and J = Jâ and Ci are Jâ(zi)-holomorphic spheres Ci : S2 â M such that for corresponding ..... and E. Zehnder, BirkhÃ¤user (1995), 555-573. [S]...

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Positivity of Symplectic Area for Perturbed J-holomorphic Curves Pawel Felcyn Department of Mathematical and Computer Sciences University of Wisconsin-Whitewater email: [email protected] June 19, 2018 Abstract In this paper we will prove that for a compact, symplectic manifold (M, ω) and for ω-compatible almost-complex structure J any properly perturbed J-holomorphic curve has a non-negative symplectic area. This non-negative property provides us with a new obstruction to the bubbling off phenomenon and thus allows us to redefine the Floer symplectic homology. In particular, in subsequent papers, we will prove the Arnold conjecture in both degenerate and non-degenerate cases with integer coefficients for general, symplectic manifolds.

1

Introduction

Let us denote by (M, ω) a compact, symplectic manifold. Here ω is a closed, non-degenerate 2-form. If H : R × M → R denotes a time-dependent Hamiltonian function then there is associated a time-dependent Hamiltonian vector fields XHt on the symplectic manifold M defined by the equality ι(XHt )ω = dHt .

(1)

We shall assume that the Hamiltonian function H is of period 1 in time. The Arnold conjecture says that the number of 1-periodic solutions of the Hamiltonian equation γ(t) ˙ = XHt (γ(t))

(2)

is estimated from below by the number of critical points of a smooth function defined over the manifold M . (See [A, Appendix 9].) 1

In trying to prove the Arnold conjecture one is usually led to study the action functional Z 1 Z ∗ Ht (γ(t)dt u ω+ aH (γ) = − 0

D

defined on the space of smooth, contractible loops γ : R → M with γ(t) = γ(t + 1). See, for example, [CZ, F5, H, HZ, S] for some special cases for both nondegenerate and degenerate 1-periodic solutions of the equation (2). In trying to extend the variational methods to the general case one encounters two difficulties: 1. The action functional aH is not uniquely defined on the space of smooth, contractible loops of M . Rather, it is well defined on the universal covering of the space of loops. 2. There is bubbling off phenomenon which causes difficulties with compactification of appropriate moduli spaces. In [F2, F3, F4, HS, O1, O2, O3] these difficulties has been overcome in some more general cases. In our approach to the Arnold conjecture which is valid for all compact symplectic manifolds we will restrict the action functional to a subset of its one special ‘branch’. Thus the first difficulty will be overcome in general. Fortunately, over the restricted set the bubbling problem will also disappear. Thus, in particular, we redefine the Floer symplectic homology to obtain a homology theory very similar to the finite dimensional Morse homology. Details of the construction of the new Floer homology and some of its consequences will be presented in subsequent papers. In this paper we present the major tool in our construction of Floer homology which is the positivity of symplectic area of properly perturbed J-holomorphic curves. For a compact Riemann surface (Σ, j) the map u : Σ → M is said to satisfy a properly perturbed Cauchy-Riemann equation if du + J ◦ du ◦ j + P (u) = 0. (See Def 3.2 for details.) Theorem 1.1 Let u : Σ → M be a properly perturbed J-holomorphic curve. Then the symplectic area of u is non-negative: Z u∗ ω ≥ 0. Σ

Acknowledgment We are very thankful J.W. Robbin for finding mistakes in the previous version of these notes. 2

2

Example

Let S 2 be a Riemannian sphere S 2 = C ∪ {∞} with the standard Kaehler metric dz ⊗ dz , (|z|2 + 1)2 where z = x + iy denotes a point of S 2 . The induced standard symplectic form is of the form dx ∧ dy ω= 2 , (x + y 2 + 1)2 and the induced metric by the standard complex structure on S 2 is of the form x b1 x b2 + yb1 yb2 , (3) hb z1 , zb2 i = (|z|2 + 1)2

where zbj = x bj + ib yj denote a tangent vectors at z = x + iy. We will consider a Hamiltonian function H : R × S 2 → R of the form H(s, z) = ψ(s)

x2 + y 2 − 1 . x2 + y 2 + 1

The gradient vector field (depending on s) of the Hamiltonian function H with respect to the metric (3) is ∇H(s, z) = 4ψ(s)z. Thus we will obtain the following equation for the properly perturbed holomorphic curve u : R × S1 × S2 → S2 : ∂u ∂u +i + 4ψ(s)u = 0. (4) ∂s ∂t Let us consider solutions of the equation (4) of the form Z s ′ ′ ψ(s )ds exp{2πk(s + it)}. u(s, t) = exp −4 (5) −∞

Indeed, itRis easy to verify that each function of the form (5), for which the s function −∞ ψ(s′ )ds′ is smooth, satisfies the perturbed Cauchy-Riemann equation (4). Assume now that the function ψ has a compact support. Under this condition we claim that the symplectic area of solutions (5) is non-negative i.e. Z u∗ ω ≥ 0.

3

Perhaps, the simplest way to see this is to homotop the curve u(s, t) to a holomorphic one. The homotopy can be done by considering the equation (4) with a parameter λ, 0 ≤ λ ≤ 1 : ∂u ∂u +i + 4λψ(s)u = 0. ∂s ∂t Then for each λ solutions of the above equation of the form Z s ′ ′ ψ(s )ds exp{2πk(s + it)} uλ (s, t) = exp −4λ −∞

will provide us a homotopy between the original solution and the holomorphic curve u0 (s, t) = exp 2πk(s + it). Now since the symplectic form ω is closed and the symplectic area of the holomorphic curve is non-negative by Stokes theorem we obtain Z Z ∗ u ω = u∗ ω0 ≥ 0. In fact, as we will see, the argument of deforming a properly perturbed Cauchy-Riemann equation to a non-perturbed one works in general and thus proving the positivity of symplectic area of perturbed J-holomorphic curves. Note, however, that if we let the function ψ to be a nonzero constant: ψ(s) = τ for all s ∈ R, then as it has been shown in [HS] the function uk (s, t) = exp(4τ s) exp{2πk(s + it)} is a solution of the equation (4) whenever πk + 2τ > 0. Moreover, for the symplectic area we have Z u∗k ω = πk. Thus, in particular, we obtain solutions uk with a negative symplectic area if k is negative. The key point why the function ψ with compact support produces only solutions with non-negative symplectic area is that the Hamiltonian part of the equation (4) represents a derivative of a global function on S 2 . In the case of the function ψ being a nonzero constant this is not so.

