Functors and Computations in Floer homology with Applications Part II

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Functors and Computations in Floer homology with Applications Part II

C Viterbo

To cite this version:

C Viterbo. Functors and Computations in Floer homology with Applications Part II. 1996. �hal-

01780167�

Functors and Computations in Floer homology with Applications

Part II

C. VITERBO ^∗ October 15, 1996,

(revised on April 25, 2018)

Abstract

The results in this paper concern computations of Floer cohomol- ogy using generating functions. The first part proves the isomorphism between Floer cohomology and Generating function cohomology in- troduced by Lisa Traynor. The second part proves that the Floer cohomology of the cotangent bundle (in the sense of Part I), is iso- morphic to the cohomology of the loop space of the base. This has many consequences, some of which were given in Part I, others will be given in forthcoming papers. The results in this paper had been announced (with indications of proof) in a talk at the ICM 94 in Zrich.

Contents

∗

D´epartement de Mah´ematiques, Bˆatiment 425, Universit´e de Paris-Sud, 91405 Orsay Cedex, FRANCE. Supported by C.N.R.S. U.R.A 1169 and Institut Universitaire de France.

Current address: DMA, ´ Ecole normale sup´erieure, PSL University, 45 rue d’Ulm, 75005

Paris, FRANCE.

1 Introduction

This paper is concerned with computations of Floer cohomology using gen- erating functions. The first part proves the isomorphism between Floer co- homology and Generating function cohomology introduced by Lisa Traynor in [?]. The statement of this theorem was given in [?] with rather precise indications of proof. However, since then, we found what we consider a sim- pler, even though less natural, proof. A proof along the original indications is given by Milinkovi´c and Oh in [?], [?]. The second part proves that the Floer cohomology of the cotangent bundle (in the sense of Part I), is iso- morphic to the cohomology of the loop space of the base. This has many consequences, some of which were given in Part I, others will be given in forthcoming papers.

We would like to point out a very interesting attempt by Joachim Weber to prove the main theorem of section ?? using a different approach, namely by considering the gradient flow of the geodesic energy as a singular perturbation of the Floer flow (see [?]).

2 Floer cohomology is isomorphic to GF-homology

Let L be a Lagrange submanifold in T

N , and assume L has a generating function quadratic at infinity, that is

L = {(x, ∂S

∂x ) | ∂S

∂ξ (x, ξ) = 0}

where S is a smooth function on N × R

, such that S(x, ξ) coincides outside a compact set with a nondegenerate quadratic form in the fibers, Q(ξ). In particular if N is non-compact, L coincides with the zero section outside a compact set.

As proved by Laudenbach and Sikorav (see [?]) this is the case for L = φ

(O

), where φ

is a Hamiltonian flow.

Moreover S is ”essentially unique” up to addition of a quadratic form in new variables and conjugation by a fiber preserving diffeomorphism (the

”fibers” are those of the projection N × R

→ N) see [?] and [?].

We may then consider, for L

and L

generated by S

, S

, the Floer

cohomology F H

(L

, L

; a, b) of the chain complex C

(L

, L

; a, b) generated

by the points, x, in L

∩ L

with A(x) ∈ [a, b] where A(γ) is R

[0,1]²

u

ω where u : [0, 1]

→ T

N is a map as in figure ?? such that

u(0, t) = γ

(t) u(s, 0) ∈ L

u(1, t) = γ(t) u(s, 1) ∈ L

where γ

is a fixed path connecting L

to L

, and x is identified with the constant path γ

, at x.

Denoting by C

the subvector space generated by the intersection points of L with the zero section having Conley-Zehnder index k, we have a differ- ential:

δ : C

(L

, L

; a, b) → C

(L

, L

; a, b)

is obtained by counting the number of holomorphic strips, that are solutions of ¯ ∂u =

+ J

= 0 where J is an almost complex structure compatible with the symplectic form

u : R × [0, 1] → T

N u(s, 0) ∈ L

u(s, 1) ∈ L

S→±∞

lim u(s, t) = x

.

