ ≤ d − m ≤ d and s ≥ 2d − m. Rearranging, we get

R − 2 ≥ 1 − 1

d.

Letting d → ∞, this inequality gives a contradiction for any R < 3 and our proof iscomplete.

Acknowledgments The authors would like to thank Dusa McDuff for pointing out the interest of theembedding problem for P(1, 2), and also thank both the referee and Chris Wendl for pointing out an errorin the original proof of Proposition 5.1, and the referee for suggesting numerous improvements to themanuscript. The second author would like to thank Klaus Mohnke, Richard Siefring and Chris Wendl forhelpful conversations, and Dietmar Salamon for raising questions that improved the exposition. His workwas partially supported by the ERC Starting Grant of Frédéric Bourgeois StG-239781-ContactMath and bythe ERC Starting Grant of Vincent Colin Geodycon.

7 Appendix A: Morse–Bott Riemann–Roch

First, we state the index formula for a real-linear Cauchy–Riemann-type operatoracting on sections of a complex line bundle over a punctured Riemann surface (S, j)(where S = S\�, with� a finite collection of punctures). We define W 1,p,δ(S, E) to bethe weighted space of sections with exponential decay (for δ > 0) at the punctures. (Seethe discussion preceding Theorem 4.2 for a review of the definition of an asymptoticoperator. We also refer the reader to [1, Sections 2.1–2.3] for a nice exposition of therelevant index computations and discussion of the asymptotic operator.)

Then, we reformulate the index theorem from [19]:

Theorem 7.1 A real Cauchy–Riemann-type operator T :W 1,p,δ(S, E) → L p,δ

(S,�0,1(E)) is Fredholm if and only if all of its δ–perturbed asymptotic operators arenon-degenerate.

For any choice of cylindrical asymptotic trivialization �, the index is given by:

index = nχ(S)+∑

z∈�+CZ�(Az + δz)−

∑z∈�−

CZ�(Az − δz)+ 2c�1 (E),

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where n denotes the (complex) rank of E, χ(S) is the Euler characteristic of thepunctured surface, CZ�(A) is the Conley–Zehnder index of the asymptotic operatorA, and c�1 (E) is the first Chern number of the bundle E relative to the asymptotictrivialization data �.

In particular then, for a punctured holomorphic curve u in an almost complexsymplectic 4-manifold (V, J ) with symplectization ends, considered modulo domainreparametrizations, the deformation index is given by

index(u) = −χ(S)+∑

z∈�+CZ�(Az + δz)−

∑z∈�−

CZ�(Az − δz)+ 2c�1 (u∗T V )

where, now, the� are given by trivializations of the contact structure over the asymp-totic limits of u and c�1 (u

∗T V ) is, as before, the first Chern number of the bundlerelative to the asymptotic trivialization given by �.

The data of the asymptotic trivialization gives a preferred class of sections that areasymptotically constant. The first Chern number of the bundle relative to an asymptotictrivialization is then defined to be the signed count of zeros of a generic asymptoticallyconstant section.

If we consider the case of a punctured holomorphic curve u in an almost com-plex manifold, asymptotic at a given puncture to a closed Reeb orbit γ that comesin a Morse–Bott family, the exponential decay condition imposed on sections of ξcorresponds to requiring that the deformations of this curve be among curves thatconverge to the same orbit. In order to consider the moduli space of curves wherethe asymptotic limit is allowed to move in the Morse–Bott family, one must considersections of E that converge exponentially fast to sections of γ ∗E that correspond todeformations in the Morse–Bott family. By an argument of elliptic regularity, this isthe same as considering the linear operator acting on sections with small exponentialgrowth. Note that the nonlinear Cauchy–Riemann operator does not make sense onthis space, even though the linearized one does. We refer the reader to [23, Section 2]for further development of this theory.

To apply Theorem 7.1 to obtain the formulas in Sect. 3.3, we must compute the rele-vant Conley–Zehnder indices and Chern numbers relative to appropriate trivializationsalong Reeb orbits in� = T 3. The results can be substituted directly into Theorem 7.1to obtain Propositions 3.1, 3.2 and 3.3. We start with the Conley–Zehnder indices.

