Construcción de la curva de calibración de la 4-Metil

Since our statements only hold up to a maximal period for the asymptotic orbits, we can- not use the same coherent Hamiltonian perturbation to compute the full contact homology. As seen above we must rescale the Hamiltonian for the cylindrical moduli spaces, which clearly affects the Hamiltonian perturbations for all punctured spheres. For showing that the graded vector space isomorphism we obtain is actually an isomorphism of graded alge- bras, we construct chain maps between the differential algebras for the different coherent Hamiltonian perturbations, which are defined by counting holomorphic curves in an almost complex manifold with cylindrical ends.

1.4.1

Moduli spaces

For a given Hamiltonian H : M → R let ˜H : R×M → R be a smooth homotopy with ˜

of stable Hamiltonian structures (ωH˜_{, λ}H˜_{) with corresponding (constant) symplectic hyper-}

plane bundlesξH˜ _{=T M andR-dependent Reeb vector fieldsR}H˜_{(s, t, p) =∂} t+X

˜

H_{(s, t, p),}

it equips R×S1 _{×M with the structure of a symplectic manifold with stable cylindrical}

ends

((−∞,−1]×S1×M, ωH/2, λH/2) and ([+1,+∞)×S1×M, ωH, λH),

where the symplectic structure on the compact, non-cylindrical part (−1,+1)×S1_{×M is}

given by

ωH˜ _=ωH˜ _+ds∧dt

with ωH˜ _{=ω+dH˜ ∧dt.}

Together with the fixedω-compatible almost complex structureJ onM, the homotopy ˜

H further equips R×S1 _{×M with an almost complex structure J}H˜ _{by requiring that it}

turns ξH˜ _{=T M into a complex subbundle with complex structure J and}

JH˜ ·∂s =R ˜ H_{(s,·) =∂} t+X ˜ H_(s,·).

It follows that (R×S1 _{×M, J}H˜_{) is an almost complex manifold with cylindrical ends}

((−∞,−1] × S1 _{× M, J}H/2_{) and ([+1,+∞) × S}1 _{× M, J}H_{). Note that J}H˜ _{is indeed}

ωH˜_-compatible.

For our applications we clearly have to replace the Hamiltonian H : M → R by the domain-dependent Hamiltonian perturbation H : `_nM0,n+1×M → R from before. It

follows that the Hamiltonian homotopy ˜H has to depend explicitly on points on the un- derlying stable punctured spheres, i.e., for the following we consider coherent Hamiltonian homotopies

˜

H :a

n

M0,n+1×R×M →R,

with corresponding domain-dependent almost complex structures

JH˜ :a

n

M0,n+1 → J(S1×M).

While it is again clear that the moduli spaces of JH˜-holomorphic curves with more than two punctures come with anS1_{-symmetry, it remains to verify nondegeneracy for the}

asymptotic orbits and transversality for the curves. Note for the first that we again have to consider rescaled versions ˜HN : `nM0,n+1×R×M → R with ˜HN(s) = ˜H(s/2N)/2N.

Since ˜HN(s) =H/2N+1 for s ≤ −2N and ˜HN(s) =H/2N for s ≥+2N, it is clear that the

nondegeneracy holds for all asymptotic orbits of period less or equal to 2N_.

While we show below that we can again achieve transversality for all JH˜-holomorphic curves with more than three punctures making use of the domain-dependency of the al- most complex structure, it remains to guarantee transversality for JH˜-holomorphic cylin- ders. Note that in analogy to proposition 1.1.6 it follows that allJH_N˜-holomorphic cylinders

connecting orbits (x+_{, T) and (x}−_{, T) withT ≤2}N _{are in natural correspondence to cylin-}

ders in M connecting the critical points x+, x−, which satisfy the R-dependent perturbed Cauchy-Riemann equation

¯

∂J,Hu·∂s=∂su+J(u)·(∂tu+T ·X

˜

H_{(T s, u)) = 0.}

While in general transversality generically only holds for t-dependent Hamiltonian homotopies ˜H, we can now make use of the following natural generalization of lemma 1.1.5:

Lemma 1.4.1: Let (H, J) be a pair of a Hamiltonian H and an almost complex structure J on a closed symplectic manifold with h[ω], π2(M)i = 0 so that (H, ω(·, J·))

is Morse-Smale. Choose ϕ ∈ C∞(R,R+) with ϕ(s) = 1/2 for s ≤ −1 and ϕ(s) = 1 for

s≥1, and let H˜ :R×M →R, H˜(s, p) = ϕ(s)·H(p). Then the following holds:

• The linearization F˜u of ∇J,H˜u=∂su+J(u)XH˜(s, u) is surjective at all solutions. • If τ >0 is sufficiently small, all finite energy solutions u:R×S1 _{→M of ∂}¯

J,H˜τu=

∂su+J(u)(∂tu+XH

τ

(s, u)) = 0with H˜τ_{(s,·) =τH˜(τ s,·) are independent of t∈S}1_.