On the other hand, we have the much simpler space H

(S

, S

), where S(x, ξ, η) = S

(x, ξ ) − S

(x, η), and S

= {(x, ξ, η) | S(x, ξ, η) ≤ λ}. This cohomology group does not depend on the choice of S according to [?] and [?], up to a shift in index, and was used in [?] as a substitute for Floer cohomology under the name GF cohomology (denoted GF

(L

, L

; a, b)).

Our first claim is Theorem 2.1.

F H

(L

, L

; a, b) ≃ GF

(L

, L

; a, b) The proof will take up the rest of this section.

Remark. : This was announced in [?], together with a sketch of the proof. Our present proof is actually simpler than the one we had in mind, in particular as far as the applications to the next section are concerned.

1

i.e. ω(ξ, J ξ) defines a Riemannian metric

γ

x

L 1

L0

u ( [0,1] ) 2

Figure 1: The map u.

We are first going to introduce a functional interpolating between A and S.

Let H(t, z, ξ) be a smooth function on R × T

N × R

, equal to some non degenerate quadratic form Q(ξ) outside a compact set.

For γ a path between L

and L

, set A

(γ, ξ) = A(γ) − R

0

H(t, γ (t), ξ)dt.

Set P (L

, L

) be the set of paths, γ, such that γ(0) ∈ L

, γ (1) ∈ L

. Then the critical points of A

on P (L

, L

) × R

are the pairs (γ, ξ) with

(

γ −X

(γ) = 0 R

0

∂

∂ξ

H(t, γ (t), ξ)dt = 0

where X

is the Hamiltonian vector field associated to the function z → H(t, z, ξ) (i.e. ξ is ”frozen”).

If we denote by φ

the Hamiltonian flow of X

(for fixed ξ ∈ R

) we have γ(t) = φ

(γ(0)), so that the critical points of A

correspond to

{z ∈ L

∩ (φ

)

(L

)|

Z

∂

∂ξ H(t, φ

(z), ξ)dt = 0}

Note that

L

= {φ

(z)|z ∈ L

, Z

0

∂

∂ξ H(t, φ

(z), ξ)dt = 0}

is an immersed Lagrange submanifold, and the above critical points corre- spond to L

∩ L

.

Consider an almost complex structure on M compatible with the sym-

plectic form, and denote by ∂ =

+ J

.

We now define the Floer cohomology of A

as usual:

- C

will be generated by the critical points of A

on P (L

, L

) - < d(x

, ξ

), (x

, ξ

) > equals the algebraic number of solutions of

 



u : R × [0, 1] ⇔ M

∂u(s, t) = −∇H(t, u(s, t), ξ(s))

d

ds

ξ(s) = − R

∂

∂ξ

H(t, u(s, t), ξ(s))dt satisfying lim

S→±∞

(u(s, t), ξ(s)) = (x±, ξ±).

Note that the set of such solutions has its image in a bounded subset of T

N × R

.

Indeed, for u outside a bounded subset, H vanishes, so the equation becomes

( ∂u = 0

d

ds

ξ(s) = − R

∂

∂ξ

H(t, 0, ξ)dt

But the first equation cannot hold in the region foliated by pseudoconvex hypersurfaces.

If on the other hand |ξ| is large the equation will be ∂u = −∇H(t, u, ξ)

d

ds

ξ(s) = Bξ where Q(ξ) =< Bξ, ξ >.

But the second equation is such that any bounded set is contained in a set with the property that if a trajectory exits the set, it will never reenter it.

Thus, the set of bounded solutions has its image in a bounded set. Then the set of solutions satisfies the same formal properties as the set of solutions of the usual Floer equation. The proofs are just verbatim translations of those in [?].

Definition. The cohomology of (C

(L

, L

, H ), d) will be denoted by F H

(L

, L

, H ).

If we restrict the complex to those solutions with A

(γ, ξ) ∈ [a, b] the coho- mology is denoted F H

(L

, L

, H ; a, b).

A first result is

Lemma 2.2. Let ψ

be a Hamiltonian flow on T

N, and H

, F

be the Hamiltonians with flows ψ

◦ φ

◦ ψ

and φ

◦ (ψ

)

.

