fredholm theory and transversality for noncompact

FREDHOLM THEORY AND TRANSVERSALITY FORNONCOMPACT PSEUDOHOLOMORPHIC CURVES IN

SYMPLECTIZATIONS

DRAGOMIR DRAGNEV

Abstract. We study pseudoholomorphic maps from a punctured Riemannsurface into the symplectization of a contact manifold. A Fredholm theoryyields the virtual dimension of the moduli spaces of such maps in terms of theEuler characteristic of the Riemann surface and the asymptotics data given bythe periodic solutions of the Reeb vector field associated to the contact form.The transversality results, establish the existence of additional structure forthese spaces. To be more precise we prove that these spaces are genericallysmooth manifolds and therefore their virtual dimension coincides with theiractual dimension.

1. Introduction

The study of pseudoholomorphic curves in symplectic geometry was initiated byGromov [13] in 1985. A pseudoholomorphic curve is a smooth map from a Riemannsurface into an almost complex manifold, which satisfies an equation of Cauchy-Riemann type. Since the appearance of Gromov’s paper, pseudoholomorphic curveshave become a powerful tool for study the geometry and the topology of symplecticmanifolds. The study of pseudoholomorphic curves in symplectizations of contactmanifolds was initiated by H. Hofer in his solution of the Weinstein conjecture[15]. It was later developed by H. Hofer, K. Wysocki and E. Zehnder in a seriesof papers, where not only the properties of these objects were established, but alsomany interesting applications to Hamiltonian dynamics and the topology of three-and four-manifolds were given [16, 17, 18, 19, 20]. Using the techniques of thepseudoholomorphic curves in the symplectizations of contact manifolds became asubject of extensive research of study the geometry and the topology of symplecticand contact manifolds which culminated in the appearance of ”the Symplectic FieldTheory” by Y. Eliashberg, A. Givental, H. Hofer, [6]. The motivation for the presentwork is the rigorous proof of some (rather technical) results, already stated andused in [19, 6]. This work is very close in spirit to [18], where the authors consideronly embedded pseudoholomorphic curves, and here we are going to prove similarresults assuming that the curves under consideration are somewhere injective, whichis more natural. The approach we use here is different from the one used in [18],but for the sake of compatibility we will adopt the notation and the terminology of[18]. With that said, we can begin defining the objects we are going to study anddescribe our results.

Consider a compact, oriented, (2n− 1)-dimensional manifold M , equipped witha 1-form λ, such that λ∧(dλ)n−1 is a volume form on M . Forms with this propertyare called ”contact”. The contact form λ determines then a (2n − 2)-dimensional

1

2 DRAGOMIR DRAGNEV

tangent distribution ξ = Ker(λ) ⊂ TM , called the associated contact structure.We point out that dλ |ξ is non-degenerate, hence the contact bundle ξ carries afiberwise symplectic structure given by dλ. Now, consider the so called Reeb vectorfield X = Xλ on M defined by iXλ = 1 and iXdλ = 0. Then the tangent bundlesplits

TM = RX ⊕ ξ

into a line bundle with a preferred section X and a symplectic vector bundle (ξ, dλ).Denote by π the projection of the tangent bundle of M along the Reeb vector field,

π : TM −→ ξ.

The symplectic vector bundle (ξ, dλ) has a class of complex multiplications J : ξ →ξ, compatible with the symplectic structure in the sense that the bilinear map

gJ(m)(h, k) = dλ(m)(h, Jk)

defines a positive definite inner product on each fiber ξm of ξ. The space of allsuch J ’s equipped with the C∞-topology is contractible [13, 3]. Choose a complexmultiplication J on ξ and define the following almost complex structure J on W =R×M

J(a,m)(h, k) = (−λ(m)(k), J(m)πk + hX(m))

where (a,m) ∈ W, (h, k) ∈ TW . Assume, now, that (S, j) is a compact Riemannsurface with conformal structure j and Γ ⊂ S is a finite set of points on S. Wecall the elements of Γ punctures and denote by S = S \ Γ the punctured Riemannsurface. We are interested in maps

u : S −→ W = R×M

which satisfy the following generalized Cauchy-Riemann equations:

(1) T u j = J T u

We can write these equations in the form ∂J u = 0 where ∂J u = du + J du j ∈Λ0,1(u∗TW ). It will be useful later on, to have more explicit form of the equations.Take local conformal coordinates s + it on S. In these coordinates the Cauchy-Riemann equations (1) are given by

(2a) πus + J(u)πut = 0,

(2b) as = λ(ut)

(2c) at = −λ(us)

We remark here that in what follows we assume Γ 6= ∅, because otherwise there areno non-constant solutions of the above equations.

Now, consider the set Σ of all functions φ which satisfy φ : R→ [0, 1] and φ′ ≥ 0.If φ ∈ Σ we define the 1-form λφ on W ,

λφ(a, m)(h, k) = φ(a)λ(m)(k)

Given a solution u = (a, u) of the above equations we consider the 2-form u∗dλφ.Taking local coordinates s + it on S, one verifies that

(3) u∗dλφ = [φ′(a)(a2s + λ(us)2) + φ(a)|πus|2]ds ∧ dt

FREDHOLM THEORY AND TRANSVERSALITY 3

Define the energy of the solution u to be

E(u) = supφ∈Σ

∫

S

u∗dλφ

Definition 1. A non-constant solution u of (1) is called finite energy surface ifE(u) < ∞. In the special case when S = S2 \∞, we call it a finite energy plane.

Definition 2. A solution u is called somewhere injective if there exists z0 ∈ S suchthat T u(z0) 6= 0 and u−1(u(z0)) = z0.

We have the following theorem proved in the appendix of [17], for finite energyplanes, which can be extended easily for finite energy surfaces. It states that a non-constant finite energy surface factors through a somewhere injective one. Comparewith Proposition 2.3.1 from [25].

Theorem 1. Let u : S → W be a non-constant finite energy surface. Then thereexists a compact Riemann surface S′ with a finite set of punctures Γ′ and a holo-morphic map φ : S → S′, Γ′ = φ(Γ) and a somewhere injective finite energy surface

v : S′ → W

so that u = v φ.

We have the following definition.

Definition 3. A subset C ⊂ W is called an unparametrized finite energy surfacefor the almost complex structure J , if there exists a compact Riemann surface (S, j)and a finite set of punctures Γ ⊂ S and a smooth map u, solving (1) and satisfying0 < E(u) < ∞ (i. e. non-constant finite energy surface) such that

C = u(S)

We call u a parametrization of C.

If u is somewhere injective, we call C somewhere injective unparametrized fi-nite energy surface. The previous theorem asserts that two somewhere injectiveparametrizations differ by a biholomorphic diffeomorphism i. e. one of the maps iscomposition of the other with a biholomorphic map.

Now, recall, some results from [15, 16], concerning the behavior of a finite energysurface near the punctures. We have the following two types of punctures:

• A puncture z ∈ Γ is said to be removable if there exists an open neighbor-hood U of z such that the image of U∗ = U \ z under u is contained in acompact subset of W .

• A puncture z ∈ Γ is said to be non-removable if the image of every punc-tured neighborhood of z is not contained in a compact subset of W .

From Gromov’s removable singularity theorem (see [13] for details), follows thatthe map u can be extended smoothly over removable punctures. That is why,from now on, when we refer to puncture we will mean a non-removable one. Wedistinguish the following two types of punctures:

• negative puncture - if the R component a of the map u = (a, u) is boundedfrom above.

• positive puncture - if the R component a of the map u = (a, u) is boundedfrom below.

4 DRAGOMIR DRAGNEV

It is a theorem, see below, that any non-removable singularity is of one of thesetwo types. Therefore Γ = Γ+∪Γ−, where Γ± is the set of positive (respectively neg-ative) punctures. Near a puncture z0 ∈ Γ we can introduce cylindrical holomorphiccoordinates σ as follows:

σ : [0,∞)× S1 −→ D \ z0 if z0 ∈ Γ+

σ : (−∞, 0]× S1 −→ D \ z0 if z0 ∈ Γ−

where D \ z0 is a closed disc punctured at z0. Let us assume that z0 ∈ Γ+,(for definiteness). Define v = u σ, v : [0,∞)× S1 → W and v satisfies

(4) vs + J(v)vt = 0

We may assume that v is a non-constant and with finite energy. We have thefollowing theorem proved in [15] and then strengthened in [16].

Theorem 2. Let v : [0,∞)×S1 → W = R×M , be as described above, solving (4)and having energy E(v) : 0 < E(v) < ∞ and its R -component is unbounded fromabove. Then,

T = lims→∞

∫ 1

0

v(s, ·)∗λexists and is a positive number. Given any sequence Rk → ∞, there exists a sub-sequence Rkj and a periodic orbit x of the Reeb vector field X so that v(Rkj , T ) →x(Tt) in C∞. If (x, T ) is non-degenerate, in the sense that it has only one Floquetmultiplier equal to 1, then the limit exists for R →∞.

In the following we shall say that v converges to x if the limit v(s, t) → x(tT )exists in C∞ as s →∞. If v converges to x we could introduce suitable coordinatesnear x. We have the following lemma proved in [16] for n = 2, which can beextended for any n ≥ 2, see [1].

Lemma 1. Let (M, λ) be a (2n− 1)-dimensional contact manifold, and let x(t) bea T -periodic solution of the equation x = Xλ(x) on M . Let τ be the minimal periodso that T = kτ for some positive integer k. Then there is an open neighborhoodU ⊂ S1 × R2n−2 of S1 × 0 and an open neighborhood V ⊂ M of P = x(t)|t ∈ Rand a diffeomorphism ϕ : U → V mapping S1 × 0 onto P such that

ϕ∗λ = f · λ0

with a positive smooth function f : U → R satisfying f(θ, 0, 0) = τ and df(θ, 0, 0) =0 for all θ ∈ S1 and λ0 denotes the standard contact form in R2n−1.

Using the coordinates from the lemma, we put (a, (θ, x, y)) = (a, (θ, z)) =(a, ϕ−1 v). Working in the universal cover of S1 × R2n−2 we may view (a, (θ, z))as a map from [0,∞) × R → R2n where θ(s, t + 1) = θ(s, t) + k. Here we assumethat x is non-degenerate and τ = T/k is its minimal period. We will say that vconverges strongly to x if it converges to x and there exist constants e, c ∈ R andd > 0 such that

|∂β [a(s, t)− Ts− c]| ≤ Me−ds

|∂β [θ(s, t)− kt− e]| ≤ Me−ds

for all multiindices β with constants M = Mβ . Moreover we have the formula forthe transversal approach to x(t), (see [16]):

z(s, t) = eR s

s0µ(τ)dτ [e(t) + r(s, t)] ∈ R2n−2


where ∂βr(s, t) → 0 as s → 0 uniformly in t for all derivatives, µ : [s0,∞) → R isa smooth function such that lims→∞ µ(s) → λ < 0. Here λ is an eigenvalue of theself-adjoint operator A in L2(S1,R2n−2) related to the linearized Reeb vector fieldX along the limit orbit x(t). The operator is defined by A = −J(t) d

dt − S∞(t),S∞(t) = S∞(t + 1) is a smooth 2(n − 1) × 2(n − 1) matrix, defined by S∞(t) =−J(t)πmdXπm, where m = (kt, 0) ∈ R × R2n−2. Moreover e(t) = e(t + 1) 6= 0 isan eigenvector corresponding to the eigenvalue λ < 0 (see [16] for details).