3

Perturbed J-holomorphic Curves

Let (M, ω) denote a 2n-dimensional compact symplectic manifold with the symplectic form ω and let (Σ, j) denotes a closed connected Riemman surface 4

with a complex structure j and with a fixed Kaehler metric. Let J denote a smooth family of ω-compatible almost-complex structure on M depending on the parameter z ∈ Σ. We will denote the space of such families by J . Let X = Map(Σ, M ; A) be the space of all smooth maps u : Σ → M which represent the homology class A ∈ H2 (M ) i.e. such that u∗ ([Σ]) = A ∈ H2 (M ), where [Σ] denotes the fundamental class of the surface Σ determined the orientation associated to the complex structure j. For simplicity in this paper we will consider only homology with integer coefficients and denote it H∗ (M ). So in this situation we will use notation [u] = A ∈ H2 (M ). We shall denote by X 1,p completion of the space X with respect to the Sobolev norm W 1,p . More precisely, X 1,p is the space of maps u : Σ → M whose first covariant derivatives with respect to Riemannian metric on M are of class Lp . Since the manifold M is compact the topology of this norm does not depend on the choice of the Riemannian metric. In order for the space X 1,p to be well-defined we must assume that p > 2. The tangent space Tu X 1,p of X 1,p at a smooth u is the completion of the space C ∞ (u∗ T M ) of all smooth sections u b ∈ Γ(u∗ T M ) in the Sobolev norm. For a family of almost complex structures J ∈ J let us consider the infinite dimensional vector bundle E → X 1,p where the fiber at u is the space Eu = Lp (Σ, Ω0,1 ⊗J u∗ T M ) of Lp -section of the vector bundle over Σ whose fiber over a point z ∈ Σ is the space of C-linear, with respect to J(z, u(z)), maps from Tz0,1 Σ to (u∗ T M )z . We note that the zero set of the section ∂ J : X 1,p → E of the infinite dimensional vector bundle E → X 1,p given by the formula ∂ J (u) = du + J ◦ du ◦ j is the set of all J-holomorphic curves in the class A ∈ H2 (M ). We shall denote by Ω0,1 Σ ⊗J T M the vector bundle over the space Σ × M whose fiber over a point (z, m) ∈ Σ × M is the space of C-antilinear maps from Tz Σ to (T M )m with respect to J(z, m). Now we want to introduce the general setting for introducing and dealing with the concept of perturbed J-holomorphic curves. Let Ω1 ⊗ C denote the space of all complex valued one form defined over the product Σ × M . On the product Σ×M there is the almost-complex structure j ×J. With respect to this almost-complex structure we have the following decomposition of the space Σ × M into the direct sum Ω1 ⊗ C = Ω1,0 ⊕ Ω0,1 5

complex linear one forms and complex antilinear one forms. Any one form in Ω1,0 can be uniquely written in the form α − iα ◦ (j × J), where α is a real valued one form in Ω1 . Similarly, any one form in Ω0,1 can be uniquely written as α + iα ◦ (j × J). Our interest will be in following subspaces: 1,0 whose restriction 1. Ω1,0 M = subspace of all complex linear one form in Ω to the horizontal subbundle TΣ (Σ×M ) of the tangent bundle T (Σ×M ) is trivial. 0,1 whose restriction 2. Ω0,1 Σ = subspace of all complex linear one form in Ω to the vertical subbundle TM (Σ × M ) of the tangent bundle T (Σ × M ) is trivial.

We will use the ∂ Σ operator in the direction of Σ in three different situations: 1. ∂ Σ : C ∞ (Σ × M ) → Ω0,1 Σ , ∂ Σ (f ) = dΣ f + i(dΣ f ) ◦ j. 2. ∂ Σ : C ∞ (Σ × M, π ∗ T M ) → C ∞ (Σ × M, Ω0,1 Σ ⊗J T M ), ∂ Σ (X) = dΣ X + J ◦ (dΣ X) ◦ j. Here dΣ denote the partial derivative in the direction of Σ. Note that it make sense to define dΣ since all tangent spaces of the form (π ∗ T M )(z,m) are canonically identified with T Mm . 3. 0,1 1,0 ∂ Σ : Ω1,0 M → ΩΣ ⊗C ΩM ,

where

∂ Σ (ψ − iψ ◦ J) = ∂ Σ (ψ) − i ∂ Σ (ψ) ◦ J, ∂ Σ (ψ) = dΣ (ψ) + i (dΣ (ψ)) ◦ j

for any real valued one form ψ which is zero on TΣ (Σ × M ). 6

We will also need the following maps: 1. ∂M : C ∞ (Σ × M ) → Ω1,0 M, ∂M (f ) = dM f − i(dM f ) ◦ J. 2. 0,1 1,0 ∂M : Ω0,1 Σ → ΩΣ ⊗ ΩM ,

∂M (ψ + iψ ◦ j) = ∂M (ψ) + i (∂M (ψ)) ◦ j, where ∂M (ψ) = dM (ψ) − i (dM (ψ)) ◦ J for any real valued one form ψ which vanishes on TM (Σ × M ). 1,0 We will often identify the tangent bundle π ∗ (T M ) with that of ΩM using the metric corresponding to J. The identification is given by the formula

X 7→ Φ(X), where Φ(X)(Y ) = ω(X, Y ) − iω(X, JY ) = hX, JY iJ + i hX, Y iJ

(6)

for any X, Y ∈ π ∗ (T M )(z,m) . Note that the map Φ is complex antilinear as it should be since the bundle 1,0 ΩM of complex linear forms is a complex dual of the bundle π ∗ (T M ). Under this identification the following diagram commutes C ∞ (π ∗(T M )) yΦ Ω1,0 M

∂

Σ −→ C ∞ (Ω0,1 Σ ⊗J T M ) yid⊗Φ

∂Σ

−→

Ω0,1 Σ

⊗

Ω1,0 M.

This is because ∂ Σ (X) ◦ Φ = ∂ Σ ω(X, .) − i ∂ Σ ω(X, .) ◦ J = dΣ ω(X, .) + idΣ ω(X, .) ◦ j

−i(dΣ ω(X, .)) ◦ J + dΣ ω(X, .) ◦ j ◦ J = ω(dΣ X, .) − iω(dΣ X, .) ◦ J +i(ω(dΣ X, .) − iω(dΣ X, .) ◦ J) ◦ j = Φ(dΣ X) + iΦ(dΣ X) ◦ j = id ⊗ Φ(dΣ X + J ◦ dΣ X ◦ j) = (id ⊗ Φ) ◦ ∂ Σ (X). 7

(7)

Definition 3.1 An element P ∈ C ∞ (Σ × M, Ω0,1 Σ ⊗J T M ) is said to be exact if there is a vector field X ∈ C ∞ (Σ × M, π ∗ T M ) of the form X = Φ−1 ◦ ∂M f

(8)

P = ∂ Σ X = dΣ + J ◦ (dΣ X) ◦ j.