Then

F H

(L

, ψ

(L

), H

) = F H

(ψ

(L

), L

, H

) = F H

(L

, L

, F

) Proof. Consider first the chain complex C

(λ) = C

(ψ

(L

), ψ

ψ

(L

), H

) for λ in [0, 1]. It is generated by points in (ψ

(L

))

∩ ψ

ψ

(L

).

But (ψ

(L

))

is defined as {φ

(ψ

(z))|z ∈ L

;

Z

∂

H

(t, φ

(ψ

(z)), ξ)dt = 0}

and since H

(t, u, ξ) = H(t, ψ

(u), ξ) we have H

(t, φ

◦ψ

(z), ξ) = H(t, φ

(z), ξ) and φ

◦ ψ

= ψ

◦ φ

, we have that (ψ

(L

))

= ψ

((L

)

) and ψ

((L

)

) ∩ ψ

(ψ

(L

)) = ψ

((L

)

∩ ψ

(L

))

Thus the generators of C

(λ) do not depend on λ.

Note also that the intersection points of (ψ

(L

))

and ψ

(ψ

(L

)) stay transverse provided they are transverse for λ = 0 (that we shall always assume), and this is sufficient, together with the fact that the value of A

on the critical points does not depend on λ, to imply that the cohomology of C(λ) will not depend on λ.

Consider now the second equality. Note that we may deform H, provided its time one flow is unchanged, and the same for the flow ψ

.

We may thus assume that ∂

H vanishes, (remember that H must be quadratic in ξ, hence it cannot vanish) for t in [0,

[, and K

, vanishes for t in [

, 1].

Then the flow (ψ

)

◦ φ

is generated by H(t, ψ

(z), ξ) + K

(t, z) = F

(t, z, ξ) and therefore

∂

F

(t, z, ξ) = ∂

H(t, ψ

(z), ξ) As a result,

(ψ

(L

))

= {φ

(ψ

(z))|z ∈ L

, Z

0

∂

H

(t, φ

◦ ψ

(z), ξ)dt = 0}

= {ψ

φ

(z)/z ∈ L

; Z

0

∂

H(t, φ

(z), ξ)dt = 0}

= {ψ

φ

(z)/z ∈ L

; Z

0

∂

F

(t, (ψ

)

◦ φ

(z), ξ)dt = 0}

= L

Thus we have

(ψ

(L

))

= ψ

((L

)

) = L

and L

∩ψ

ψ

(L

) is independent of λ, and by the same argument as above, F H

(L

, ψ

ψ

(L

), F

) does not depend on λ. Using again the invariance of the critical levels, we proved the second equality.

In particular for L

= O

, L = φ

(O

) we get

F H

(O

, L, 0) = F H

(O

, O

, H ) Now let S(q, ξ) be a g.f.q.i. for L. The flow of X

is given by

q ˙ = 0

˙

p =

(q, ξ) or

q(t) ≡ q(0)

p(t) = t

S(q, ξ) + p(0).

and (O

)

= {q,

(q, ξ))| R

0

∂

S(q(t), ξ)dt = 0} but since q(t) ≡ q(0), we have that R

0

∂

S(q(t), ξ)dt = ∂

S(q, ξ), hence (O

)

is the Lagrange sub- manifold generated by S.

Now we claim

Lemma 2.3. For S a g.f.q.i we have F H

(O

, O

, S; a, b) ≃ H

(S

, S

).

Proof. Indeed the generators on both sides are critical points of S, and con- necting trajectories solve:

∂u = −∇

S(q(s, t), ξ(s))ds

d

ds

ξ(s) = − R

0

∂

S(q(s, t), ξ(s))dt.

where u(s, t) = (q(s, t), p(s, t))

Now the function f(q, p) = |p| is pluri-subharmonic for our J, and since

∂u is tangent to ker df (note that ∇

S is horizontal), we have that f ◦ u

satisfies the maximum principle. Since p(s, 1) = p(s, 0) = 0, we must have

p ≡ 0, and thus

∂

∂s q(s, t) = −∇

S(q(s, t), ξ(s))

∂

∂t q(s, t) = 0

In other words, q only depends on s, and satisfies q ˙ = −∇

S(q(s), ξ(s))

ξ(s) = ˙ −∂

S(q(s), ξ(s)) and lim

s→±∞

(q(s), ξ(s)) = (q

, ξ

) are critical points for S.