Let us now recall briefly, the Conley-Zehnder index for symplectic maps see[28, 9, 10] for more details. Consider R2n = R2 ⊕ . . . ⊕ R2 with coordinates(x1, y1, . . . , xn, yn) and the standard symplectic form ω0 =

∑ni=1 dxi ∧ dyi. De-

note by Σ(n) ⊂ Sp(n) the space of all arcs of symplectic matrices starting at Idand ending at some element which does not have 1 in its spectrum and by G(n)the space of continuous 1-periodic loops, starting and ending at Id. The homotopyclasses of G(n) generate the fundamental group of Sp(n). It is well known thatit is isomorphic to Z. A particular isomorphism, called the Maslov isomorphismµM : π1(Sp(n), Id) → Z is characterized by µM ([t → (e2πitIdC) ⊕ IdCn−1 ]) = 1,where we have identified R2 = C (see [2]). We have the following natural maps fora loop α ∈ G(n) and arcs Φ,Ψ ∈ Σ(n) for some n.

• G(n)× Σ(n) → Σ(n)(α(t), Φ(t)) → α(t)Φ(t) = (αΦ)(t)

• Σ(n) → Σ(n)Φ(t) → Φ−1(t)

• Σ(n)× Σ(m) → Σ(n + m)(Φ(t),Ψ(t)) → Φ(t)⊕Ψ(t) = (Φ⊕Ψ)(t)

The Conley-Zehnder index µ is then uniquely defined by the following naturalaxioms [17].

Theorem 3. There exists a unique family of maps µ = µn : Σ(n) → Z, n ∈ Nhaving the properties:

• Homotopic maps have the same index• If α ∈ G(n), Φ ∈ Σ(n) then µ(αΦ) = µ(Φ) + 2µM (α)• µ(Φ−1) = −µ(Φ)• µ1(γ) = 1 where γ(t) = eπitIdC ∈ Σ(1)• µm+n(Φ⊕Ψ) = µn(Φ) + µm(Ψ)

Consider a finite energy surface u = (a, u). According to Theorem 1, at thepunctures u converges to periodic orbits of the Reeb vector field,assuming they arenon-degenerate. Following [17] we can introduce Conley-Zehnder index µ(z) at eachpuncture z ∈ Γ, and define the Conley-Zehnder index of u as follows. CompactifyS by adding at each puncture the circle S1 at infinity. Then u can be extendedcontinuously over the circles at infinity. Let u be the M - part of the extendedmap. Since Γ 6= ∅, the bundle u∗ξ → S is trivializable as a symplectic vectorbundle. Take some trivialization Ψ, then we have induced trivialization over theasymptotic limits. Assume that (x, T ) is a periodic orbit of the Reeb vector fieldcorresponding to a positive puncture (say z0) i. e. x = X(x) and x(0) = x(T ).Denote by φt the flow generated by X and consider the map

Z(t) = Tφt(x(0)) |ξx(0) : ξx(0) → ξx(t)

6 DRAGOMIR DRAGNEV

This is a symplectic linear map (it is easy to check that φ∗t λ = λ ). Then the map

Φ(t) = Ψ(t)Z(t)Ψ(0)−1

where Ψ(t) : ξx(t) → R2n−2, t ∈ [0, T ], defines an arc of symplectic matrices startingat Id and ending at some element which does not have 1 in its spectrum sincewe assumed the asymptotic limits to be non-degenerate. Now we can define theConley-Zehnder index for positive puncture z0 with asymptotic limit (x, T ) to be

µ(z0) = µ(x) = µ(Φ)

Similarly we can define the Conley-Zehnder index corresponding to a negative punc-ture. We then proceed and define the Conley-Zehnder index of a finite energysurface u to be

µ(u) =∑

z∈Γ+

µ(z)−∑

z∈Γ−µ(z)

We remark here that the Conley-Zehnder index at a puncture depends on the chosentrivialization. However, it is shown in [17] that the index µ(u) is independent ofthe choice of the trivialization.

We want to study maps u : S = S \ Γ → R × M = W where u is somewhereinjective finite energy surface, (S, j) - compact Riemann surface and Γ - finite set ofpunctures. We allow Γ (which we assume ordered) and j to vary and are interestedin the image C = u(S \Γ). As we know from Theorem 1 if (v, (S′, j′), Γ′) is anotherparametrization then there exists φ s. t. φ(S) = S′, φ(Γ) = Γ′, u = v φ, and φ isbiholomorphic. We call (u, (S, j),Γ) and (v, (S′, j′),Γ′), equivalent if φ, in addition,respects the order of Γ and Γ′. Our objective is to study the equivalence classes ofsomewhere injective maps.

Now we want to define a suitable Banach space on which to vary J (resp.J).We follow the approach of A. Floer ,[8], and introduce Floer’s Cε -space. Considera compatible complex multiplication J0 : ξ → ξ and let J0 be the correspond-ing extension over W . Consider the space of all smooth maps Y (m) : ξm → ξm

satisfyingY (m)J0(m) + J0(m)Y (m) = 0

dλ(Y h, k) + dλ(h, Y k) = 0, (h, k ∈ ξ).

Let ε = εn∞n=1 be a sequence of positive numbers such that limn→∞ εn = 0. Wedefine the space Cε to be

Cε =

Y ∈ HomR(ξ), Y ∈ C∞ | ‖Y ‖ε =

∞∑n=1

εn‖Y ‖n < ∞

If ε → 0 sufficiently fast then (Cε, ‖·‖ε) is a separable Banach space, which lies densein C∞. For ∆ > 0, denote by U∆ =

J | J = J0 exp(−J0Y ), Y ∈ Cε, ‖Y ‖ε < ∆

.

The map Y → J ∈ U∆ provides a global chart for U∆, equipped with a separableBanach manifold structure.

Consider the set of all pairs (C, J), where C is an unparametrized, somewhereinjective finite energy surface for the complex structure J ∈ U∆, asymptotic at thepunctures to non-degenerate periodic orbits of the Reeb vector field (xj , Tj)ν

j=1.Denote this set by M(x1, . . . , xν).

Our main result is the following.


Theorem 4. The set M(x1, . . . , xν) carries a structure of a separable Banachmanifold. The projection map η

η : M(x1, . . . , xν) → U∆

η(C, J) = J

is a Fredholm map with Fredholm index near C

Ind(C) = µ(C) + (n− 3)(χS −#Γ)

Here µ(C) = µ(u), χS is the Euler characteristic of a Riemann surface S, suchthat (u, (S, j), Γ) parametrizes C and u is the M -part of u.

We have several important corollaries from the above theorem.

Corrolary 1. For regular values J of η, η−1(J) is a smooth finite dimensionalmanifold whose dimension agrees with the Fredholm index.

Corrolary 2. There exists a dense subset S ⊂ U∆ such that for every J ∈ S ifu : S = S \ Γ → W is a somewhere injective finite energy surface for J then

µ(u) + (n− 3)(χS −#Γ) ≥ 1

provided πTu does not vanish identically.

As a consequence of Corollary 2 we get Theorem 2.1 in [19].

Remark 1. Our results remain valid if we consider symplectic cobordisms (see [6],section 1.3). The only difference is that the inequality from Corollary 2 reduces to

µ(u) + (n− 3)(χS −#Γ) ≥ 0

due to the fact that in this case the almost complex structure J is not R - invariant.

Let us now give a brief synopsis of what follows next. The proof of the abovetheorem and the corollaries exploits results and ideas from [12, 23, 13, 25, 30], wheresimilar results are stated and proved for maps from a Riemann surface into a com-pact symplectic manifold. Let us point out some of the differences and difficultieswe are going to encounter here. The first one is the lack of the compactness of thesource and the target spaces of the maps involved. Another difference from thesymplectic case is that is that the linearizations of the Cauchy-Riemann operators(∂J) are asymptotically degenerate. To overcome this problem we have to find asuitable functional analytic setting for ∂J . This set up involves Sobolev spaces withsuitable weights derived from the non-degeneracy properties of the periodic orbitsand is the content of Section 2. In Section 3 we develop the Fredholm theory forparametrized pseudoholomorphic maps. As a guideline in this analysis we use thework of M. Schwarz, [30]. Further, in Section 4, we discuss transversality type re-sults. The subtlety in the contact case is that we vary the complex structure J ona codimension 2 subbundle of the tangent bundle. Nevertheless the transversalityresults are proven by examining carefully the form of the pseudoholomorphic curveequation in this case. In section 5 we complete the proof of our main theorem- Theorem 4 and the corollaries. We do this following the approach of [12, 23],adapting the results from the previous sections to the case of unparametrized pseu-doholomorphic curves. A delicate point, discussed here is the question of smoothcompatibility of the coordinate charts given by the Implicit Function Theorem.

8 DRAGOMIR DRAGNEV

2. The set up for Fredholm theory

In this section we are going to build the foundation for our further analysis.The Fredholm analysis on which the construction of the moduli spaces is based,significantly depends on the appropriate definition of Banach spaces involved. Asit is mentioned in the Introduction, these will be weighted Sobolev spaces withweights determined from the nondegeneracy properties of the periodic orbits of theReeb vector field. We proceed further in this section by defining the nonlinear, firstorder elliptic operator, whose zeroes will be precisely the pseudoholomorphic mapswe are interested in.

2.1. The definition of (δ, 1, p) -convergence. We already know from the intro-duction the behavior of a finite energy surface near a puncture. Let u : S → Wbe a J-holomorphic curve, whose M -part - u converges near a puncture (assumepositive one) to a nondegenerate periodic orbit of the Reeb vector field. Recall thatnear a periodic orbit we introduced special coordinates given by Lemma 1. Wewill use these coordinates to define the notion of (δ, 1, p) -convergence. Introducecylindrical coordinates near the puncture and assume that

u : [R,∞)× S1 → W = R×M

such thatu(s, t) → x(Tt + c)

uniformly for t ∈ S1. Here x is a T -periodic orbit for the Reeb vector field. UsingLemma 1, we write with ϕ,

u(s, t) = ϕ(θ(s, t), x(s, t), y(s, t)) = ϕ(θ(s, t), z(s, t))

Now, for constants c, d ∈ R we define

ad(s, t) = a(s, t)− Ts− d

θc(s, t) = θ(s, t)− kt− c

Here T = kτ , where τ is the minimal period of x. Let 0 < δ < ∞ and p > 2. Wehave,

Definition 4. We say that u is (δ, 1, p) -convergent to a periodic orbit (x, T ) of theReeb vector field if

(5) (s, t) → eδsad(s, t), eδsθc(s, t)

are in W 1,p([R,∞)× S1,R) for some c, d, R ∈ R and

(6) (s, t) → eδsz(s, t)

is in W 1,p([R,∞)× S1,R2n−2).