(9)

such that

Definition 3.2 Let ∂ J,f : X → E be a section of the form ∂ J,f (u) = du + J ◦ du ◦ j + P (u),

(10)

−1 ◦ ∂ f , and where P ∈ C ∞ (Σ × M, Ω0,1 M Σ ⊗J T M ) is exact, P = ∂ Σ ◦ Φ P (u)(z) = P (z, u(z)). The equation ∂ J,f (u) = 0 (11)

will be called a perturbed Cauchy-Riemann equation. Solutions of the equation (11) will be called perturbed J-holomorphic curves. Definition 3.3 The equation (11) is said to be a properly perturbed Cauchy-Riemann equation if there is a constant z ∈ C such that the function f satisfies the equation f =zg

(12)

for some real function g ∈ C ∞ (Σ × M, R). The perturbation term (9) is said to be properly exact if the function f in the equation (8)satisfies the equation (12)

4

Properties of Perturbed J-holomorphic Curves

Lemma 4.1 Let P = ∂ Σ ◦ Φ−1 ◦ ∂M f be an exact perturbation. Then it can be written as (13) P = (id ⊗ Φ)−1 ◦ ∂M ◦ ∂ Σ f. On the other hand, any perturbation of the form (13) is exact.

8

Proof: Let us first verify that the following diagram commutes ∂

Σ C ∞ (Σ× M ) −→ ∂ y M

Ω1,0 M

∂Σ

−→

Ω0,1 Σ ∂ y M

Ω0,1 Σ

⊗

(14)

Ω1,0 M.

Indeed, we may assume that the function f is real. Then ∂M (∂ Σ f ) = ∂M (dΣ f + i(dΣ f ) ◦ j) = ∂M (dΣ f ) + i(∂M (dΣ f )) ◦ j = dM (dΣ f ) − i(dM (dΣ f )) ◦ J +i(dM (dΣ f )) ◦ j + (dM (dΣ f )) ◦ J ◦ j and ∂ Σ (∂M f ) = ∂ Σ (∂M f − i(∂M f ) ◦ J) = dΣ (dM f ) + i(dΣ (dM )) ◦ j −i(dΣ (dM )) ◦ J + (dΣ (dM f )) ◦ j ◦ J. Since partial derivatives commutes the commutativity of the diagram follows. To finish the proof of the lemma combine the two commutative diagrams (7) and (14). Proposition 4.2 Let Σ = S 2 be the Riemannian sphere and let ψ ∈ Ω0,1 Σ . Then the perturbation (id ⊗ Φ)−1 ◦ ∂M ψ is exact. Proof: Using the Lemma 4.1 it is enough to show that ψ = ∂ Σ (f ) for some function f ∈ C ∞ (Σ × M ). For every m ∈ M we have ψm = ψ(., m) ∈ Λ0,1 (S 2 ). Since Λ0,2 (S 2 ) = 0, ∂ψm = 0. Moreover, since H 0,1 (S 2 ) = 0 and H 0 (S 2 ) = C there is the unique function fm : S 2 → C such that ∂fm = ψm and fm (z0 ) = 0 for the fixed point z0 ∈ S 2 . Define f by the formula f (z, m) = fm (z). Then one verifies that ψ = ∂ Σ (f ) and this finishes the proof of the Proposition. We note that if g : R × S 1 → S 2 , g(s, t) = z = s + it is a holomorphic coordinate system and a function H : R × S 1 × M → R has a compact support then the Proposition 4.2 implies that the perturbation P = ∇Hds − J∇Hdt 9

(15)

is exact. This is so because id ⊗ Φ(∇Hds − J∇Hdt) = (idH + dH ◦ J) ⊗ (ds − idt) = i(dH − idH ◦ J) ⊗ ds + (dH − idH ◦ J) ⊗ dt = ∂M (iHds + Hdt). In our construction of a new Floer symplectic (co)homology it will be very important to note that the Hamiltonian perturbation term (15) is in fact properly exact. It follows from the equation ∂ Σ (ig) = iH ds + H dt,

(16)

where the function g is defined as g(z, m) =

Z

(z,m)

H ds + H dt,

(∞,m)

where the integration is taken over any smooth path contained in the set Σ×m Theorem 4.3 n Let u : Σ → M be a smooth map from a Riemannian surface Σ to a symplectic manifold M with compatible family of almostcomplex structures J. Assume that du(z0 ) 6= 0 and that du(z0 ) is complex anti-linear at a single point z0 ∈ Σ. Then u is not a perturbed J-holomorphic curve. Proof: Assume that u is a perturbed J-holomorphic curve. Then by the Definition 3.2 and the Lemma 4.1 it satisfies the equation du + J ◦ du ◦ j + P (u) = 0, where P = (id ⊗ Φ)−1 ◦ ∂M ◦ ∂ Σ f.

(17)

1,0 Consider the tensor (id⊗Φ)◦P ∈ Ω0,1 Σ ⊗ΩM . For any two vectors v, w ∈ Tz0 Σ we have the following formula

θ(v, u) := (id ⊗ Φ) ◦ P (v ⊗ du(w)) 1

1

= − ∂ J u(v), J ◦ ∂ J u(w) − i ∂ J u(v), ∂ J u(w) . 2 2 10

This is because of the formula (6) and the fact that, since du(z0 ) is complex anti-linear, at z0 we have ∂ J u(z0 ) = −P (u)(z0 ) = 2du(z0 ). In particular, since vectors ∂ J u(v) and J ◦ ∂ J u(v) are orthogonal we obtain that the quadratic function 1

θ(v, v) = − i ∂ J u(v), ∂ J u(v) 2 is purely imaginary. Moreover, one easily verifies that θ(j ◦ v, j ◦ v) = θ(v, v).

(18)

Therefore, because of (17) the expression ∂M ◦ ∂ Σ f (v, du(v)) is also purely imaginary at z0 . Thus one computes at z0 ∂M ◦ ∂ Σ f (v, du(v)) = i(djv ddu(v) f1 + dv ddu(jv) f1 ) +i(dv ddu(v) f2 − djv ddu(jv) f2 , where we have written f = f1 + if2 . Now, one can easily see that at z0 ∂M ◦ ∂ Σ f (v, du(v)) = −∂M ◦ ∂ Σ f (j ◦ v, du(j ◦ v)). This is a contradiction because of the equation (18). This proves that u can not be a perturbed J-holomorphic curve. Remark 4.4 The Theorem 4.3 implies that the antipodal map u : S 2 → S 2 where S 2 is equipped with the standard complex structure is not a perturbed J-holomorphic curve.