Thus our connecting trajectories, are just bounded gradient trajectories for S, and thus the coboundary map is the same on both complexes, hence the cohomologies are the same.

We may finally conclude our proof. Let L = φ

(0

), so that it has a generating function S.

Now let S

be the generating function of L

= φ

(L), so that S

= 0 and S

= S. We claim that the modules F H

(L

, 0

, S

, a, b) are all isomorphic.

Indeed, L

∩ (0

)

= φ

(0

) ∩ φ

(L) = φ

(0

∩ L) is constant and the usual argument proves the constancy of the Floer cohomology ring. It remains to show that

F H

(φ

(0

), 0

, 0) = F H

(0

, φ

(0

), 0) Again this follows from the equality

F H

(φ

(0

), 0

, 0) = F H

(φ

◦φ

(0

), φ

(0

), 0), because φ

φ

(0

)∩

φ

(0

) = φ

(φ

(0

) ∩ 0

).

We thus proved that

F H

(O

, L, 0; a, b) ≃ F H

(O

, O

, S; a, b) Since

F H

(O

, L, 0; a, b) ≃ F H

(L; a, b) and the above lemma proves states that

F H

(O

, O

, S; a, b) ≃ H

(S

, S

) = GH

(L; a, b)

this concludes our proof.

3 The isomorphism of F H ^∗ ( DT ^∗ N ) and H ^∗ (Λ N )

We will now compute F H

(M ) for M = DT

N = {(q, p) ∈ T

N | |p|

= 1}

for some metric g.

Let us denote by ΛN the free lopp space of N (i.e. C

(S

, N ) ).

We have:

Theorem 3.1.

F H

(DT

N ) ≃ H

(ΛN )

where ΛN is the free loop space of N . The same holds for S

equivariant cohomologies with rational coefficients.

The proof will take up the rest of this section. The next two lemmata are valid in any manifold satisfying the assumptions of section ??. First let us consider the diagonal ∆ in M × M , where M is the manifold M endowed with the symplectic form −ω, and ϕ

is the flow of H on M .

Our first result is Lemma 3.2.

F H

(H; a, b) ≃ F H

(∆, (Id × ϕ

)∆; a, b).

Proof. Remember that F H

(H, a, b) = F H

(0, 0, H, a, b).

We first point out that the cochain spaces associated to both sides are the same, and generated by the fixed points of ϕ

.

The connecting trajectories are, for the left hand side given by

∂v = −∇H(t, v) v : R × S

→ M

while for the right hand side, we first notice that F H

(∆, (id × ϕ

)∆; a, b) ≃ F H

(∆, (Id × ϕ

)∆, 0; a, b) ≃ F H

(∆, ∆, 0 ⊕ H; a, b) (Note: by K = 0 ⊕ H we mean the Hamiltonian on M × M defined by K(t, z

, z

) = H(t, z

)).

The second isomorphism follows from lemma ??.

Now the coboundary map for this cohomology is obtained by counting solutions of

∂u = −∇K

where u = (u

, u

) : R × S

⇔ M × M and ∂u

= 0; ∂u

= −∇H(t, u

).

Moreover, since t → u(s, t) connects ∆ to itself, we have u

(s, 0) = u

(s, 0) and u

(s, 1) = u

(s, 1).

Therefore we may glue together u

and u

to obtain a map e v : S

× R → M such that

∂ e v = −∇ H(t, e e v)

where H(t, u) = 0 for e t in [0, 1]

= H(t, u) for t in [1, 2]

(the circle being identified with R /2 Z ). We shall make the simplifying as- sumption that H(t, z) = 0 for t close to 0 or 1, so that ˜ H is continuous in t.

This is of course not a restriction on the time one map.

But the time 2 map of H e coincides with the time 1 map for H, hence we may continuously deform one equation into the other, and the two cohomolo- gies are isomorphic. (Remember that if we have a family of Hamiltonians depending continuously on some parameter, but having the same time 1 map, then the corresponding Floer cohomologies are alos independnt from the parameter).