A priori, this definition depends on the choice of ϕ as given by Lemma 1. Itturns out, however, that this is not the case.

Lemma 2. The definition of (δ, 1, p) -convergence is independent of the choice ofϕ.

Proof: Indeed, take another map ψ which satisfies Lemma 1. Consider the mapφ = ψ ϕ−1. The map φ satisfies obviously

φ(θ, 0) = (θ + e, 0)


for some constant e ∈ R. We must show that if eδsθc(s, t) ∈ W 1,p([R,∞)× S1,R)and eδsz(s, t) ∈ W 1,p([R,∞)×S1,R2n−2), then eδsθc(s, t) and eδsz(s, t) are of classW 1,p, where

(θ(s, t), z(s, t)) = φ(θ(s, t), z(s, t)) = (φ1(θ, z), φ2(θ, z))

We have that

θ = θ + e +∫ 1

0

D2φ1(θ, τz)dτ · z = θ + e + 〈A(s, t), z(s, t)〉

z = (∫ 1

0

D2φ2(θ, τz)dτ)z = B(s, t)z(s, t)

It is clear that A and B are bounded maps and therefore in view of the above equa-tions eδsθc(s, t) and eδsz(s, t) are of class Lp. Differentiating the above equationswith respect to s, we get

∂

∂sθ =

∂

∂sθ + 〈As(s, t), z(s, t)〉+ 〈A(s, t), zs(s, t)〉

∂

∂sz = Bs(s, t)z + B(s, t)zs

Again it is easy to see that eδs ∂∂s θc and eδs ∂

∂s z(s, t) are of class Lp. Similarly weobtain that the t-derivatives with the corresponding weights are of class Lp. Thisconcludes the proof of the lemma.¤

2.2. The appropriate function spaces. We begin with the definition of theframework for the later analysis. We are going to exploit the concept of manifoldsof maps as presented in Eliasson’s paper [7]. Most of our constructions are similarto the ones used by M. Schwarz [30], and we will refer to them frequently. Let S bea closed Riemann surface with finite set of punctures Γ = Γ+ ∪ Γ− and S = S \ Γ.Near each puncture we have cylindrical coordinates σiν

i=1, where ν = #Γ. Letx1, . . . , xν be smooth periodic orbits of the Reeb vector field. Define the space

C∞x1,...,xν(S, W ) =

h = (a, h) ∈ C∞(S,W ) | limεis→∞

(h σi)(s, t) = xi(Tit + ei),

limεis→∞

((a σi)(s, t)− Tis)/s = 0, ei ∈ R, i = 1, . . . , νwhere εi = ±1, depending on the charge of the puncture. Our next task is to findan appropriate completion of the space C∞x1,...,xν

(S,W ) to a Banach manifold ofmaps of certain Sobolev type class.

We proceed with defining the metrics gJ on M and gJ on W as follows

(7) gJ(h, k) = λ(h)λ(k) + dλ(πh, Jπk)

(8) gJ((a, h), (b, k)) = ab + λ(h)λ(k) + dλ(πh, Jπk)

where π is the projection onto the contact structure along the Reeb vector field.Denote by ∇ and ∇ the Levi-Civita connections associated with gJ and gJ

respectively and by exp and ˜exp the corresponding exponential maps. We have thefollowing lemma.

Lemma 3. Let ∇ and gJ be as defined above, Z a section of TM, Y - a section ofthe contact structure ξ and X the Reeb vector field. Then the following holds.

10 DRAGOMIR DRAGNEV

• ∇XX = 0• ∇ZX and ∇XY are sections of ξ.

Proof: Differentiating the identity gJ(X, X) = 1 in the direction of Z we get:

(9) gJ(∇ZX, X) = 0

and therefore ∇ZX is a section of ξ. Differentiating the equality gJ(X,Y ) = 0 inthe direction of X we get

(10) gJ(∇XX,Y ) + gJ(X, ∇XY ) = 0

Consider now the Lie bracket [X,Y ], we have that

(11) 0 = dλ(X, Y ) = X(λ(Y ))− Y (λ(X)) + λ([X, Y ])

Recall that λ(X) = 1 and λ(Y ) = 0 consequently in view of (11) we get

λ([X, Y ]) = 0

This implies that [X,Y ] is a section of ξ.Since the Levi-Civita connection is torsion free i. e.

∇XY − ∇Y X − [X, Y ] = 0

we deduce that ∇XY is a section of ξ and therefore gJ(X, ∇XY ) = 0. Using(10), we obtain gJ(∇XX, Y ) = 0. This combined with (9) for Z = X yields that∇XX = 0. ¤

We now continue with the definition of Sobolev space structures on the pull-backbundles u∗TW , for maps u such that the M -part of u - u converges at the puncturesto periodic orbits of the Reeb vector field. Assume u0 : [R,∞)×S1 → W is (δ, 1, p)-convergent to a periodic orbit (x, T = kτ). Let γ : [R,∞) × S1 → TW such thatγ(s, t) ∈ Tu0(s,t)W . We assume γ is of class W 1,p and if we write

γ(s, t) = (b(s, t), h(s, t)X(u0(s, t)) + Q(s, t))

where Q(s, t) ∈ ξu0(s,t), we assume in addition that

(12a) eδs(b, h) ∈ W 1,p([R,∞)× S1,R2)

(12b) eδsQ ∈ W 1,p(u∗0ξ)

where the latter means that

(13)∫| eδsQ |p +

∫| ∇se

δsQ |p +∫| ∇te

δsQ |p< ∞

with | Q |2= gJ(Q,Q) = dλ(Q, JQ), Q ∈ ξ and the integration is taken over[R,∞)× S1, see Lemma 3.

Definition 5. We say that γ ∈ W 1,pδ (u∗0TW ) if γ ∈ W 1,p

loc (u∗0TW ) and it is asdescribed above near the punctures.

Define for γ ∈ W 1,pδ (u∗0TW ) and c, d ∈ R, assuming that | γ(s, t) | and | c | are

small enough, u = (a, u) by:

a(s, t) = b(s, t) + d

u(s, t) = expu0(s,t)((h(s, t) + c)X(u0(s, t)) + Q(s, t))

where γ = (b, γ), γ = hX + Q and Q ∈ ξ. We have the following proposition.


Proposition 1. Assume u0 is (δ, 1, p) -convergent to a periodic orbit (x, T ) andγ, c, d are as described above. Then u is (δ, 1, p) -convergent to (x, T ). Moreover ifu is (δ, 1, p) -convergent to (x, T ) for some γ then γ ∈ W 1,p

δ (u∗0TW ).

Proof: Let γ ∈ W 1,pδ (u∗0TW ) and u0 is (δ, 1, p) -convergent to (x, T ). We will

show first that the M -part of u - u converges to x. Indeed we have

lims→∞

u(s, t) = lims→∞

expu0(s,t)((h(s, t) + c)X(s, t) + Q(s, t)) =

lims→∞

expx(Tt+e)(cX(x(Tt + e)) = x(Tt + c + e)

The last equality is a consequence of Lemma 3.Now we want to show that u satisfies conditions (5) and (6) of Definition 4. First

we notice that (5) for the R-part a of u is obviously satisfied so we have to check (5)and (6) for u(s, t). Working in the local coordinates given by Lemma 1, we write

(θ0, z0) = ϕ−1(u0(s, t))

(θ, z) = ϕ−1(u(s, t)) = ϕ−1(expϕ(θ0,z0)((h + c)X)) + Q = F (θ0, z0;h + c, Q)

where Q = ϕ−1∗ Q. The crucial observation is that

(14) F (θ0, 0; h + c, 0) = (θ0(s, t) + h(s, t) + c, 0)

as it is easily seen from Lemma 1 and Lemma 3. Therefore we can write

F (θ0, z0; h + c, Q) =

F (θ0, 0;h + c, 0)+∫ 1

0

d

dτF (θ0, τz0;h + c, τQ)dτ =

(θ0 + h + c, 0)+(∫ 1

0

D2F (θ0, τz0;h + c, τQ)dτ)z0

+(∫ 1

0

D4F (θ0, τz0;h + c, τQ)dτ)Q =

(θ0 + h + c, 0)+A(s, t)z0 + B(s, t)Q

Thus we obtain that

(15) (θ, z) = (θ0 + h + c, 0) + A(s, t)z0 + B(s, t)Q

Combining (15) with the assumptions on γ and u0 it follows that eδsθc and eδszare of class W 1,p. Now assume that u is (δ, 1, p) -convergent to (x, T ) for some γ.Then the equation (15) holds and we conclude easily that γ ∈ W 1,p

δ (u∗0TW ).¤On W we consider the metric gJ given by (8) and the Levi-Civita connection

associated with it. Denote by D ⊂ TW the associated injectivity neighborhood ofthe zero section. Pick ε > 0 such that 2ε is less than the corresponding injectivityradius of the zero section. Define Dε ⊂ D by Dε = (w, ξ) | w ∈ W, ξ ∈ TwW, ‖ξ‖ <ε. Denote by B2

ε (0) ⊂ R2 the disk with radius ε and center 0 ∈ R2 and by D2νε (0)

the polydisk D2νε (0) = B2

ε (0)×. . .×B2ε (0). Let (c, d) = (c1, d1, . . . , cν , dν) ∈ D2ν

ε (0).Take R >> 1 and consider a smooth function κ : R → [0, 1] such that, κ(s) = 0for | s |≤ R + 1/2 and κ(s) = 1 for | s |≥ R + 1. Let ZR = [R,∞) × S1 andZ−R = (−∞,−R]× S1. Now for h ∈ C∞x1,...,xν

(S,W ) we define h(c,d) as follows:

h(c,d) = h

12 DRAGOMIR DRAGNEV

on S \⋃νi=1 σi(ZεiR) and on σi(ZεiR) we define h(c,d) in the coordinates given by

Lemma 1 to have the form

h(c,d)(s, t) = (a(s, t) + κ(s)di, θ(s, t) + κ(s)ci, (1− κ(s))z(s, t))

where εi = ±1 depending on the charge of the puncture for i = 1, . . . , ν. Nowwe are ready to define the desired completion of the space C∞x1,...,xν

(S,W ). Wecomplete it with maps from S to W which are (δ, 1, p)-convergent at the puncturesto periodic orbits xiν

i=1 of the Reeb vector field.

Definition 6. Given periodic orbits xiνi=1 we define

B = P1,p,δx1...xν

(S,W ) = ˜exp γ | γ ∈ W 1,pδ (h∗(c,d)Dε)

where h ∈ C∞x1,...,xν(S, W ), (c, d) ∈ D2ν

ε (0) and

W 1,pδ (h∗(c,d)Dε) = γ ∈ W 1,p

δ (h∗(c,d)TW ) | γ(z) ∈ Dε, z ∈ S˜exp γ = ˜exph(c,d)(z)γ(z).

We have the following theorems, whose proofs go over the same lines as theproofs of Theorem 2.1.7 and Theorem 2.2.1 from [30] and therefore are omitted.

Theorem 5. B = P1,p,δx1...xν

(S, W ) is endowed with the differentiable structure of aninfinite dimensional, separable Banach manifold.