5

Hermitian Structures

Here we will review basic properties of Hermitian structures on almostcomplex manifolds and apply them in our context. Apart from other sections of these notes we will consider almost-complex manifold (M, J) with a fixed almost-complex structure J. Let E → M be a complex vector bundle over M . Then the almostcomplex structure J induces the following decomposition M Ωk (M, E) = Ωp,q (M, E), p+q=k

11

where Ωk (M, E) denotes the space Ωk (M )⊗C E of smooth E-valued k-forms and Ωp,q (M, E) denotes its subset of all k-form complex linear with respect to p arguments and complex anti-linear with respect q arguments. Let ∇ : C ∞ (M, E) → Ω1 (M, E) be a covariant derivative on the vector bundle E. We can decompose ∇ as ∇ = ∂∇ + ∂ ∇ ,

(19)

into complex linear part and complex anti-linear part, respectively. The complex linear part ∂∇ : C ∞ (M, E) → Ω1,0 (M, E) is given by the formula 1 (20) ∂∇ = (∇ − i ∇ ◦ J) 2 and the complex anti-linear part ∂ ∇ : C ∞ (M, E) → Ω0,1 (M, E) is given by the formula 1 (21) ∂ ∇ = (∇ + i ∇ ◦ J) . 2 Definition 5.1 An operator D ′′ : C ∞ (M, E) → Ω0,1 (M, E) is said to be a Cauchy-Riemann operator if D ′′ (f s) = ∂ f ⊗ s + f D ′′ s,

(22)

for every f ∈ C ∞ (M, C) and any s ∈ C ∞ (M, E). Lemma 5.2 The complex anti-linear part ∂ ∇ of the covariant derivative ∇ is a Cauchy-Riemann operator on E. Definition 5.3 A Hermitian metric h in E is a smooth family of Hermitian inner products in the fibers of the vector bundle E. As a example, let (M, J) be a symplectic manifold and let J be ωcompatible almost-complex structure on M . Then the tensor h·, ·i given by the formula hu, vi = ω(Ju, v) + i ω(u, v) (23) defines a Hermitian metric on the tangent bundle T M with respect to the almost-complex structure J. Compare this with the formula (6). Any Hermitian metric h on E determines the Hermitian connection ∇h = ∇ on E. It is a unique connection ∇ : C ∞ (M, E) → Ω1 (M, E) 12

which preserves both the complex structure of E and the metric Re h ( the real part of h) induced by the Hermitian structure h. Alternatively, the Hermitian connection is a unique connection ∇ such that d h(u, v) = h(∇ u, v) + h(u, ∇ v). (24) Here is the basic fact about Hermitian connections: Proposition 5.4 For every Cauchy-Riemann operator D ′′ : C ∞ (M, E) → Ω0,1 (M, E) there exists a unique Hermitian connection ∇ such that its complex anti-linear part 1 ∂ h = (∇ + i ∇ ◦ J) 2 ′′ is equal to D . A Hermitian metric h on a complex bundle E induces the Hermitian metric h∗ on the complex dual bundle E ∗ . Namely, If e = (ei ) is a unitary frame for E, e∗ = (e∗i ) the dual frame for E ∗ , then set h∗ (e∗i , e∗j ) = δij . If we identify E with E ∗ ( in complex anti-linear way) via formula s 7→ s∗ = h(·, s)

(25)

then the formula (24) implies that the Hermitian connection ∇∗ on the dual bundle E ∗ is uniquely determined by the requirement: d ht, si = h∇∗ t, si + ht, ∇ si

(26)

for t ∈ C ∞ (M, E ∗ ) and s ∈ C ∞ (M, E). This is so because under the identification (25) the connection ∇ corresponds to a connection which is both preserving the metric and the complex structure on E ∗ and thus it corresponds to ∇∗ Theorem 5.5 Let (M, J) be an almost-complex manifold and let h be a Hermitian metric on the tangent bundle E = T M . Then the complex antilinear part ∂ h∗ : Ω1,0 (M ) → Ω0,1 Ω1,0 (M ) ∼ = Ω0,1 (M ) ⊗ Ω1,0 (M ) of the dual Hermitian connection ∇∗ is given by the following formula

1 ιX (∂ η) = − (ιX (d η) + i ιJ X (d η)) 2 for any vector field X. 13

(27)

Note that we have identified the tangent space T M with bundle Ω1,0 (M ) of complex linear forms via formula (25) Proof: The formula (26) implies that for any complex linear form η ∈ Ω1,0 (M ) and for any vector field X we have d(ιX η) = ιX (∇∗ η) + η(∇X).

(28)

Let us fixed an arbitrary point m ∈ M and let Xm ∈ Tm (M ) and Ym ∈ Tm (M ) be any two tangent vectors at m. Choose a smooth map σ : (−ǫ, ǫ)3 → M with σ(0, 0, 0) = m satisfying the following three conditions: 1. d σ(0,0,0) ([1, 0, 0]) = Xm , d σ(0,0,0) ([0, 1, 0]) = J Xm , 2. d σ(0,0,0) ([0, 0, 1]) = Ym , 3. η(d σ(s,t,0) ([0, 0, 1])) = const. Let X1 , X2 , and Y denote vector fields d σ([1, 0, 0]), d σ([0, 1, 0]), and d σ([0, 0, 1]), respectively. With this notation the condition 3. above implies that d(ιY η)(Xm ) = d(ιY η)(J Xm ) = 0. Therefore combining general identities for exterior derivative ιXi (d η)(Y ) = d η(Xi , Y ) = d(ιY η)(Xi ) − d(ιXi η)(Y ) − η([Xi , Y ]) with the identity (28) and noting that commutators [Xi , Y ] are trivial we obtain ιXi (d η)(Y ) = −ιXi (∇∗ η)(Y ) − η(∇ Xi )(Y ) for any tangent vector Y at m. Thus at the point m we have ιXm (d η) = −ιXm (∇∗ η) − η(∇ Xm )

(29)

ιJ Xm (d η) = −ιJ Xm (∇∗ η) − η(∇ (J Xm )).

(30)

and Now notice that the expression η(∇ X) is complex linear with respect to the variable X since the metric ∇ is Hermitian and the 1-form η is chosen to be complex linear. Therefore combining identities (29) and (30), at the point m ∈ M we obtain the equation 1 ιXm (∂ η) = − (iXm (d η) + i ιJ Xm (d η)) . 2 Since this equality is true for any point m ∈ M and any tangent vector Xm ∈ Tm M the theorem follows. 14

Proposition 5.6 Let (M, ω) be a symplectic manifold of the dimension 2n with a compatible almost-complex structure J. Denote by h′ a Hermitian metric on the tangle bundle T M by the equation (23) and by Φ : T M ∼ = Ω1,0 (M ) the isomorphism given by the equation (6). Let f : M → C be a complex function of the form f = zζ for some complex number z and a real valued function ζ such that ∂ (f )(m) 6= 0 for m ∈ M . Then there is an open set U with m ∈ U and a Hermitian metric h on the set U with the properties: • h(X1 , X2 ) = h′ (X1 , X2 ) for any two vector fields Xi , i = 1, 2 with h′ (Xi , Φ−1 ◦ ∂ (f )) ∼ = 0 on U . Moreover, h′ (X, Φ−1 ◦ ∂ (f )) = 0 if and −1 only if h(X, Φ ◦ ∂ (f )) = 0 for any vector field X on U . • The map U