Assume ϕ

is equal to ψ

. We denote by Γ

the graph of ϕ (i.e. Γ

= (id×

ϕ)∆), and Γ

= {(z

, ψ(z

), z

, ψ(z

) . . . , z

, ψ(z

), z

, ψ(z

))} in (M ×M )

Set ∆

= ∆ × ×∆ (r times)

Lemma 3.3. . For ϕ = ψ

we have

F H

(∆, Γ

; a, b) ≃ F H

(∆

, Γ

; a, b)

Proof. Clearly, we may identify Γ

∩ ∆ and ∆

∩ Γ

, since a point in this intersection is given by (z

, ψ(z

), . . . z

, ψ(z

)) = (z

, z

, . . . z

, z

) that is z

= ψ(z

); z

= ψ(z

), . . . , z

= ψ(z

) or else

z

= ψ

(z

), z

= ψ

(z

) for i ≥ 2

Note that there is a Z /r action on the set of such points, induced by

z → ψ (z) on Γ

∩ ∆ and by the shift (z

) → (z

) on Γ

∩ ∆

, and these

two actions obviously coincide. We shall not mention this point in the proof,

but all our results hold for Z /r equivariant cohomology with any coefficient

ring, and this eventually allows us to recover the S

equivariant cohomology (with rational coefficients), due to the Lemma in appendix 2 of [?].

Now let us compare the trajectories.

For the second cohomology (u

, v

) : R × [0, 1] → M satisfying

 



∂u

= ∂v

= 0

u

(s, 0) = v

(s, 0) = 0 u

(s, 1) = ψ(v

(s, 1)) If we set w

(s, t) = ψ

(v

(s, t)) so that

 



∂w

= ∇K(s, t) ∂u

= 0 u

(s, 0) = w

(s, 0)

u

(s, 1) = w

(s, 1)

We may now glue together the u

and w

as in figure ?? to get a map u : R × R /2r Z → M

such that

u(s, t) = u

(s, t − 2j ) for 2j ≤ t ≤ 2j + 1 u(s, t) = w

(s, t − 2j − 1) for 2j + 1 ≤ t < 2j + 2 It satisfies ∂u = ∇F (u) where

F (t, u) = K(t − 2j, u) for 2j ≤ t ≤ 2j + 1

= 0 otherwise

Thus the time 2k map of X

is equal to ψ

= ϕ.

We thus identified the Floer trajectories defining F H

(∆, Γ

; a, b) with those defining

F H

(H; a, b) ≃ F H

(∆, Γ

; a, b) This concludes our proof.

Let us now prove theorem ??. Let H be a Hamiltonian equal to H(t, q, p) =

c|p| for |p| large, where c is some constant (that differs from the length of a

closed geodesic, but will eventually become large), and ϕ

its flow.

j-1 u

v (s,1)=u ( s,1) j j-1

j

v j-1

v (s, 0 )=u ( s, 0 )

j-1 j-1

v j

Figure 2: The glueing of the maps u

and v

For some integer r (that will eventually become large) set ψ = ϕ

, so that ψ

= ϕ

= ϕ.

Then according to lemmata ?? and ??, we have that F H

(H; a, b) ≃ F H

(∆

, Γ

; a, b)

Now we may assume that H(t, q, p) = h(|p|) with h increasing and convex, and k is the Legendre dual of h. Let q(t) be a loop in M , and E(q) = R

S¹

k(| q|)dt. Then we claim ˙

F H

(H; a, b) ≃ H

(E

, E

)

We know from [?] that there is a subset U

in ∆

, defined by U

= {(q

, P

)

|d(q

, q

) ≤ ε/2}

for ε small enough, independent from r, such that Γ

(that is Γ

in the notation of [?]) is the graph of dS

over U

.

From proposition 1.8 in [?], we have that there is a pseudogradient vector field ξ

of S

such that, denoting by HI

the cohomological Conley index (see [?]), we have, for some integer d,

HI

(S

, S

; ξ

) ≃ H

(E

, E

) where

E

= {q = (q

) ∈ N

|d(q

, q

) ≤ ε/2 and E

(q) < b}

(Note: in [?], E

is denoted Λ

)

(E

(q) = sup

S

(q, P ) and d is some normalizing constant (equal in fact to rn).