Remark 2. We point out that with the definition of B above, a Banach neighbor-hood U of a map u ∈ B is described as a bundle over the polydisk D2ν

ε (0),

U =⋃

(c,d)∈D2νε (0)

˜expu(c,d)γ | γ ∈ W 1,p

δ (u∗(c,d)TW )

We may identify U with W 1,pδ (u∗TW )×D2ν

ε (0) by choosing a suitable trivializationas follows. Let Π(c,d) : u∗TW → u∗(c,d)TW denote parallel transport along theshortest geodesic from a point of u to a point of u(c,d), then we identify (γ, (c, d)) ∈W 1,p

δ (u∗TW )×D2νε (0) with ˜expu(c,d)

Π(c,d)γ.

Theorem 6. The vector spaces W 1,pδ (h∗TW ) and Lp

δ(h∗TW ) are well defined for

every h ∈ B0. Moreover

W 1,pδ (B∗TW ) = ∪h∈BW 1,p

δ (h∗TW )

Lpδ(B∗TW ) = ∪h∈BLp

δ(h∗TW )

are smooth vector bundles over B. There is a natural identification:

ThB ∼= W 1,pδ (h∗TW )⊕ R2ν

Our next construction is of the bundle X J over S ×W . We define it as follows :

X J = Λ0,1S ⊕J TW

X Jz,w = φ ∈ Hom(TzS, TwW ) | φ j(z) = −J(z, w) φ

We defineE = Lp

δ(B∗X J) = ∪u∈Bu × Lpδ(u

∗X J)We have

Eu = Lpδ(Λ

0,1S ⊕J u∗TW )i.e. E is a Banach space bundle over the Banach manifold B.


2.3. The nonlinear Cauchy-Riemann operator. We proceed to define the non-linear, first order, elliptic operator of Cauchy-Riemann type whose zero loci will giveus the moduli space of pseudoholomorphic curves as a subspace of B. We define itas a smooth section of a Banach bundle over a Banach manifold as follows

∂J : B → E∂J(u) = du + J du j

The rest of this subsection is devoted to the linearization of ∂J at a solution u ∈∂−1

J(0). Actually, by linearization we mean the projection of the tangent space at

the point of the bundle on its vertical subspace, which is identified with the fibreof the bundle. This is well defined since the point is contained in the zero sectionof the bundle.

d∂J(u) : TuB → T(u,0)ET(u,0)E = TuB ⊕ Eu

Denote by Π the projection Π : T(u,0)E → Eu and define

Fu : TuB → Eu

Fu = Π d∂J(u)

We will call Fu the linearization of ∂J at the solution u. It is determined foreach pseudoholomorphic curve, however its definition for general u depends on thechoice of the connection. Here we work with the Levi-Civita connection ∇ of themetric gJ . Our next task is to find a representation for Fu. Using the identificationprovided by Theorem 6 and Remark 2, define for ξ ∈ u∗(c,d)TW the parallel transport

Φ(c,d)u (ξ) : Tu(c,d)W → T ˜expu(c,d)

ξW and consider the map

Pu(γ, (c, d)) = Π−1(c,d) Φ(c,d)

u (Π(c,d)γ)−1 ∂J ˜expu(c,d)(Π(c,d)γ)

Then the linearization Fu viewed as an operator

Fu : W 1,pδ (u∗TW )⊕ R2ν → Lp

δ(u∗X J)

is given by

Fu(γ, (c, d)) =d

dλ|λ=0 Pu(λγ, λ(c, d))

From this expression it is easily seen that

(16) Fu(γ, (c, d)) = Du(γ, (0, 0)) + Ku(0, (c, d))

We have the following explicit formula for Du viewed as operator from W 1,pδ (u∗TW )

to Lpδ(u

∗X J).

Proposition 2.

(17) Duξ = ∇ξ + J(u) ∇ξ j +∇ξJ(u) d(u) j

Proof : Let ξ ∈ u∗TW . Denote by Φu(ξ) : TuW → T ˜expu(ξ)W the paralleltransport along the geodesic ˜expu(τξ). Consider the map

Gu(λξ) = Φu(ξ)−1 ∂J (expu(λξ))

The differential of Gu at 0 is just Du. Let (s, t) be local conformal coordinates onS. We compute :

14 DRAGOMIR DRAGNEV

d

dλ|λ=0 Gu(λξ) · ∂

∂s=

d

dλ|λ=0 (Φu(ξ)−1 (

∂

∂s˜expu(λξ) + J( ˜expu(λξ))

∂

∂t˜expu(λξ)) =

∇λ(∂

∂s˜expu(λξ) + J( ˜expu(λξ))

∂

∂t˜expu(λξ)) |λ=0=

(∇sd

dλ˜expu(λξ) +∇λJ( ˜expu(λξ))

∂

∂t˜expu(λξ) + J( ˜expu(λξ))∇t

d

dλ˜expu(λξ)) |λ=0

= ∇sξ+J(u)∇tξ + (∇ξJ(u))∂

∂tu

Here we used the fact that the Levi-Civita connection is torsion free and that∇λ |λ=0 J( ˜expu(λξ)) = ∇ξJ(u) and the following simple lemma.

Lemma 4. Given any w ∈ W , consider the smooth curve γ : [0, 1] → W , γ(λ) =˜expw(λξ), for ξ ∈ TwW . For v ∈ C∞(TW ) we have

d

dλ|λ=0 Φγ(ξ)−1 · v(λ) = ∇λv |λ=0

Proof : Indeed, let e1, . . . , e2n be an arbitrary parallel frame along γ, in otherwords we have ∇λej = 0 for every j = 1, . . . , 2n. Write v(λ) =

∑2nj=1 aj(λ)ej(λ).

We have on the one hand

Φγ(ξ)−1(2n∑

j=1

aj(λ)ej(λ)) =2n∑

j=1

aj(λ)ej(0)

and on the other hand

∇λ(2n∑

j=1

aj(λ)ej(λ)) = (2n∑

j=1

daj(λ)dλ

ej(λ))

The last two equalities imply the lemma. ¤Now consider the operator Ku. It is a finite dimensional operator with compact

support. Notice that Ku = 0 on σi(Zεi(R+1)) for i = 1, . . . , ν, because of theconstruction of u(c,d). Therefore by homotoping Ku to 0 we may conclude thatthe Fredholm property and transversality for the operator Fu are satisfied providedthey can be established for the operator Du. Moreover in this case we have thatthe Fredholm indices of the operators are related as follows.

(18) Ind(Fu) = Ind(Du) + 2ν

Because of this in the sequel we will sometimes abuse the notation and call Du thelinearization of ∂J .

3. Fredholm Theory

In this section we are going to study the linearized Cauchy - Riemann operator,Du. We will establish the Fredholm property for this operator by using results dueto R. Lockhart and R. McOwen, [24] and M. Schwarz, [30]. The computation of theindex exploits an index formula derived by M. Schwarz, [30], plus some observationsabout the properties of the Conley-Zehnder index.


3.1. Generalized Cauchy - Riemann operator on Hermitean bundles. Herewe will sum up some general results concerning the Cauchy -Riemann operators onpunctured Riemann surfaces. We mention that, in general, for operators on bundlesover noncompact manifolds ellipticity does not imply the Fredholm property. In[24] the authors deal with questions concerning the Fredholm property for generalelliptic operators on bundles over manifolds with cylindrical ends, while in [30],the author, concentrates on the same questions for generalized Cauchy - Riemannoperators. For our needs the latter will be sufficient. To the end of this subsectionwe will follow more or less the exposition of M. Schwarz, ([30], chapter 3).

Let S be a compact Riemann surface with a finite, nonempty set of puncturesΓ and denote S = S \ Γ. We already mentioned that near each puncture we canintroduce special cylindrical coordinates. We recall that construction. Let z0 ∈ Γ+

be a positive puncture. Take any holomorphic chart h : D ⊂ C→ U ⊂ S around z0

satisfying h(0) = z0 and introduce holomorphic polar coordinates in a puncturedneighborhood of z0 as follows

σ : Z+ = [0,∞)× S1 → U \ z0(s, t) → h(e−2π(s+it))

Then lims→∞ σ(s, t) = z0 and Z+ is equipped with the standard complex structure.Similarly for negative puncture z0 we introduce

σ : Z− = (−∞, 0]× S1 → U \ z0(s, t) → h(e+2π(s+it))

and lims→−∞ σ(s, t) = z0. Denote by ZT (respectively Z−T ) the cylinder [T,∞)×S1 (resp. (−∞,−T ]× S1).

Consider the standard symplectic vector space (R2n, ω0) with the standard com-plex structure J0 = i⊕ . . .⊕ i, where we identified R2n ∼= C⊕ . . .⊕ C.

Definition 7. A smooth loop of real matrices S ∈ C∞(S1,LR(Cn)) is called regular,if the differential equation

x(t) = J0S(t)x(t)x(0) = x(1)

only admits the trivial solution x = 0. We call S admissible if it is regular andpointwise symmetric, i.e. S(t) = S(t)T for all t ∈ S1.

We remark that it is equivalent to define S as admissible iff the operator

A = J0∂t + S : C∞(S1,R2n) → C∞(S1,R2n)

is injective and L2 - selfadjoint. Note that here we refer to L2 inner productassociated with the standard Euclidean structure on R2n, 〈·, ·〉 = ω0 (Id× J0).

We next proceed with the definition of the generalized ∂ - operator on Hermiteanbundles. Let E → B be a rank - 2n bundle over a smooth manifold. By Hermiteanstructure we understand a pair (ω, J) of a smooth bilinear form ω on E such thateach fibre (Ep, ω(p)), p ∈ B is a symplectic vector space, together with an ω -compatible complex structure J on E i. e. J2 = Id and ω(Id×J0) is a Riemannianmetric. This is tantamount to the existence of a Hermitean metric (·, ·)ω,J on thecomplex vector bundle (E, J) related to (ω, J) as follows

(v, w)ω,J = ω(v, Jw)− iω(v, w)

for all v, w - sections of E.

16 DRAGOMIR DRAGNEV

Definition 8. A unitary trivialization of a Hermitean bundle E → B is a bundleisomorphism

Φ : B × R2n → E

withJ(p)Φ(p) = Φ(p)J0

ω(p) (Φ(p), Φ(p)) = ω0

for all p ∈ B.

In view of this definition,

A : C∞(S1,R2n) → C∞(S1,R2n)

A = J(t)∂t + S(t)

is regular and selfadjoint if it is injective and S = J + ST , with J, S ∈ C∞(S1).Under a unitary trivialization Φ this is the case iff

J0Φ−1Φ + Φ−1SΦ

is admissible in the sense of Definition 7.Further we observe that any Hermitean vector bundle ζ over S possesses a unitary

trivialization near the punctures (cylindrical ends). Since we assume Γ 6= ∅ then Shas the homotopy type of a 1-dimensional cellular complex. Thus we could alwaysfind a unitary trivialization over the entire surface. Denote by SZ = ∪ν

i=1σi(Zεi),where εi = ± depending on the charge of the puncture.

Definition 9. Let ζ be a smooth Hermitean vector bundle over S with fixed unitarytrivializations near the punctures.