→ R

u 7→ h(Φ−1 ◦ ∂ (f ), Φ−1 ◦ ∂ (f )) = h(Φ−1 ◦ ∂ (f )(m), Φ−1 ◦ ∂ (f )(m)) is constant. • ∇(Φ−1 ◦ ∂ (f )) ∈ T M ⊗J Ω1,0 (U ), where ∇ denotes the Hermitian connection of the metric h. Proof: We first note that the first two conditions determine the metric h uniquely. Thus we only need to show that ∇(Φ−1 ◦ ∂ (f )) is complex linear. Without loss of generality we may assume that f is a purely imaginary function: f = iζ. Next, we note that h′ ( · , Φ−1 ◦ ∂(f )) = ∂ζ. Let η := h( · , Φ−1 ◦ ∂(f )). By the construction the forms ∂ζ and η differ only by a factor of real function and thus have the same zero sets. Using the Theorem 5.5 it is enough to show that d (η) = 0. Define function g : M → R 0 if x ∈ ζ −1 (m) g(x) = t where t is a time needed to travel from the level ζ −1 (m) to x along the flow of Φ−1 ◦ ∂ (f ). 15

By construction of the form η if η(Y ) = 0 then the vector Y is tangent to a level set of the function g. Now since g is a real function we have I ∂g = 0 γ

over any closed (contractible) loop γ. This implies that the function ρ : M →C Z x ∂g ρ(x) = m

is well defined in a (contractible) neighborhood of m. Moreover simple computation shows that dρ = η. This finishes the proof of the Proposition since we have now dη = d dρ = 0. For vector field whose connection is of type (1, 0) we have the following: Lemma 5.7 Let (M, J) be an almost-complex manifold of dimension 2n and let h be a Hermitian metric defined on the tangle bundle T M . Let X be a vector field on M . Assume that ∇(X) ∈ T M ⊗ Ω1,0 (M ). Then for any point m ∈ M and any tangent vector Ym ∈ Tm (M ) the torsion T (X, Ym ) is trivial. Proof: Choose a vector field Y which agree with the tangent vector Yp at the point m such that commutators [X, Y ] and [JX, Y ] are trivial. Using the fact that the Torsion tensor T (X, Y ) is complex anti-linear with respect to the two variables X and Y (see the Section 7) we have JT (X, Ym ) = −T (JX, Y ) = −∇JX (Y ) + ∇Y (JX) = −J(∇X (Y ) − ∇Y (X)) = −JT (X, Ym ). This shows that T (X, Ym ) = 0.

16

6

Compactness

Let us consider the following weak version of the Gromov’s Compactness Theorem [G] Theorem 6.1 Let (M, ω) be a compact symplectic manifold and let Jk be a sequence of ω-tame almost complex structures which converge to J∞ in C ∞ -topology. Then for any sequence uk : Σ → M of Jk -holomorphic curves with uniformly bounded energy there are subsequence (still denoted by uk ), a finite collection (u1 , ..., um ) of J∞ -holomorphic spheres ui : S 2 → M , and a J∞ -holomorphic curve u∞ : Σ → M such that for corresponding homology classes in H2 (M ) we have [uk ] = [u∞ ] + [u1 ] + ... + [um ] for all k large enough. We want to prove a similar theorem for perturbed J-holomorphic curves satisfying the equation (11). Let us describe an extension of Gromov’s nice trick to perturbed J-holomorphic curves. Consider a solution u : Σ → M of the differential equation (11). To such u we associate a map u e : Σ → Σ×M given by the formula u e(z) = (z, u(z)), for z = (s, t) ∈ Σ. Then u e satisfies the nonlinear Cauchy-Riemann equation u) = de u + Je ◦ de u ◦ j = 0, ∂ Je(e

(31)

where Je is almost-complex structure on the product Σ × M given by the formula j 0 e J= . −P j J To check that u e satisfies the equation (31) note that 0 u) = ∂ Je(e . ∂ J (u) + P (u)

Choose a symplectic form ω0 on Σ such that the complex structure j on Σ is compatible with ω0 and such that ω0 ([Σ]) = 1. Define a symplectic structure ω e on Σ × M by the formula where N is a positive number.

ω e = N ω0 + ω, 17

Lemma 6.2 If N is large enough, then the almost-complex structure Je is ω e -tame. Proof: Define

sup

f = ||ω||L∞

||P (z, m)||

(z,m)∈Σ×M

and compute a a ω e Je , = N ω0 (j(a), a) + ω(P (a) + Jv, v) v v ≥ N |a|2 − |a|f |v|Je + |v|2Je 1

1

= ((N ) 2 |a| − |v|)2 + (2(N ) 2 − f )|a||v| 1

≥ (2(N ) 2 − f )|a||v|. Here we have used the notation |v|2 = |v|2Je = sup ω(J(z)v, v). z

To finish the proof of the lemma it is enough to choose N >

2 f 2

.

So we choose N large enough so that the almost-complex structure Je is ω e -tame. In this situation there is a Riemannian metric defined on the product Σ × M via the formula 1 e ω e (J v, w) + ω e (Jew, v) , hv, wiJe = 2

for tangent vectors v, w. The energy of u e is defined as

1 E(e u) = 2

Z Z

|De u|2Je .

In fact the energy depends only on a homology class [e u] of u e and we have ω e (e u) = E(e u) ≥ 0.

Theorem 6.3 Let (M, ω) be a compact symplectic manifold and let Jν be a sequence in J of ω-compatible families of almost complex structures which converge to J∞ in C 1 -topology. Let Pν be a sequence of perturbations of the form Pν ∈ C ∞ (Σ × M, Ω0,1 Σ ⊗J ν T M ) 18

which converge to P∞ in C 1 -topology. Then for any sequence uν : Σ → M of solutions of the equation (11) with P = Pν , J = Jν such that [uν ] = A ∈ H2 (M ) there is subsequence (still denoted by uν ) and a finite collection (u∞ ; C1 ..., Cm ), where u∞ is a of solution of the equation (11) with P = P∞ and J = J∞ and Ci are J∞ (zi )-holomorphic spheres Ci : S 2 → M such that for corresponding homology classes in H2 (M ) we have A = [u∞ ] + [C1 ] + ... + [Cm ].