On the other hand E

≃ E

where E

= {q ∈ ΛM|E(q) ≤ b} (see e.g.

[?]), so we only need:

Lemma 3.4.

F H

(∆

, Γ

; a, b) ≃ HI

(S

, S

, ξ

)

Proof. We would like to find an almost complex structure J = J

, on (T

N ×

T

N )

such that the holomorphic maps corresponding to Floer trajectories

are also in one to one correspondence with bounded trajectories of ξ

.

δ u h

h=cu-δ

Figure 3: The hamiltonian h

We first choose h so that the critical values of S

are in [−δ, 0]. Indeed, we may impose that h

(u)u − h(u) is in [−δ, 0] (see Figure ??).

Then the Floer trajectories used to define the left hand side will have area less than 2δ.

Now, let us show that for δ small enough, such a Floer trajectory must stay inside T

U

.

Indeed, since all critical points are inside T

(U

), we have, for a Floer trajectory exiting from T

U

, that it defines a J-holomorphic curve Σ such that

(i)

∂Σ ⊂ ∆

∪ Γ

. (ii)

Σ is in T

(U

− U

).

Now we are in the following abstract situation:

Let V

⊂ V

be convex sets, L

, L

be disjoint Lagrange submanifolds in

T

(V

− V

). We set S

= ∂V

= {x ∈ V

| (1 − t)d(x, V

) = td(x, V

)}, and

B

= ∂T

V

= T

V .

Let Σ be a holomorphic curve with boundary in L

∪ L

. Then we claim that the area of Σ is bounded from below.

First let us consider the case of a pseudo-holomorphic curve, Σ closed in T

(V

− V

) (and in particular with boundary in T

V ∪ T

V ), such that

∂Σ ∩ T

V = ∂Σ ∩ B

is non-empty.

Then B

must intersect Σ, since otherwise, as ∂Σ ∩ B

6= ∅, there would be an interior tangency between some B

(for 0 < t < 1/2) and Σ, and this is impossible by the pseudoconvexity of B

.

Thus for α =

d(V

, V

), and x

∈ B

∩ Σ we have that B (x

, α) is in T

(V

− V

), and thus, we have

area(Σ) ≥ area(Σ ∩ B(x

, α) ≥ πα

exp(η(cα))

where c is an upper bound for the sectional curvatures of the metric g

associated to ω and J (see Appendix).

Now consider the case where Σ has a boundary contained in the union of the two Lagrange submanifolds L

, L

.

Let U

, U

be tubular neighbourhoods of L

, L

respectively, and assume that they are disjoint, symmetric (i.e. there is an anti-holomorphic diffeo- morphism of U

, fixing L

), and pseudoconvex. This can be easily achieved, through a perturbation of J near the L

.

Consider now ∂Σ ∩ L

= γ

. Then either γ

and γ

are both contained inside B

or one of them is not.

In the first case, consider Σ ∩ T

(V

− V

). Then this intersection does not have a boundary in T

(V

− V

), except on B

∪ B

, and we thus have again a lower bound on the area of Σ as in the case of a closed curve, except that α is to be replaced by α/2.

In the second case, assume for instance that B

intersects γ

.

Let us then consider the symetrization of Σ inside U

, where J is inte- grable, (see for instance [?]). This will be a closed curve Σ in b U

, that has a point x

in B

, and we also have a ball B (x

, α) in U

∩ T

(V

− V

) for α ≤ inf{

d(V

, V

),

d(L

, ∂U

)}, and again, we get a lower estimate of the area of Σ. Since area(Σ) b ≥ 1/2 area(b Σ), we also get an estimate on the area of Σ.

This proves our abstract statement. We now claim that we are in the above framework, with α ≃ ε

/4.

Indeed, the diameter of T

(U

− U

) is ε/2, and we have to show that

∆

has a pseudoconvex symmetric neighbourhood of radius ε/4.

But we will show that

(ii) Γ

∩ T

(U

− U

) ∩ {|X

| ≤ ε

/4|Y

| ≤ ε

/4} = ∅.

This implies that {|X

| ≤ ε/4|Y

| ≤ ε/4} is a tubular neighbourhood of the zero section, is disjoint from Γ

in the region we are considering. Thus we have our U

, with radius ε/4. A similar fact would hold for Γ

.