Φ : SZ × R2n → ζ |SZ

Ψ : SZ × R2n → ζ |SZ

We call Φ and Ψ equivalent if

Φ−1Ψ : SZ → U(n) ⊂ GLR(Cn)

is homotopically trivial.

We would like to define the general class of Cauchy - Riemann operators onpunctured Riemann surfaces. The difficulty is that we cannot do this explicitlybecause of the lack of global conformal coordinates on S. Let E be a Hermiteanbundle over S and denote

XJ (E) = Λ0,1 ⊗J E = HomC(T S, E)

Let j be a conformal structure on S and consider any volume form α on S compat-ible with j,

α |SZ =ν∑

i=1

(σ−1i )∗ds ∧ dt.

Then we can define a Hermitean structure (ω, J) on XJ(E) by

Jφ = J φ

ωz(φ, φ′) =ω(φv, φ′v)α(v, jv)


for all z ∈ S, φ, φ′ ∈ XJ (E)z, where v ∈ TzS can be chosen arbitrarily. Com-puting in local conformal coordinates one can see that the induced L2

J -metric onL2

S(XJ(E)) is independent of α. It only depends on (ω, J) on E. The 2-form

〈s, s〉J = ω(s, J s)α

on S associated to s, s ∈ L2loc(X

J(E)) is intrinsically well-defined. Let ζ, η besmooth rank-2n vector bundles over S and

F : C∞(ζ) → C∞(η)

a first order linear differential operator.

Definition 10. Given unitary trivializations Φ and Ψ of the bundles ζ |SZand

η |SZover SZ . We choose an extension to a family of local complex trivializations

ΦU , ΨU where (U,ϕ) are conformal coordinate charts on S including σi : Zεi →S. We call F a ∂-operator if the local complex trivializations can be chosen sothat the local representation FU on ζ |U with respect to ΦU : U × R2n → ζ |U ,ΨU : U × R2n → η |U and the conformal coordinates (s, t) on U is given by FU =Ψ−1

U F ΦU ,

FU =∂

∂s+ J0

∂

∂t+ SU

with SU ∈ C∞(U,LR(Cn)). Moreover, we call such a ∂-operator F admissible if onthe cylindrical ends Ui = σi(Zεi) the zero order terms SUi , yield admissible loops

S∞i = limεis→∞

Si(s) ∈ C∞(S1,LR(Cn))

in the sense of Definition 7. For fixed unitary trivializations Φ,Ψ over SZ we call

Ω = (Φ |Γ, Ψ |Γ, (Si |Γ)i=1,...,ν)

the asymptotic data of (F, Φ, Ψ) and we denote by FΩ(ζ, η) the set of all ∂-operatorsF with asymptotic data Ω.

We remark here that the above definition allows us to consider the formallyadjoint operator F ∗ of a ∂-operator F , which turns out to be a ∂-operator itself.Recall the following theorem from [30].

Theorem 7. Let ζ, η be rank-2n Hermitean vector bundles over S and F be anadmissible ∂-operator.

F : C∞(ζ) → C∞(η)

Then the induced operators

Fp : W 1,p(ζ) → Lp(η)

for p ≥ 2 are Fredholm withkerFp = ker Fq

IndFp = IndFq

for all p, q ≥ 2. Furthermore, operators F, F ′ ∈ FΩ(ζ, η) with the same asymptoticdata have the same index, IndF = IndF ′.

18 DRAGOMIR DRAGNEV

3.2. Application to the linearization of ∂J . Our task is to apply Theorem 7and the constructions summarized in the previous subsection to the operator Du

viewed as an operator

Du : W 1,pδ (u∗TW ) → Lp

δ(u∗X J)

We first consider the Hermitean structure on these bundles. u∗TW inherits itsHermitean structure from (ωλ, J), where

ωλ(w)((h, k), (h′, k′)) = hλ(k′)− h′λ(k) + dλ(k, k′)

for (h, k), (h′, k′) ∈ TwW . Using (ωλ, J) we can define, following the discussionin the previous section, an Hermitean structure on u∗X J . Now the bundle u∗ξ,being a symplectic vector bundle over S gives rise to unitary trivializations Φi ofu∗ξ |σi(Zεi ) for all i = 1, . . . , ν. We will refer to such a trivialization as Φ of u∗ξ |SZ

.We can extend Φ to a trivialization Φ of u∗TW |SZ

in an obvious way using thesplitting TW = R⊕ RX ⊕ ξ. Our first theorem in this section is the following.

Theorem 8. Let u ∈ P1,p,δx1...xν

(S, W ). Then the linearization Fu of ∂J

Fu : TuB → Eu

is an admissible ∂-operator, and therefore is a Fredholm operator.

Before we continue with the proof we mention that in what follows next we viewthe periodic orbits xi, Tiν

i=1 as 1-periodic orbits i.e. we consider xi(Tit) witht ∈ [0, 1].

Proof: As we already mentioned at the end of section 2, it will be sufficientto establish that Du : W 1,p

δ (u∗TW ) → Lpδ(u

∗X J) is an admissible operator. Weknow that u∗TW and u∗X J possess suitable local unitary trivializations, so it isenough to consider Du near the punctures (cylindrical ends) σi(ZεiT ), for T À 0and i = 1, . . . , ν with respect to trivializations ΦZ and ΨZ as given by Definition10. Consider the operator

Diu : W 1,p

δ (u∗TWσi(ZεiT )) → Lpδ(u

∗X Jσi(ZεiT ))

and denote by Di = Diu · ∂

∂s where ∂∂s is actually Tσi( ∂

∂s ). Then

Di : W 1,pδ (u∗TWσi(ZεiT )) → Lp

δ(u∗TWσi(ZεiT )).

For (s, t) ∈ ZεiT , we have

(Di · ζ)(s, t) = ∇sζ + J(s, t)∇tζ + (∇ζ J(s, t))∂u

∂t

where ∇ is as before the Levi-Civita connection associated with the metric gJ .Using the unitary trivialization Φ : ZεiT × R2n → (u σi)∗TW with JΦ = ΦJ0

where J0 is the standard complex structure on R2n, we compute Ditriv = Φ−1DiΦ

to beDi

triv · v = ∂sv + J0∂tv + Cv + Cv

whereCv = Φ−1(∇sΦ)v

Cv = J0Φ−1(∇tΦ)v + Φ−1(∇ΦvJ)∂tu.


It is not hard to see that Cv → 0 uniformly in t ∈ S1 as εis →∞. Then we mustshow that C∞ = limεis→∞ C(s, t) is admissible. Observing that limεis→∞

∂u∂t =

limεis→∞(∂a∂t , ∂u

∂t )T = X(xi(t)) ⊂ TW , we have

C∞ = limεis→∞

C(s, t) = Φ(t)−1(J(t)(∇tΦ(t)) ·+(∇Φ·J(t))X(xi)).

We will show first that C∞ is symmetric.

Lemma 5. The operator J0∂∂t + C∞ is L2-symmetric, i. e.

〈J0∂tv + C∞v, w〉L2 = 〈v, J0∂tw + C∞w〉L2

for all v, w ∈ C∞(S1,R2n).

Proof: Consider smooth vector fields along (0, xi), ζ = Φv and η = Φw. Thenwe have to prove that

〈J∇tζ + (∇ζ J)X(xi), η〉L2 = 〈ζ, J∇tη + (∇ηJ)X(xi)〉L2

where now the L2 product is on u∗TW . Write ζ = (ζ1, ζ2) in accordance with thesplitting TW = R ⊕ TM . Recall that on W we have the metric gJ given by (8)and on M the metric gJ given by (7), together with their respective Levi-Civitaconnections ∇ and ∇. Observe that

(19) ∇ζη = (∂

∂ζ1η1, ∇ζ2η2)T

for any ζ, η - sections of TW . Consider the equation JX = 1 ∈ R. Differentiatingin the direction of ζ we get:

0 = ∇ζ(JX) = (∇ζ J)X + J∇ζX

and therefore

(∇ζ J)X = −J∇ζX

and in view of our previous discussion, (19) and Lemma 3, we get

−J∇ζX = −J∇πζ2X

where π : TM → ξ is the projection along X. Now examining the form of theHermitean structure on u∗TW and writing ζ2 = πζ2 + λ(ζ2)X the lemma willfollow easily if we can show that

〈J02∂t(ζ1, λ(ζ2))T , (η1, λ(η2))T 〉L2

R2= 〈(ζ1, λ(ζ2))T , J0

2∂t(η1, λ(η2))T 〉L2R2

where J02 is the standard complex multiplication on R2 and

〈J∇tπζ2 − J∇πζ2X, πη2〉L2 = 〈πη2, J∇tπη2 − J∇πη2X〉L2

where the L2 - inner product is the one associated with dλ (Idξ × J). The formerequality is obvious, while the latter follows easily by observing that the operatorJ∇tY − J∇Y X corresponds to the Hessian of the functional A(z) =

∫S1(z∗λ)dt at

the critical point xi. Indeed, let α(t, µ1, µ2) = expxi(t)(µ1Y +µ2Z) with µj ∈ (−ε, ε)

20 DRAGOMIR DRAGNEV

for j = 1, 2 and ε small enough. We compute∂

∂µ1(A α) =

∫

S1(−dλ(α)(αt, αµ1) +

∂

∂tλ(α)(αµ1))dt

= −∫

S1dλ(α)(αt, αµ1)dt

=∫

S1gJ(αt − λ(α)(αt)X(α), J(αµ1 − λ(α)(αµ1)X(α)))dt

= 〈αt − λ(α)(αt)X(α), J(αµ1 − λ(α)(αµ1)X(α))〉L2

and∂2

∂µ1∂µ2(A α) |µ1=µ2=0=

〈∇µ2αt − ∂

∂µ2(λ(α)(αt))X(α)− λ(α)(αt)∇µ2X(α), J(αµ1 − λ(α)(αµ1)X(α))〉L2

+〈αt − λ(α)(αt)X(α), ∇µ2(J(αµ1 − λ(α)(αµ1)X(α)))〉L2 |µ1=µ2=0

The second summand is zero when µ1 = µ2 = 0 because xi solves the equationx = X(x) and using that ∂

∂µ2(λ(α)(αt)) = −dλ(α)(αt, αµ2) + ∂

∂tλ(α)(αµ2) weconclude that

∂2

∂µ1∂µ2(A α) |µ1=µ2=0= 〈∇tZ − ∇ZX, JπY 〉L2

similarly∂2

∂µ2∂µ1(A α) |µ1=µ2=0= 〈∇tY − ∇Y X,JπZ〉L2

This proves the lemma.¤From the proof of the lemma we have seen that the asymptotic operator

Ai∞(t) = −J(t)∇t · −J(t)∇·X(xi)

is L2-symmetric and it splits:

Ai∞ = Ai

1∞ ⊕Ai2∞

respecting the splitting of the pull-back of the tangent bundle x∗i TW with xi =(0, xi) and TW splits as

TW = β ⊕ ξ

whereβ → R×M = W

ξ → R×M = W

are bundles with fibersβ(a,u) = R⊕ RX(u)