(32)

Proof: Consider corresponding elements u eν , Jeν , and Je∞ . Since Jν → J∞ , Pν → P∞ , and all uν represent the same homology class A the energy of the sequence u eν is uniformly bounded for almost all ν and we may apply the Gromov’s theorem 6.1. Thus there is a collection (e ui : S 2 → Σ × M ), i = 1, ..., m, of Je∞ -holomorphic spheres Je∞ -holomorphic curve u e∞ : Σ → Σ × M such that m X [e ui ] A + [Σ] = [e uν ] = [e u∞ ] + i=1

in H2 (Σ) ⊕ H2 (M ) ⊆ H2 (Σ × M ). For every i, we can write a unique decomposition [e ui ] = Ai + B i , and also a unique decomposition [e u∞ ] = A∞ + B ∞ , where Ai , A∞ ∈ H2 (M ) and B i , B ∞ ∈ H2 (S 2 ). In particular, we have ∞

A

+

m X

i

A = A,

B

∞

+

m X

B i = [Σ].

i=0

i=0

Now, the class [Σ] is indecomposable in H2 (Σ). Therefore B ∞ = [Σ] and B i = 0. Thus the map pr1 ◦ u e∞ : Σ → Σ, where pr1 : Σ × M → Σ is the natural projection on the first factor, is a holomorphic of degree one. Eventually reparametrizing the map u e∞ we may assume that the map pr1 ◦ u e∞ is the identity map on Σ. With this reparametrization define u∞ = pr2 ◦ u e∞ ,

where pr2 : Σ × M → M is the natural projection on the second factor. It is easy to verify now, that u∞ such defined satisfies the equation (11) with P = P∞ and J = J∞ . 19

Consider now maps u ei , for i = 1, ..., m. The projections pr1 ◦ u ei are 2 holomorphic maps from S to Σ of degree zero. Therefore they must be constant: pr1 ◦ u ei (S 2 ) = zi ∈ Σ.

This easily implies that pr2 ◦ u ei : S 2 → M is J(zi )-holomorphic sphere. Define Ci = pr2 ◦ u ei : S 2 → M.

This finishes the proof of the theorem since the equation (32) is obvious now. Remark 6.4 If Σ is a Riemann sphere S 2 then the above theorem is true if we replace homology classes in H2 (M ) by homotopy classes in π2 (M ). Remark 6.5 Note that we have the following identity:

7

Linearization

ω(Ci ) = ω e (e ui ).

We will examine the moduli space M(A, J, P ) of all solutions of the equation (11) which represent a given homology class A ∈ H 2 (M ). In order to do it we need study the linearization of the perturbed Cauchy-Riemann operator. In general, there is no unique way to construct such linearization since we have to define a way to identify, for each fixed z ∈ Σ, the fiber Lp Ω0,1 (z)⊗J(z) Tu(z) M with that of the form

Lp Ω0,1 (z)⊗J(z) Tm M

for any m close to u(z). Perhaps, it will be the best for us if we choose such identification based on the family of Hermitian connections ∇(z) relative to the family of almost complex structures J(z). Recall that a Hermitian connection is a connection that preserves a Hermitian metric with respect to the J(z). In contrast to

20

the Levi-Civita connection its torsion tensor T is, in general, nontrivial and, moreover, it is complex anti-linear in two variables, i.e. T (Jξ, η) = T (ξ, Jη) = −JT (ξ, η). To describe the linearization based on Hermitian connections let us choose u ∈ M(A, J, P ) and z ∈ Σ. Since the hermitian connection ∇(z) preserves the almost-complex structure J(z) the map Φub (z) : Lp ((expu (b u))∗ (Ω0,1 ⊗J T M )) → Lp (Ω0,1 ⊗J u∗ T M )

induced by the parallel transport along the geodesic curve t → exp(tb u) corresponding to the Hermitian metric at z on M is well-defined. Thus in the neighborhood of u the perturbed Cauchy-Riemann ∂ J,P is represented by the map F : W 1,p (u∗ T M ) → Lp (Ω0,1 ⊗J u∗ T M ) defined by The linearization at u

u))). F(b u) = Φub (z)(∂ J,P (exp(b

Du : W 1,p (u∗ T M ) → Lp (Ω0,1 ⊗J u∗ T M ) is defined as Du (b u(z)) = dF(0)(b u(z)). It is not hard to compute Du (See [MS] for J-holomorphic curves). Proposition 7.1 If u : Σ → M satisfies the equation (11) with the exact b ∈ C ∞ (u∗ T M ) then the operator Du (b u) perturbation term P = ∂ Σ X and u can be written as b = ∇∗ (b u) + T (du, u b) + ∇ub (dΣ X) Du u

+J ◦ (∇∗ (b u) + T (du, u b) + ∇ub (dΣ X)) ◦ j,

where ∇∗ denotes the induced connection on u∗ (T M ) via the map (id, u). Lemma 7.2 For u ∈ M(A, J, P ) the operator Du : W 1,2 (u) → L2 (u) is elliptic. Here W 1,2 (u) and L2 (u) denote spaces W 1,2 (u∗ T M ) and W 0,2 (Ω0,1 ⊗J u∗ T M ), respectively. Thus the operator Du is Fredholm. Proof: Since the vector bundles W 1,2 (u∗ T M ) and W 0,2 (Ω0,1 ⊗J u∗ T M ) are defined over the compact manifold Σ it is enough to notice that main symbol of the operator F (u) is elliptic. From general theory of elliptic operators defined on vectors bundles over a compact manifold follows that F (u) is also Fredholm. 21

Remark 7.3 By the elliptic regularity it follows that the map Du : W 1,p (u∗ T M ) → Lp (Ω0,1 ⊗J u∗ T M ) is also Fredholm. Here is the basic fact about properly exact perturbations. Theorem 7.4 Let the perturbation term P be properly exact, P = ∂ Σ X. Then there exists a family of Hermitian connections such that Du ((id, u)∗ X) is closed to (id, u)∗ (∂ Σ X). so (id, u)∗ (∂ Σ X) is in the range of Du . Proof: If X(z, u(z)) = 0 then choose a Hermitian connection ∇(z) corresponding to the Hermitian metric determined by (J(z), ω). If ||X(z, u(z))|| > ǫ then choose the perturbed Hermitian connections (in the neighborhood of u(z)) given by the Proposition 5.6. For the latter connections the Proposition 5.6 and the Lemma 5.7 imply that J∇Y (X) = ∇JY (X). Therefore for these connections we have Du ((id, u)∗ X) = (id, u)∗ (∂ Σ X) + ∇du+J◦du◦j+P (u) (X). The last expression is zero. Now make ǫ small enough and choose Hermitian connections for z satisfying 0 < ||X(z, u(z))|| < ǫ so that we get a smooth bounded (by a constant independent of the choice of ǫ) family of Hermitian connections satisfying the conclusion of the Theorem.

8

Compactness Properties of the space V

We will study the space of smooth functions (real valued, for simplicity, but all applies to the space of function of the form zf where z is a fixed complex number and f is arbitrary real valued function) defined over the set Σ × M . Choose a decreasing sequence ǫk > 0 and consider the subspace Cǫ∞ (Σ × M ) of smooth functions f ∈ C ∞ (Σ × M ) which satisfy ||f ||2ǫ =

∞ X k=0

D E ǫk ∇k f, ∇k f < ∞, 22

where ∇k denotes k-th hermitian covariant derivative determined by the metric on the manifold Σ × M . This defines a separable Hilbert space of the subspace of smooth functions defined on Σ × M and induces topology on the space C ∞ (Σ × M ). For a given sequence ǫk we will call this topology the ǫ-topology, and the space C ∞ (Σ × M ) with the ǫ-topology will be denoted by V . Following Floer [F1] one can choose a sequence ǫk such that the space V is a dense subset of Lp (Σ × M ), for p > 2. Proposition 8.1 For every positive number K the ǫ-open set V (K) = {X ∈ V | ||f ||ǫ < K} is relatively compact in the C ∞ -topology. Moreover, if fk → f∞ in C ∞ topology and ||fk ||ǫ ≤ K for all k, then ||f∞ ||ǫ ≤ K. Proof: To show that the set V (K) is relatively compact in the C ∞ -topology we will use the method of diagonal subsequence. For each natural n let us introduce a norm ||.||ǫ,n by the formula ||f ||2ǫ,n =

n X k=0

D E ǫk ∇k f, ∇k f .