Let us now prove our last claim.

Indeed we only have to show that |

j

| ≥ ε/4 for (q, P ) in U

− U

. But

≃ q

− q

−

+ η(q

− q

, P

) (see [?], proof of 1.2) where η(0, P ) = dη(0, P ) = 0, and since for some j , d(q

, q

) ≥

, we have

|X

| = |

j

| ≥

−

≥

for r large enough.

It is particularly important to notice that our lower bounds are indepen- dent from r, since they only depend on an upper bound for the sectional curvatures of the metric, and this quantity stays bounded as r goes to infin- ity.

If we choose δ <

ε we get that all Floer trajectories for J

must stay in T

U

.

We claim that

(i) for a suitable choice of J

, the J

holomorphic curves defining the Floer cohomology are in one to one correspondence with trajectories of S

.

(ii) there is a family J

of almost complex structures connecting J

to J

, taming ω and making T

U

pseudo convex. Note that our almost complex structures will be time dependent, but this is not important.

Lemma 3.5. Let U be a manifold with boundary, and f : U → R be a smooth function. Then for L =graph(df ), φ

the Hamiltonian flow (q, p) → (q, p + tdf (q)), set J(t, u) = (φ

)

J

(u).

Then, there is a one to one correspondence between solutions of ( q(t) = ˙ −∇f (q(t))

s→±∞

lim q(t) = x

and

 



 

∂

v = 0

v(s, 0) ∈ 0

, v(s, 1) ∈ L

s→±∞

lim v(s, t) = x

Proof. We have, setting v (s, t) = φ

u(s, t)

∂

v = dφ

(u) ∂

∂s u(s, t) +J ( ∂

∂t φ

)(u)

= dφ

(u)[ ∂

∂t + dφ

(u)[ ∂

∂S + dφ

(φ

(u))Jdφ

(u) ∂

∂t u +∇H(u)]

= dφ

(u)[∂

u + ∇H(u)

= dφ

(u)[∂

u + ∇H(u)].

Thus

∂

u = −∇H(u)

u(S, 0) ∈ O

u(s, 1) ∈ O

.

Now since H(q, p) = f (q), ∇H(q, p) = ∇f (q), and in local coordinates, we have dp

· ∇f (q) = 0.

Hence d(|p|) · ∂

u =

X

j=1

dp

· ∇f(q) = 0 and |p ◦ u| satisfies the maximum principle. But since u(s, 0) ∈ O

, u(s, 1) ∈ O

, we have p ◦ u ≡ 0, hence u(s, t) = q(s, t), 0). Now

+

= 0 is the second half of ∂

u =

−∇H(u), hence q(s, t) = q(t), and the first half becomes ˙ q(t) = −∇f (q).

Note that if J

makes T

U pseudoconvex, the same holds for J

the linear interpolation between J

and J

.

From this lemma, and the previous arguments, we may conclude that F H

(∆

, Γ

; a, b) is isomorphic to the cohomology of the Thom-Smale- Witten complex of ∇S

restricted to S

− S

.

According to [?] this last cohomology equals HI

(S

, S

; ∇S

).

Our proof will be complete if we are able to show that HI

(S

, S

; ∇S

) ≃ HI

(S

, S

; ξ

)

where ξ

is as in [?]. This may be explained by looking at figure ??.

We represented there a pseudogradient vector field η for S

, equal to ∇S

in a neighbourhood of U

− U

and to ξ

in a neighbourhood of U

.

ξφ ξφ

∇ S

∇ S φ φ

U _{r ,ε} U _{r ,ε /2}

Figure 4: The phase portrait of the pseudo-gradient ξ

of S

Now, all critical points of S

are inside U

, and a heteroclinic trajectory for η stays inside U

since η = ξ

on ∂U

is tangent to ∂U

.

Thus I

(U

, η) ≃ I

(U

, η).

But I

(U

, η) ≃ I

(U

, ∇S

) since η = ∇S

near ∂U

, and I

(U

, η) ≃ I

(U

, ξ

).