ξ(a,u) = ξu

Now with (h, k) ∈ C∞(x∗i TW ), we have that (h, k) = (h, λ(k)X(xi))⊕ πk and

Ai∞(h, k) = (−λ(∇tk),−J(xi)∇tπk + hX(xi) + J(xi)∇kX(xi))

= (− d

dtλ(k), (

d

dth)X(xi))⊕ (−J(xi)∇tπk + hX(xi) + J(xi)∇kX(xi))

= (− d

dtλ(k), (

d

dth)X(xi))⊕ (−J(xi)∇tπk −B(xi)πk)


= Ai1∞(h, λ(k)X(xi))⊕Ai

2∞(πk)

where B(xi)πk = −J(xi)∇kX(xi) = −J(xi)∇πkX(xi) according to Lemma 3.From the above formulae we can conclude that the spectrum of Ai

1∞ as an operatoron L2(x∗i β) is 2πZ and therefore the operator Ai

∞ is not regular in the sense that0 is in its spectrum. It is not hard to see that Ai

2∞ as an operator on L2(x∗i ξ) isregular if xi is non-degenerate. Indeed the equation Ai

2∞Y = 0 is the equation ofthe linearized flow of the equation xi = X(xi) restricted to ξ. Since we assumed(x1, . . . , xν) to be non-degenerate then it follows that Ai

2∞ are regular for i =1, . . . , ν. Recall that we set up our operators to act on weighted Sobolev spacesand here we are going to specify the choice of the weight δ > 0. We chooseδ < minπ/τ1, π/τ2, . . . , π/τν , π so small so that the spectra of the operators

Ai2∞δ = Ai

2∞ + εiδId

viewed as operators on L2(x∗i ξ) do not intersect the interval (−δ, δ) for i = 1, . . . , ν,where εi = ±1 depending on whether xi corresponds to a positive or a negativepuncture and τi is the minimal period of xi. We remark that from the abovearguments follows that Ci

∞(t) has the form

Ci∞(t) =

(02×2 0

0 Ci2∞

)

where Ci2∞ is a (2n− 2)× (2n− 2) symmetric matrix.

Consider a smooth function a : R → [0, 1] defined as follows : a(s) = 0 fors ∈ [−R, R], a(s) = δs for s > R + 1 and a(s) = −δs for s < −R − 1 with R À 0.Now define for each i = 1, . . . , ν and with T > R, (h, k) ∈ u∗TW |σi(ZεiT ) the maps

φi(h, k) = (e−a(εis)h, e−a(εis)λ(k)X)⊕ (e−a(εis)πk)

It is clear that φi provides isomorphism between the spaces

φi : W 1,pδ (u∗TW |σi(ZεiT )) → W 1,p(u∗TW |σi(ZεiT ))

as well as for the corresponding Lp spaces

φi : Lpδ(u

∗TW |σi(ZεiT )) → Lp(u∗TW |σi(ZεiT ))

Consider the operator

F i = φ−1i Di φi : W 1,p(u∗TW |σi(ZεiT )) → Lp(u∗TW |σi(ZεiT ))

Then F i is isomorphic to Di. Denote F i∞ = limεis→∞(Φ−1 F i Φ). From the

above discussion, we conclude that F i∞ has the form

F i∞ = ∂s + J0∂t + Ci

∞

with Ci∞ of the form

Ci∞ =

(εiδId2×2 0

0 Ci2∞ + εiδId(2n−2)×(2n−2)

)

Consequently, the operators J0∂t + Ci∞ are injective and L2-self-adjoint for i =

1, . . . , ν. Therefore the operators F i are admissible and because of the isomorphismabove so are the operators Di for i = 1, . . . , ν. Thus Du is an admissible ∂-operator.This proves Theorem 8.¤

22 DRAGOMIR DRAGNEV

Next we want to compute the index of the operator Du. Given admissible loopC(t), t ∈ [0, 1], of symmetric matrices we consider the differential equation

(20)d

dtΨC(t) = J0C(t)ΨC(t)

ΨC(0) = Id

Then the solution ΨC(t) defines an arc of symplectic matrices starting at Id andending at some element of Sp(n) which does not have 1 in its spectrum. We defineµ(C) = µ(ΨC).

Proposition 3. With δ as described above we have,

µ(Ci∞) = µ(Ci

2∞)− εi

Proof : From the properties of the Conley-Zehnder index (Theorem 3) we havethat

µ(n)(Cj∞) = µ(1)(εjδId2×2) + µ(n−1)(Cj

2∞ + εjδId(2n−2)×(2n−2))

It follows from Theorem 3.3 in [28] that

(21) µ(1)(εjδId2×2) = −εj

Actually the above theorem is not applied immediately in our case because theauthors of the above reference use different conventions than we do here. Namelyin [28] the standard complex structure is given by

J0 =(

0 Id−Id 0

)

while here we work with J0 = i ⊕ . . . ⊕ i. Nevertheless, if we denote by µSZ theindex from [28], then µ = −µSZ and with that said, (21) follows easily.

Consider the homotopy τ → Cj2∞ + τεjδId where τ ∈ [0, 1] we cannot have any

nontrivial solution of J0x+(Cj2∞+τεjδId)x, with x(0) = x(1) since otherwise there

would be an eigenvalue for J0x + Cj2∞ in (−δ, δ). This shows that the associated

homotopy of arcs is admissible and µ(n−1)(Cj2∞) = µ(n−1)(Cj

2∞ + τεjδId). ¤Now we are in the position to state the theorem concerning the index of Du.

Theorem 9. The index of Du is given by the formula

(22) Ind(Du) = n(χS − ν) + µ(u)− ν

where χS is the Euler characteristic of S and ν = #Γ and therefore the index ofthe operator Fu is

(23) Ind(Fu) = n(χS − ν) + µ(u) + ν

Proof: The proof follows from the index formula derived by M. Schwarz, ([30],Theorem 3.3.11) combined with the previous proposition and the observation thatµ(ΨCi

2∞) = µ(zi). For the second formula we refer to (18). ¤


4. Transversality

In this section we establish transversality results for the Cauchy-Riemann typeoperators. We will allow the almost complex structure J to change in the set U∆,defined in the introduction. Let u ∈ B and consider the map (u, J) → ∂J u. Itdefines a smooth section F

F : B × U∆ → EF(u, J) = ∂J(u)

The differential DF(u, J) at a zero of this section i. e. (u, J) s. t. F(u, J) = 0 isgiven by

DF(u, J) : TuB × TJU∆ → Eu

(24) DF(u, J)((ξ, (c, d)), Y ) = Fu(ξ, (c, d)) + Y (u) du j

where (ξ, (c, d)) ∈ TuB ∼= W 1,pδ (u∗TW ) ⊕ R2ν , Y ∈ TJU∆. We have the following

theorem

Theorem 10. The operator DF(u, J) is surjective.

Proof: Observe first that DF(u, J) has a closed range since Fu is Fredholm.Therefore it is enough to show that its range is dense. We notice that the rangeof this operator will be dense if the range of the operator DF ′(u, J)(ξ, (c, d), Y ) =DF ′(u, J)(ξ, Y ) = Duξ+Y (u)duj is dense. Here we view Du as an operator act-ing on TuB ∼= W 1,p

δ (u∗TW )⊕R2ν by letting it act trivially on the second summand.In the sequel we abuse the notation and will write DF(u, J) instead of DF ′(u, J).Now, arguing indirectly, assume that the range of DF(u, J) is not dense. Thenby the Hahn-Banach theorem, there exists a section l ∈ Lq

δ(Λ0,1T ∗S

⊗J u∗TW ) ,

where 1p + 1

q = 1, so that l 6= 0 and

(25) 〈l, DF(u, J)(ξ, Y )〉L2S,J

= 0

for all (ξ, Y ). The last equality implies that

(26) 〈l, Du(ξ)〉L2S,J

= 0

and

(27) 〈l, Y (u) du j〉L2S,J

= 0

But the first of these equations means that l is a weak solution of the equationD∗

ul = 0, where D∗u is the formally adjoint operator of Du. In local coordinates

D∗ul = 0 has the form

(28) ∇s l − J(s, t)∇t l + C(s, t)l = 0

where l = l · ∂∂s . In other words D∗

u is a first order elliptic operator and thereforeit follows (by the elliptic regularity) that l is a smooth solution of the equationD∗

ul = 0. Thus we can apply to l the Unique Continuation principle, which meansthat l ≡ 0 iff it vanishes on a non-discrete subset of S (see [11, 22, 25, 30]). We willneed the following theorem from [18].

24 DRAGOMIR DRAGNEV

Theorem 11. Let u : S → W be a somewhere injective finite energy surfacewith non-degenerate asymptotic limits. Assume there exists z0 ∈ S such that π ·Tu(z0) 6= 0 (where u is the M -part of u). Then u is somewhere injective and theset z ∈ S|π · Tu(z) 6= 0, u(z) /∈ uS \ z is open and dense in S.

Remark 3. We argue that we can apply the above theorem here. Indeed assumethat πTu ≡ 0 or equivalently that dλ-energy of the map u -

∫S

u∗dλ = 0,(see(3). Applying Theorem 6.11 from [17] together with our somewhere injectivity as-sumption, we can conclude that the only possibility for such map is to be a ”triv-ial cylinder” over a periodic orbit of the Reeb vector field x, i.e. of the form:u(e2π(s+it)) = (Ts + c, x(Tt + d)) for suitable constants c, d and in this case,S ∼= S2 \ 0,∞. Since we have finitely many periodic orbits which are isolated, be-ing non-degenerate, we conclude that the moduli space M(x, x) ∼= U∆ and Theorem4 is easily satisfied. Therefore we can assume further, that πTu does not vanishidentically.