The key tool we will consider is the alternative description of Conley–Zehnderindices in terms of winding numbers of eigenvectors of the corresponding asymptoticoperators [8, Section 3]. (Further exposition is also in [7, Theorem 3.9].) The mainresult is that the asymptotic operator A, which is a self-adjoint unbounded operatoron L2(γ ∗E), has spectrum given by eigenvalues, each eigenvalue has multiplicity atmost 2, and each eigenvector is nowhere vanishing. After choice of a trivialization� of γ ∗E , an eigenvector has a well-defined winding number. Furthermore, eacheigenvector associated with the same eigenvalue has the same winding number, whichwe denote w(λ). By a perturbation theoretic argument (see [8] or references in [7]),each integer is attained by w(λ) and each winding number is attained twice—eitherby an eigenvalue with multiplicity 2, or by two simple eigenvalues. Furthermore,

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R. Hind, S. Lisi

the winding number w(λ) is monotone non-decreasing in λ. Non-degeneracy of A isequivalent to saying that 0 is not an eigenvalue of A. For a non-degenerate operatorA, we consider the largest negative eigenvalue λ− < 0 and the smallest positiveeigenvalue 0 < λ+. Then,

CZ�(A) = w�(λ+)+ w�(λ−). (4)

In T 3, the contact structure is globally trivial. Given explicitly, writing T 3 =(R/2πZ)3 with circle valued coordinates θ, q1, q2, we have α = cos(θ)dq1 +sin(θ)dq2 and a trivialization of ξ by sin(θ)∂q1 − cos(θ)∂q2 and ∂θ . With respectto this trivialization, the asymptotic operator at a T -periodic orbit is

−J0d

dt− T

(0 00 1

)

We note that the spectrum includes 0 and −T . Each of these eigenvalues has mul-tiplicity 1 and has eigenvectors of winding 0 with respect to our trivialization.

The perturbed operator corresponding to a negative puncture allowed to move in theMorse–Bott family is the one coming from allowing asymptotic exponential growth.At the level of the asymptotic operator, this corresponds to considering the operatorA + δ, thus pushing the entirety of the spectrum to the right. Thus, the eigenvalues ofthe δ-perturbed operator, on either side of 0 are −T + δ < 0 < 0+ δ. These both havewinding number 0, so by Eq. (4), we have Conley–Zehnder index 0.

For a positive puncture allowed to move in the Morse–Bott family, we again considera δ-perturbed operator, but now consider the shifted operator A − δ. The spectrum isthen shifted to the left, so we now have that −T − δ < 0 and 0 − δ < 0 are both in thespectrum of A−δ, and have winding number 0. Thus, the smallest positive eigenvaluemust have winding 1. It follows then that for a positive puncture allowed to move inthe Morse–Bott family, the Conley–Zehnder index is 1.

Now we move to the Chern numbers. Here we need a separate analysis for each ofthe possible ranges X\L , T ∗L and R ×� as described in Sect. 3.1.

For a curve contained in R ×�, it is clear that the Chern number of u∗(T (R ×�))with respect to this asymptotic trivialization is 0, since this trivialization of the contactstructure is global.

For a curve contained in T ∗T 2, we claim that the Chern number of u∗T (T ∗T 2)

relative to this trivialization is also 0. Consider� as the boundary of the unit disk bundlein u∗T (T ∗T 2). Let p1, q1, p2, q2 be canonical coordinates on T ∗T 2, where q1, q2 areT 2 valued coordinates as before and p1, p2 the corresponding cotangent coordinates.The trivialization of T (R × �) induced by the trivialization of ξ corresponds tothe trivialization of T (T ∗T 2)|� by p1∂q1 + p2∂q2 , p1∂q1 + p2∂q2 , p2∂q1 − p1∂q2 andp1∂p2 − p2∂p1 . This defines a map� → U (2) by, at each point, considering the matrixto write this frame with respect to the standard trivialization ∂p1 , ∂q1 , ∂p2 , ∂q2 . Thematrix has constant determinant 1 ∈ U (1) and thus when restricted to any Reeb orbitrepresents a contractible loop in U (2). From this, we obtain that any asymptoticallyconstant section of u∗T (T ∗T 2) over a curve can be deformed to a nowhere vanishingsection, establishing the claim.

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Finally, we need to compute c�1 (u∗T X) for a punctured sphere in X\L , of degree

d and of intersection e with the exceptional divisor E , and asymptotic ends on orbitsin classes (ki , li ). First, observe that this is additive with respect to connect sums,so we only need to compute this for a curve with d = e = 0, since the first Chernnumber of T X |CP1(∞) is 3 and the first Chern number of E is 1 (and a closed curveof degree d and intersection e with E represents the class d[CP1] − e[E]). Any suchcurve can now be deformed and decomposed to a collection of s disks of degree 0 ande = 0, each with boundary on a geodesic of class (ki , li ), i = 1, . . . , s, and thus, thesemay be taken to be model disks contained in C

2 with boundary on L . Now, observethat the relative first Chern number for a rank k complex vector bundle V , c�1 (V ),can be computed by computing the relative first Chern number for the determinantbundle�k

C(V ). Our trivialization of ξ induces a trivialization of T (T ∗T 2)|� and thus

of�2C(T (T ∗T 2))|� . A straightforward computation shows that this trivialization has

winding number ki + li with respect to the obvious trivialization of �2(T C2), and

thus, the relative Chern number is ki + li .

References

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