Each norm ||.||ǫ,n is equivalent to the corresponding Sobolev norm. Since all functions have support in the compact set Σ×M then the natural embedding W (ǫ, n) → W (ǫ, m) is compact if n > m. Here W (ǫ, n) denotes the completion in the norm ||.||ǫ,n . Let fk be a sequence of smooth functions such that ||fn ||ǫ < K for every natural number k. Then it is bounded in the ||.||ǫ,2 -norm and by the above remark there exists a subsequence fk1 convergent in the ||.||ǫ,1 -norm. Next, the sequence fk1 is bounded in ||.||ǫ,3 -norm so we can choose a subsequence fk2 of the sequence fk1 convergent in the ||.||ǫ,2 -norm. Continuing this process we will obtain for each l > 1 a subsequence fkl of the sequence fkl−1 which is convergent in the ||.||ǫ,l -norm. Choose the diagonal subsequence fkk . It is convergent in the ||.||ǫ,l -norm for every l. Therefore it is convergent in the C ∞ -topology. This proves relative compactness in the C ∞ -topology. To show that if fk → f∞ in C ∞ -topology and ||fk ||ǫ ≤ K for all k, then ||f∞ ||ǫ ≤ K, it is enough to notice that ||f ||2ǫ = lim ||f ||2ǫ,n n→∞

for every f . This proves the proposition. 23

9

Universal Moduli Space

We will study the universal moduli space M(A, J) = {(u, f ) ∈ X 1,p × V | ∂ J,f (u) = 0}, where u is solution of the equation (11) corresponding to homology class A ∈ H2 (M ) and to the family of almost complex structures J. Here V denotes the space of functions f as described in the Proposition 8.1. Consider the infinite dimensional vector bundle E → X 1,p × V whose fiber at the point (u, f ) is the space E(u, f ) = Lp (Ω0,1 ⊗J u∗ T M ) of Lp -sections of the vector bundle Ω0,1 ⊗J u∗ (T M ) over Σ. Then the moduli space M(A, J) is a zero set F −1 (0) of the section of the vector bundle given by the formula F(u, f ) = ∂ J,f (u). If the point (u, f ) is zero of the section F then the differential at this point , D(F)(u, X) : W 1,p (u∗ T M ) × V → E(u, f ), is given by the formula D(F)(u, f )(b u, fb) = Du u b + (id, u)∗ ∂ Σ (Φ−1 ◦ ∂M fb) ,

(33)

since ∂ Σ is linear. Let L denote the codimension one subspace of the space V orthogonal to the vector f and define the operator DL (F)(u, f ) : W 1,p (u∗ T M ) × L → E(u, f ) as the restriction of the operator D(F)(u, f ) to the subspace W 1,p (u∗ T M )× L. Proposition 9.1 If the point (u, f ) is zero of the section F and u is a map with u∗ (ω) 6= 0 then the linear operator DL (F)(u, f ) is onto for suitable chosen family of Hermitian connections. Proof: We claim that the operators D(F)(u, f ) and DL (F)(u, f ) have the same range. Indeed, let ξ = Du (b u) + ∂ Σ (Φ−1 ◦ ∂M fb). Write fb = tf + g, where g ∈ L. By the Theorem 7.4 we have Du ((id, u)∗ Φ−1 ◦ ∂M f ) = (id, u)∗ (∂ Σ Φ−1 ◦ ∂M f ). 24

Therefore, ξ = Du (b u) + (id, u)∗ (∂ Σ (tΦ−1 ◦ ∂M f ) + (id, u)∗ (∂ Σ (Φ−1 ◦ ∂M g)) = Du (b u + (id, u)∗ (tΦ−1 ◦ ∂M f )) + (id, u)∗ (∂ Σ (Φ−1 ◦ ∂M g)) = DL (F)(u, f )(b u + (id, u)∗ (tΦ−1 ◦ ∂M f ), g). Thus it is enough to show that the range of D(F)(u, f ) is equal to Lp (Ω0,1 ⊗J u∗ T M ). Since Du is a Fredholm operator, by the Remark 7.3 the operator D(F) has a closed range and it is enough to prove that the range is dense. Using the Hahn-Banach Theorem it is enough to show that if η ∈ Lq (Ω0,1 ⊗J u∗ T M ) with p1 + 1q = 1 satisfies Z hη, Du u bi = 0

and

Z D E η, ∂ Σ (Φ−1 ◦ ∂M fb) = 0

(34)

for every u b ∈ W 1,p (u∗ T M ) and every fb) ∈ V, then η ≡ 0. From the first equation we obtain that η is a week solution of Du∗ η = 0. However, the coefficients of the first order terms of Du are of class C ∞ and the same is true for the adjoint Du∗ . Thus by elliptic regularity η satisfies the equation Du∗ η = 0 in the strong sense and, moreover, η is of class C ∞ . Hence we can write Du Du∗ η = ∆η + lower order terms = 0 and using the Aronszajn’s theorem [Ar] it is enough to show that η vanishes at some open set. By the assumption there is an open set U ⊂ Σ such that the map u restricted to the set U is an embedding. Choose z0 ∈ U such that η(z0 ) 6= 0. Let Y be a vector field on M with support in a small neighborhood of u(z0 ). Choose polar coordinates z = r exp (2πiθ) on Σ with the property that if Y (r exp (2πθ)) 6= 0 then r1 < r < r2 for some positive r1 and r2 . Next, R ∞ choose a function g : (0, ∞) → R with a compact support such that 0 g(r)dr = 0, Rand g(r) > 0 for r1 < r < r2 . r Let f (r) = 0 g(s)ds. After complexification it means ∂ Σ (f ) = gdz and ∂ Σ (Y f ) = gdz ⊗J Y . We may assume that if gdz ⊗J Y (z, u(z)) 6= 0 then hη, gdz ⊗J Y i (z) > 0. Thus if (34) holds η must be zero. Now we are ready to prove the following theorem 25