Pick now z0 ∈ S together with a neighborhood V on which u is somewhereinjective. Consider a neighborhood U of zero in (R2n−1, λ′0 = dθ +

∑n−1i=1 xidyi −

dy1). By Darboux’ Theorem there exists a diffeomorphism φ such that

φ : U → u(V )

φ∗λ = λ′0Introduce new coordinates (a, θ, x, y) = (a, φ−1 u). In these coordinates theCauchy-Riemann equations (1) take the form

as = λ′0(θt, xt, yt) = θt +n−1∑

i=1

xiyit − y1t

at = −λ′0(θs, xs, ys) = −θs −n−1∑

i=1

xiyis + y1s

(xy

)

s

+ J

(xy

)

t

= 0

where s + it are conformal local coordinates on V ⊂ S, J = dφ−1 J dφ. Wewant to show that l vanishes on V . The idea is that we have a particular form ofthe Cauchy-Riemann operator. The linearized operator in local coordinates (s, t)on V ⊂ S and coordinates (a, θ, x, y) has the form

Duξ · ∂

∂s=

(D1

D2

)ξ · ∂

∂s

where

D1ξ · ∂

∂s=

(αβ

)

s

+(

0 1−1 0

)(αβ

)

t

−n−1∑

i=1

xi

(ηit

−ηis

)

−n−1∑

i=1

ζi

(yit

−yis

)+

(η1t

−η1s

)

and

D2ξ · ∂

∂s= ∂sυ + J∂tυ +∇υJ dw j


with ξ · ∂∂s = (α, β, ζ, η)T , υ = (ζ, η)T , w = (x, y). Now for sections ξ and Y (u)

supported on V we can rewrite the equations (26) and (27) in the form

(29) 〈l,Du(ξ)〉L2V,J

= 0

(30) 〈l, Y (u) du j〉L2V,J

= 0

and we can use the local coordinate expressions. Write l = (l, l), then the equation(30) becomes

〈l, Y dw j〉L2V,J

= 0

Now, we are going to use standard arguments to show that l ≡ 0. We provide thedetails for completeness. Since in V we have that dw(z0) 6= 0, we can find Y0 suchthat

〈l, Y0(u(z0)) dw(z0) j(z0)〉 6= 0

Using a cut-off function, we can find Y that is supported near w(z0). Then

〈l, Y dw j〉L2V,J

6= 0

which is a contradiction. Therefore l = 0 on V and by the unique continuationprinciple it follows that it vanishes on S. Then we have from (29)

〈l, D1ξ〉L2V,J0

= 0

and we obtain the following two equations:

〈l, ∂J0

(αβ

)〉L2

V,J0= 0

and

〈l,−n−1∑

i=1

xi

(ηit

−ηis

)−

n−1∑

i=1

ζi

(yit

−yis

)+

(η1t

−η1s

)〉L2

V,J0= 0

where J0 is the standard complex multiplication on R2. From the first equation weconclude again that the unique continuation principle can be applied to l. Fromthe second one we get by setting ζ = 0 and ηi = 0 for i > 1

〈l, (−x1 + 1)(

η1t

−η1s

)〉L2

V,J0= 0

Now since (θ, x, y) is in a neighborhood of zero in R2n−1, we may assume that−x1 + 1 > 1/2 and choosing η1t = l1 and η1s = −l2 (we can solve these equationslocally in a neighborhood V ′ ⊂ V of z0) with l = (l1, l2). Now, perhaps aftershrinking V to V ′ and using again cut-off function supported on V ′ we get

0 = 〈l, (−x1 + 1)l〉L2V ′,J0

≥ 1/2〈l, l〉L2V ′,J0

≥ 0

which is only possible if l = 0 on V ′ and by the unique continuation principle weget l = 0 on S.

Thus we showed that l ≡ 0 and therefore the range of DF(u, J) is dense and theoperator is surjective. ¤

26 DRAGOMIR DRAGNEV

5. The proof of the Main Theorem

In this section we will complete the proof of our main theorem - Theorem 4 andthe corollaries. We recall from the Introduction that two somewhere injective finiteenergy surfaces (u, (S, j), Γ) and (v, (S′, j′),Γ′) are called equivalent if there exists abiholomorphic map φ : S → S′ such that Γ′ = φ(Γ), preserving the numbering, andu = v φ. We denote this equivalence relation by (u, (S, j), Γ) ∼ (v, (S′, j′),Γ′) andby [u, (S, j),Γ] the corresponding equivalence class. We call the set of equivalenceclasses of such curves moduli space of unparametrized finite energy surfaces. Inother words we will allow the set of punctures Γ and the conformal structure j tovary on S and will be interested only in the image C = u(S) ⊂ W = R×M . Thuswe have to adapt the machinery we developed so far for maps from fixed domainsto maps from varying domains. Similar problems are studied by Liu and Tian, [23],and Fukaya and Ono, [12], who consider pseudoholomorphic maps from noddedRiemann surfaces into a compact symplectic manifold. Here we follow more or lesstheir approach. To simplify the situation, we will be dealing only with smoothRiemann surfaces. Our task is further facilitated by our assumption that the mapswe consider are somewhere injective. In fact, that is why the space M(x1, . . . , xν)possesses a smooth manifold structure. In the general case when multiply-coveredmaps are considered as well this space will have an orbifold structure, near thepoints at which the Cauchy-Riemann operator is transversal to the zero section,(see [29] for the definition of an orbifold).

Definition 11. Let S = S \Γ be a punctured Riemann surface and denote by g thegenus of S and ν = #Γ. We call S stable if 2g + ν ≥ 3 and unstable otherwise.

Equivalently we could define S to be stable as follows. Denote by Aut(S) theset of automorphisms of S preserving Γ. Then S is stable if Aut(S) is finite.

In view of Definition 11 and our assumption that Γ 6= ∅, the only possibleunstable Riemann surfaces we have to consider are : i) S = S2 and #Γ = 1 and ii)S = S2 and #Γ = 2.

We start proving Theorem 4, by considering first the case of maps from stablepunctured Riemann surfaces into the symplectization of a contact manifold. Ourfirst task is to find a nice parametrization for the space of isomorphism classesof punctured Riemann surfaces of genus g with ν punctures - Mg,ν . In whatfollows next we will need some facts and constructions from the world of algebraicgeometry([4, 14]). In this world (punctured) Riemann surfaces are also recognizedunder the name (pointed) complex curves. We start with a definition.

Definition 12. Let (S0, Γ0 = p1, . . . pν) be a pointed curve. A versal deformationof (S0, p1, . . . pν) is a flat family f : X → B with an isomorphism ψ : S0

∼= f−1(b0)and disjoint family of sections σi : B → X such that σi(b0) = ψ(pi) for i = 1, . . . , ν,having the versality property that any other deformation f ′ : X ′ → B′, σ′i : B′ → X ′

is analytically isomorphic in a neighborhood U of each point of B′ to the pullbackof f : X → B by a map ϕ : U → B. A versal deformation f : X → B is calleduniversal if the map ϕ : U → B is uniquely determined.

It is a known fact [14], that stable pointed complex curves possess universaldeformations with smooth base spaces B.

Next we would like to understand the role the automorphism group of the centralfiber f−1 plays in the story. Let f : X → B , σi : B → X, f−1(b0) ∼= S0,


σi(b0) = pi be an universal family and α a nontrivial automorphism of (S0, Γ0).Then we can construct another deformation of (S0, Γ0) by composing with α thegiven identification ψ : f−1(b0) ∼= S0. By the universal property of the family thereexists (possibly after shrinking B) an extension of α to an action α on X making f- Aut(S0,Γ0) - equivariant. Furthermore if b and b′ parametrize isomorphic curvesthen there exists β ∈ Aut(S0, Γ0) so that b′ = βb. From this we conclude thatthe space Mg,ν has the structure of a smooth orbifold and (B, Aut(S0,Γ0)) →B/Aut(S0,Γ0) → Mg,ν provide local uniformizing system, [29], around [S0, Γ0].For our purposes we will need to make this construction more explicit. We pointout first, that the vector space H1(S0, TS0−

∑νi=1 pi) which parametrizes the first

order deformations of (S0, Γ0) is of complex dimension 3g− 3 + ν, according to theRiemann-Roch formula. Let V be sufficiently small neighborhood of 0 in C3g−3+ν .We would like to construct an universal family with base space V of the formV × S0 so that we get a parametrization of a neighborhood of [(S0, j0), Γ0] in Mg,ν

by deforming the conformal structure j0 fiberwise away from the punctures. Thefiber over σ ∈ V - Sσ is just S0 equipped with conformal structure jσ so that jσ = j0on a G0 -invariant neighborhood of the punctures with G0 = Aut(S0). Let

(31) ρ : C3g−3+ν → H1(S0, TS0 −ν∑

i=1

pi)

be a linear isomorphism and π be the lift of the map Λ0,1(T S0) → H1(S0, TS0 −∑νi=1 pi) such that π ≡ 0 on a G0-invariant neighborhood of the punctures. Then

we define

(32) jσ = (Id− j0π(ρ(σ))

2)j0(Id− j0

π(ρ(σ))2

)−1

so that[

djεσ

dε |ε=0

]= ρ(σ) ∈ H1(S0, TS0 −

∑νi=1 pi).

Let J be an almost complex structure on W and u0 be a J -pseudoholomorphicmap from a punctured Riemann surface (S0, j0, Γ0) into W . Denote as beforeby G0 the group of automorphisms of (S0, j0, Γ0) and by Aut(u0) the group ofautomorphisms of u0 i. e. Aut(u0) = g ∈ G0 | u = u g ⊂ G0. The crucialobservation is that if u0 is somewhere injective then Aut(u0) = Id.

Let C0 be an unparametrized, somewhere injective, J-holomorphic curve withfinite energy and (u0, (S0, j0), Γ0) be a parametrization of C0. We would like todescribe a neighbourhood of (C0, J) in M(x1, ..., xν). Namely, we will say thatanother J-holomorphic curve, C is close to C0 if C is represented by (u, (S0, jσ), Γ0),where u : S0 → W is a (J , jσ)-holomorphic curve, converging at the punctures toperiodic orbits xiν

i=1 of the Reeb vector field and u is close to u0 in W 1,ploc . The

last means that u = ˜expu0,(c,d)γ, where γ ∈ W 1,p

loc (u∗0,(c,d)TW ) is sufficiently smalland (c, d) sufficiently close to 0 ∈ R2ν . Moreover we assume that jσ is C∞-closeto j0. Since Aut(C0) is the trivial group one concludes that (u, (S0, jσ), Γ0) is theunique representative of C with u close to u0 in W 1,p

loc . This motivates our interestin the set of all triples (σ, u, J) | ∂J,jσ

u = 0, where u : S0 → W is a map ofsuitable Sobolev type class, converging at the punctures to non-degenerate periodicorbits xiν

i=1 of the Reeb vector field, σ ∈ V and J ∈ U∆. The idea is again torepresent the operator ∂J,jσ

u = 0 as a smooth section of some Banach bundle overa Banach manifold so that the zero set of this section will give us precisely the set of

28 DRAGOMIR DRAGNEV

solutions of the Cauchy-Riemann equations. Recall the space B we constructed inSection 2. Denote by B0 = P1,p,δ

x1,...,xν(S0, W ) and recall the bundle E over B0 which

we defined earlier as well. We can extend this bundle to a bundle E over V ×B0 bydefining the fiber over (σ, u) to be

E(σ,u) = Lpδ(Λ

0,1(T Sσ)⊗J u∗TW )

This way we may view the Cauchy-Riemann operator

(33) ∂J,jσ: V × B0 × U∆ → E

as a smooth section of a Banach bundle over a Banach manifold. Assume that wehave a solution u0 of the equation ∂J0,j0

u0 = 0. We compute the linearization of(33) at the solution (0, u0, J0) as before and get an operator

DH(0, u0, J0) : V × Tu0B0 × TJ0U∆ → E(0,u0)

(34) DH(0, u0, J0)(σ, ξ, (c, d), Y ) = DF(u0, J0)(ξ, (c, d), Y ) + J0 du0 djεσ

dε|ε=0

where DF(u0, J0) is the operator defined in Section 4 and given by (24). Beforewe proceed we will take a moment to observe that we can replace at no extra costthe last term in (34) which is a finite dimensional operator with J0 du0 π(ρ(σ)).This change will not affect the range of DH(0, u0, J0) since from Theorem 10 weknow that DF(u0, J0) is surjective. Here we abuse the notation and denote thenew operator by DH. We want to construct a coordinate chart around the element(C0, J0) ∈ M(x1, . . . , xν), where C0 = [u0, (S0, j0), Γ0], which will provide thestructure of an infinite dimensional separable Banach manifold for M(x1, . . . , xν).Our main tool in this effort is the infinite dimensional implicit function theorem (see[26]), applied to the operator DH. The surjectivity of DH(0, u0, J0) follows easilyfrom the surjectivity of the operator DF(u0, J0). We need the following lemma.