Therefore

I

(U

, ∇S

) ≃ I

(U

, ξ

)

and our argument obviously extends if we restrict ourselves to S

−S

. Thus I

(S

, S

, ∇S

) ≃ I

(S

, S

, ξ

).

Note that these indices do not depend on ε, for r large enough, hence it

is not important that the previous inequality relates the index of U

to the

index of U

.

4 Appendix

Let (M, ω) be a symplectic manifold with contact type boundary, and J

be an admissible almost complex structure on M.

Given U a domain in M, and x

in U, we consider w

(U ) = inf {

Z

C

ω | C is J

- holomorphic, x

∈ C}

and w(U ), the usual Gromov width, is then given by

w(U ) = sup{w

(U) | x

∈ U, J

is admissible } A natural question is to compute the limits

k→+∞

lim w(U

) = w(U )

k→+∞

lim w

(U

) = w

(U )

Here we still denote by J

the almost complex structure J

× . . . × J

on M

, and by x

the point (x

, . . . , x

) in M

.

Note that the sequence w

(U ) is obviously decreasing.

On the other hand, w(U ) is not, a priori, equal to sup {w

(U)|x

∈ U, J

is admissible} = w(U e ) since there are many more almost complex struc- tures on M

than those of the type J

× . . . × J

. Clearly, we have w(U e ) ≤ w(U ).

While it is clear, if U contains a symplectic ball of radius r, that w(U) ≥ πr

, no such lower bound holds for w

(U ).

In fact, it is not a priori obvious that w

(U) is non zero. This is what we prove in this appendix.

Proposition 4.1. Let g

be the metric associated to (ω, J

) . Then, if the sectional curvature of g

is bounded by c , and injectivity radius at x

bounded by ρ, we have

w

(U ) ≥ πρ

δ(c, ρ)

where δ(c, ρ) is a continuous positive function, such that δ(c, 0) = 1 .

Lemma 4.2. Let B

(r) be the ball of radius r, centered at x

, for the metric g

. Let C be a minimal surface for g

through x

. Then

area (C ∩ B

(r)) ≥ πr

exp(η(cr))

where η is continuous, η(0) = 0, and c is the upper bound of the sectional curvature of g

.

Proof. It is similar to the case where g

is the euclidean metric. Set a(r) = area (C ∩ B

(r))

then a

(r) = length(C ∩ ∂B

(r)) and since C is minimal, a(r) must be less than the area of the cone through x

, spanned by C ∩ ∂B

(r).

Let us then compute the area of such a cone. It is clearly given by R

0

length(γ

)ds where γ

(t) = exp(

· exp

(c

(t))) and c

(t) is the curve C ∩ ∂B

(r), the exponential being taken at x

.

Let M be a bound on the sectional curvature of g

. Then we have, by classical comparison theorems ([?] p.117, remark 8.14b)

kD exp

(u)k ≤ sinh(Mr)

Mr for kuk ≤ r kD exp

(y)k ≤ Mr

sin(Mr) for y ∈ B(x

, r) Thus

length (γ

) ≤ s r

sinh(Mr)

sin(Mr) · length(Cr)

and Z

0

length (γ

)ds ≤ r 2

sinh(Mr)

sin(Mr) · length(Cr) and we have

a(r) ≤ r

2 ϕ(Mr)a

(r) so that

a

(r)

a(r) ≥ 2

rϕ(Mr) log( a(r)

a(ε) ) ≥ log r

ε

+

Z

2

uϕ(Mu) (1 − ϕ(Mu))du

≥ log( r

ε

) +

Z

2 1 − ϕ(v))

vϕ(v) dr

Since 1 − ϕ(u) ∼ u as u goes to zero, the quantity R

2

u(ϕ(u))

(1 − ϕ(u))du converges to η(Mr) as ε goes to zero, with η continuous and η(0) = 0.

Then, since lim

ε→0

a(ε)

ε

≥ π, we have a(r) ≥ πr

exp(η(Mr)).

Now, replacing U by U

, the sectional curvature of the induced metrics stays bounded (even though the bound may change as we go from k = 1 to k = 2), and since U

contains (B

(ρ))

, we get

ω

(U

) ≥ πρ

δ(c, ρ).

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