Theorem 9.2 Let (u, f ) with ||f ||ǫ = K satisfies F(u, X) = 0 and du is of maximum rank at some point. Then there is a pair (u1 , f1 ) with ||f1 ||ǫ < K such that F(u1 , f1 ) = 0. Proof: We will need the following version of the implicit function theorem for Banach spaces. Theorem 9.3 Let f : E1 × E2 → F be a smooth map between Banach spaces. Assume that the partial derivative D1 f is surjective at the point (e1 , e2 ) ∈ E1 × E2 and admits a bounded right inverse. Then for every f2 ∈ E2 near e2 there exists f1 ∈ E1 such that f (e1 , e2 ) = f (f1 , f2 ). Let L denote a tangent space at X to the sphere S(K) of radius K in the Hilbert space V . Then by the Proposition 9.1 the partial derivative of F at (u, f ) in the direction W 1,p (u∗ T M ) × L is onto. It has also a bounded right inverse since its restriction to the space W 1,p (u∗ T M ) is Fredholm by the Remark 7.3. Thus we apply the Implicit Function Theorem 9.3 to obtain a pair (u1 , f1 ) with ||f1 ||ǫ < K such that F(u1 , f1 ) = 0. Definition 9.4 The extended universal moduli space is the space Mex (A, J) ⊂ W 1,p (u∗ T M ) × V × R of all triples (u, f, λ) with f ∈ V such that (u, f ) is a solution of the equation F(u, f ) = 0 such that homology class [u] is equal to A and ||f ||ǫ = λ. The Theorem 9.2 implies Proposition 9.5 Let Λ(A, J) be a set defined as the image of the projection π : Mex (A, J) → R on the third factor. Then Λ(A, J) is open in the set of all real numbers R.

10

Nonnegative Properties of Symplectic Area

In this section we will prove the main theorem of these notes. Consider a solution u of the nonlinear partial differential equation du + J ◦ du ◦ j + ∂ Σ (X)(u) = 0, with properly exact perturbation term. 26

(35)

Theorem 10.1 Let u be a solution of the equation (35) for some J ∈ J . Then the symplectic form ω evaluated on the class [u] is nonnegative: Z u∗ ω ≥ 0. Σ

Proof: We will prove the theorem by arriving at a contradiction. Thus assume that Z u∗ ω < 0. Σ

Without loss of generality we way assume that ||f ||ǫ = 1. Let us choose a constant h such that ω(C) > h for any J(z)-holomorphic sphere C in M Consider the set Λ0 = Λ(A, J) ∩ [0, 1]. where A is a homology class of u, A = [u]. (See notation of the Proposition 9.5). Then the set Λ0 is not empty since 1 ∈ Λ0 . By the Proposition 9.5 it is also open in [0, 1]. Let λ1 = inf Λ0 . By the definition of λ1 there exists a sequence (un , fn , λn ) of elements of the extended universal moduli space Mex (A, J) (Definition 9.4) such that λn → λ1 . By the compactness properties (Proposition 8.1) we can assume that fn → f 1 , where f 1 satisfies ||f 1||ǫ ≤ λ1 . By the compactness Theorem 6.3 there exists a subsequence of (un , fn , λn , ) 1 ) where u1 is a solution of the equation and the collection (u1 ; C11 , ..., Cr(1) 1 (35) with ||f 1 ||ǫ ≤ λ1 and C11 , ..., Cr(1) are J(zi )-holomorphic spheres. We have 1 [u] = [u1 ] + [C11 ] + ... + [Cr(1) ] so 1 ω e ([e u]) = ω e ([e u1 ]) + ω([C11 ]) + ... + ω([Cr(1) ]).

See the Remark 6.5. We have

ω([u1 ]) ≤ ω([u]) < 0. For the energy of u e1 , ω e ([e u1 ]), we obtain the following equality: ω e ([e u1 ]) = ω e (A) − ω([C11 ]) − ... − ω([Cr1 ]). 27

We claim that λ1 > 0 since otherwise u1 there would be a holomorphic curve with negative symplectic area which is impossible. Moreover we claim that r(1) ,the number of J(zi )-holomorphic spheres, is strictly positive since otherwise because of the inequality ||f 1 ||ǫ ≤ λ1 the Theorem 9.2 would imply that there was an element of the universal moduli space Mex (A, J) corresponding to the parameter λ1 . But this would contradict the fact that Λ0 is an open set. Assume that for a natural number k ≥ 1 we have the following data which we will call k-data: 1. For each 1 ≤ i ≤ k there exists a triple (ui , f i , λi ) ∈ Mex (Ai , J). 2. For each 1 < i ≤ k numbers λi > 0 satisfies (a) λi−1 > λi , (b) λi = inf Λi−1 , where Λi−1 = Λ(Ai−1 , J) ∩ [0, 1], 3. For each 1 < i ≤ k there exists a natural number r(i) > 0 and a i ) of non constant J(z)-holomorphic spheres sequence (C1i , C2i , ..., Cr(i) such that: (a) i ω(Ai−1 ) = ω(Ai ) + ω([C1i ]) + ... + ω([Cr(i) ]),

(b) i ω e (e ui ) = ω e (Ai−1 ) − ω([C1i ]) − ... − ω([Cr(i) ]).

28

If λk > 0 then, using the above method, we can produce the (k + 1)-data. Assume that this process never stops i.e. for any natural number n ∈ N there is the n-data. Then, by the definition of the constant h and by the condition (3) of n-data, we obtain ω e (e un ) ≤ ω e (A) − ω(C11 ) − ... − ω(C1n ) < ω e (A) − nh.

Choose

ω e (A) . h Then we obtain that ω e (e un ) < 0. But this is a contradiction since the energy of a holomorphic curve can never be negative. Therefore, the process must stop somewhere i.e. λm = 0 n>

for some m. However this can not happen by the same reason as λR1 could never be zero and we have arrived at a contradiction. Therefore, Σ u∗ ω ≥ 0, and this completes the proof of the theorem.

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[Ar]

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[CZ]

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[F3]

A. Floer, Morse theory for Lagrangian intersections, J. Differential Geom. 28 (1988), 513-611. 29

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A. Floer, Symplectic fixed points and holomorphic spheres, Comm. Math. Phys. 120 (1989), 575-611.

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[H]

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[HZ]

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[M]

D. McDuff, Elliptic methods in symplectic geometry, Bulletin AMS 23(2) (1990), 311-358.

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Y.-G. Oh, Floer cohomology of Lagrangian intersections and pseudoholomorphic disks I, Comm. Pure Appl. Math. 46 (1993), 949-994.

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Y.-G. Oh, Floer cohomology of Lagrangian intersections and pseudoholomorphic disks II: (CP n , RP n ) Comm. Pure Appl. Math. 46 (1993), 995-1012.

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Y.-G. Oh, Floer cohomology of Lagrangian intersections and pseudoholomorphic disks III: Arnold-Givental Conjecture, The Floer Memorial Volume, edited by H. Hofer, C. Taubes, A. Weinstein, and E. Zehnder, Birkh¨auser (1995), 555-573.

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30