Lemma 6. Let X, Y, Z be Banach spaces and consider the linear map Φ : X×Y →Z, Φ(x, y) = Fx + Gy, where F : X → Z is a Fredholm map and G : Y → Z is abounded map. If Φ is surjective, then kerΦ has a closed topological complementingsubspace Q in X × Y , i. e. X × Y = kerΦ⊕Q.

Proof: Since F is a Fredholm operator we have that a) kerF = X1 is a finitedimensional space and b) range(F ) = Z2 is of finite codimension. Then we havethe decompositions X = X1 ⊕ X2 and Z = Z1 ⊕ Z2 with dim Z1 < ∞. Nowusing the surjectivity of Φ we can find a finite dimensional subspace Y2 in Y sothat G |Y2 : Y2 → Z2 is an isomorphism. Decompose Y = Y1 ⊕ Y2 and considerthe space Q = X2 × Y2. We claim that Q complements kerΦ in X × Y . Indeed,let x2 ∈ X2 and y2 ∈ Y2 then Φ(x2, y2) = Fx2 + Gy2 = 0 if and only if x2 = 0and y2 = 0, thus kerΦ ∩ Q = ∅. Now let x ∈ X, y ∈ Y and z = Φ(x, y) ∈ Z.Write z = z1 + z2 according to the splitting of Z. Using the surjectivity of theoperator F |X2 : X2 → Z1 we can find x2 ∈ X2 and y2 ∈ Y2 such that Fx2 = z1 andGy2 = z2. Therefore z = Φ(x2, y2) and (x, y) − (x2, y2) ∈ kerΦ, which proves thelemma.¤

Now applying Theorem 2.7.3 from [26] and the previous lemma to the operatorDH(0, u0, J0), we get a smooth map

Ψ : kerDH(0, u0, J0) → V × Tu0B0 × TJ0U∆


Ψ(σ, ξ, (c, d), Y ) = (ϕ(σ, ξ, (c, d), Y ), Φ(σ, ξ, (c, d), Y ), Υ(σ, ξ, (c, d), Y ))

so that with u′ = ˜expu0Φ(σ, ξ, (c, d), Y ) and J is the extension to W of J =

J0 exp(−J0Υ(σ, ξ, (c, d), Y )), the map u′ is (J , jϕ(σ,ξ,(c,d),Y ))-holomorphic. Thuswe constructed a coordinate chart around (C0, J0) in M(x1, . . . , xν). Next wewant to show that the charts we just constructed are smoothly compatible. Let(C0, J0) ∈M(x1, . . . , xν) and (u0, (S0, j0), Γ0) be a parametrization of C0. Assume(C1, J1) ∈M(x1, . . . , xν) and C1 = [v0, (T0, k0), Γ′0]. Then by the above discussionwe get maps

Ψ : U ⊂ kerDH(0, u0, J0) →M(x1, . . . , xν)

Ψ(ξ) = (σ(ξ), u(ξ), J(ξ))for ξ ∈ U ,

Φ : V ⊂ kerDH(0, v0, J1) →M(x1, . . . , xν)

Φ(η) = (τ(η), v(η), J(η))

for η ∈ V . These maps provide coordinate charts for (C0, J0) and (C1, J1) respec-tively with U, V - small neighborhoods of 0 in the respective kernels. Now assumethat Ψ(U) ∩ Φ(V ) 6= ∅, then there exist ξ0 ∈ U and η0 ∈ V so that

(35) (u(ξ0), (S0, jσ(ξ0)), Γ0) ∼ (v(η0), (T0, kτ(η0)), Γ′0)

Therefore there exists a biholomorphic map ψσ(ξ0) : (S0, jσ(ξ0)) → (T0, kτ(η0)) suchthat ψσ(ξ0)(Γ0) = Γ′0 and v(η0) = u(ξ0) ψ−1

σ(ξ0). From the local structure of the

space Mg,ν we know that we can extend ψσ(ξ0) to a family of biholomorphic mapsψσ(ξ) : (S0, jσ(ξ)) → (T0, kω(ξ)) for ξ close to ξ0 and ω(ξ0) = τ(η0). Consider thetriples (ω(ξ), u(ξ)ψ−1

σ(ξ), J(ξ)). They provide a family of pseudoholomorphic maps

in a neighborhood of (0, v0, J′0) and by the uniqueness of the map given by the

implicit function theorem they describe a neighbourhood of (τ(η0), v(η0), J(η0)).The delicate question is about the smoothness of these maps. Notice that for mapsu(ξ) which are of class W l,p for some l ≥ 1 the map u(ξ) → u(ξ) ψ−1

σ(ξ) is notsmooth but only continuous, because the map ψ−1

σ(ξ) acts from behind which leadsto a loss of a derivative. However in our case this map is smooth since u(ξ) issmooth being a solution of an elliptic differential equation. Thus the transitionmaps are smoothly compatible and therefore M(x1, . . . , xν) carries the structureof a smooth, separable, Banach manifold.

Consider the projection map

η : M(x1, . . . , xν) → U∆

The tangent space T([u0],J0)M(x1, . . . , xν) at the point ([u0], J0) is identified with

kerDH(0, u0, J0) and

dη : T([u0],J0)M(x1, . . . , xν) → TJ0U∆

is given bydη(0, u0, J0)(σ, ξ, (c, d), Y ) = Y

We have thatker dη = ker(Fu0 + J0 du0 π(ρ(σ)))

andIm(dη) = Y | Y (u0) du0 j0 ∈ Im(Fu0 + J0 du0 π(ρ(σ)))

30 DRAGOMIR DRAGNEV

is a closed subspace of TJ0U∆. It is not hard to see that Im(dη) has the same finitecodimension as the Im(Fu0 + J0 du0 π(ρ(σ))), since DH(0, u0, J0) is surjective.Therefore η is a Fredholm operator and

Ind(η) = Ind(Fu0 + J0 du0 π(ρ(σ))) = Ind(Fu0) + dimR V

= µ(u0) + (n− 3)(χS0 − ν)

This completes the proof of the Theorem 4 for the case of stable Riemann surfaces.Let us now consider the cases of unstable punctured Riemann surfaces. We

will consider in detail the case S = S2 and #Γ = 1, the other case being treatedsimilarly. First, without loss of generality we may assume that Γ = ∞ andS ∼= C. The group of automorphisms of C is a subgroup of the group of Mobiustransformations preserving ∞, Aut(C) = G1 = az + b | a, b ∈ C. Let u : C→ Wbe a somewhere injective finite energy plane and Aut(u) = ϕ ∈ G1 | u = u ϕ.Using the assumption that u is somewhere injective leads to the conclusion thatAut(u) = Id. Therefore the group G1 acts freely on the solutions of the equation∂J u = 0. Recall from Section 2 that we can describe the set of solutions of thisequation as the zero locus of a smooth section of a Banach bundle over a Banachmanifold i. e. ∂J : B → E . The group G1 acts on B making ∂J a G1 -equivariantsection, ∂J(u ϕ) = ϕ∗∂J(u). Unfortunately this action is not smooth becauseof the same reasons as before. To overcome this problem, consider the set of allpairs (u, J) such that ∂J u = 0 or in other words the zero locus of the sectionF : B×U∆ → E . Denote this set by M(x1). The linearization of F at a zero of thissection (u, J) is DF(u, J) given by (24). We know from section 4 that this operatoris surjective. Using an implicit function theorem argument as above we can showthat M(x1) is an infinite dimensional, smooth, separable Banach manifold whosetangent space T(u,J)M(x1) is identified with kerDF(u, J) and the projection map

η : M(x1) → U∆ is a Fredholm operator with index near u, Indη = IndFu =µ(u)+ (n+1). The space of unparametrized finite energy planes M(x1) is just thequotient M(x1)/G1. The action of G1 on M(x1) is smooth because the elementsof M(x1) are smooth being solutions of an elliptic equation. We want to showthat M(x1) = M(x1)/G1 possesses the structure of a smooth separable Banachmanifold. Using results of R. Palais, [27], this will be the case if we can show thatthe action of G1 on M(x1) is proper, in the sense that the map

G1 × M(x1) → M(x1)× M(x1)

(g, u) → (u g, u)

is a proper map. Indeed, assume that uk ⊂ M(x1) and gk ⊂ G1 are sequencessuch that the sequence (uk gk, uk) possesses a convergent subsequence. We have toshow that then gk possesses a convergent subsequence. Without loss of generalitywe may assume that uk → u and uk gk → v. By the explicit description of G1

then gk(z) = akz + bk for some ak, bk ∈ C. If the sequences ak and bk arebounded then there will exist uniformly convergent subsequence gkj → g ∈ G1 andwe are done. So assume that at least one of the sequences is unbounded. Thenthere exists z ∈ C, such that gk(z) → ∞. Denote by ur

k and vr the R-parts of uk

and v. Then we have on the one hand that vr(z) = limk→∞ urk gk(z) and on the

other that limk→∞ urk gk(z) = ∞ which is a contradiction. Thus the action of G1

is proper and M(x1) has a smooth separable Banach manifold structure. Then the


projection map η is a Fredholm operator whose index near u is

Ind(dη) = Ind(dη)− dimR(g1) = µ(u) + (n− 3)

where g1 is the Lie algebra of G1.For the case S = S2 and #Γ = 2 the proof is the same. The only difference is

that the automorphism group is of real dimension 2.Proof of Corollary 1 : If J is a regular value of η, then H J

u = Fu + J du π(ρ(σ)) is a surjective Fredholm operator. From the implicit function theoremfollows that MJ

g,ν - the moduli space of unparametrized finite energy surfaces is afinite dimensional manifold whose dimension is equal to Ind(H J

u ). In the unstablecase the operator H J

u is the operator Fu restricted to the tangent bundle of thecorresponding quotient.

Proof of Corollary 2 : The proof of this corollary is an application of the Sard-Smale Theorem for Fredholm maps, [31]. In other words the set S of regular valuesof η is a residual in the sense of Baire and is in particular dense in U∆. Now for J ∈S, the operator H J

u is surjective and therefore Ind(H Ju ) ≥ dim(ker(DJ

u)) ≥ 1 dueto the R-invariance of J , provided of course that πTu does not vanish identically,see Remark 3.

Acknowledgments. I would like to thank my Ph.D. advisor, Prof. Helmut Hoferfor his continuous support and encouragement, during the preparation of this work.I am also very grateful to Ilya Ustilovski and Kris Wysocki for many stimulatingdiscussions and their interest in this paper.

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Department of Mathematics, University of Southern California, Los Angeles, CA90089

E-mail address: [email protected]

fredholm theory and transversality for noncompact

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