• In this case, the linearization D˜u = D˜uτ of ∂¯J,H˜τ is onto at any solution

u:R×S1 _→M.

Proof: The proof is a simple generalization of the arguments given in [SZ] and we just show the first statement. Let ˜ϕ : R → R+ with ∂sϕ˜ = ϕ. Then ˜u(s) = u( ˜ϕ(s)) satisfies ∇_J,H˜u˜= 0 whenever u:R→M is a solution of ∇J,Hu= 0, since

∂su˜+∇H˜(s,u˜) = ∂sϕ˜(s)·∂su+ϕ(s)· ∇H(u).

For ˜η ∈ Lp_(˜u∗_{T M) we find η ∈ L}p_(u∗_{T M) so that ˜η(s) = η( ˜ϕ(s)). Assuming that}

hFu˜ξ,˜ η˜i = 0 for all ˜ξ ∈ H1,p(˜u∗T M), it follows that hFuξ, ηi = 0 for all ξ ∈ H1,p(u∗T M)

by identifying ˜ξ(s) = ξ( ˜ϕ(s)), where ˜Fu˜, Fu denote the linearizations of ∇J,H˜, ∇J,H at

˜

u, u, respectively. The regularity of (H, J) provides us with the surjectivity of Fu at any

solutionu:R→M, so thatη and therefore ˜η must vanish. ¤

With the fixed HamiltonianH(2) _{:M →Rfor the cylinders we choose the Hamiltonian}

homotopy for the cylinders ˜H(2) _{:R×M →R to be}

˜

H(2)_{(s, p) =ϕ(s)·H}(2)_(p),

so that ˜H(2)_{(s,·) = H}(2)_{/2 for s≤ −1 and ˜H}(2)_{(s,·) = H}(2)_{. After possibly rescaling H}(2)_,

fixedJ and the chosenH(2)_{, ˜H}(2)_{, respectively.}

Before we prove transversality in the next subsection, let us state the following analogue of theorem 1.1.6. Denote byJH_N˜ the domain-dependent almost complex structure onR×S1_{×M induced by ˜H}

N.

Theorem 1.4.2: Depending on the number of punctures n we have the following result about the moduli spaces of JH_N˜-holomorphic curves in R×S1_×M:

• n= 0: All holomorphic spheres are constant.

• n= 1: Holomorphic planes do not exist.

• n = 2: For T ≤ 2N _{the automorphism group Aut(CP}1_{) acts on the moduli space of}

parametrized curves M0(S1_×M,(x+_{, T),(x}−_{, T), J}H˜

N) of holomorphic cylinders with

constant finite isotropy groupZT and the quotient can be naturally identified with the

space of gradient flow lines of H(2) _{with respect to the metric ω(·, J·) on M between}

the critical points x+ _{and x}− _{of H}(2)_{. In particular, we have}

]M(R×S1×M; (x+, T),(x−, T);JH_N˜) = δx−_,x+

since the zero-dimensional components are empty for x+ _6=x− _{and just contain the}

constant path for x+ _=x−_.

• n ≥ 3: For P+ _{⊂ P(H}(2)_/2N_{,≤ 2}N_{) and P}− _{⊂ P(H}(2)_/2N+1_{,≤ 2}N_{) the action of}

Aut(CP1) on the moduli space of parametrized curves is free and the moduli space is given by the product

S1× {(s0, u, z) :s0 ∈R, u:CP1− {z} →M : (∗1),(∗2)}/Aut(CP1) with (∗1) : du+XH˜N z (z, h01+s0, u)⊗dh02 +J(u)·(du+XH˜N z (z, h01+s0, u)⊗dh02)·i= 0, (∗2) : u◦ψ±_k(s, t)s→±∞−→ x±_k.

In particular, it remains a free S1_{-action on the moduli space.}

Proof: The proof is completely analogous to the one of theorem 1.1.6. Note that it follows by lemma 1.1.3 that h : CP1 − {z} → R×S1 _{can be identified with (s}

that the map u now satisfies an s0-dependent perturbed Cauchy-Riemann equation. For

n = 2 observe that by lemma 1.3.1 we can identifyM(S1_{×M; (x}+_{, T),(x}−_{, T);J}H˜ N) with

the space of allu:R→M satisfying∇_J,H˜(2)u= 0, u(s, t)→x±, which following the proof

of lemma 1.3.1 can be identified with the space of ˜u(s) = u( ˜ϕ(s)) satisfying∇_J,H(2)u= 0. ¤

1.4.2

Transversality

For the remaining part of this section we discuss transversality, where we again restrict ourselves to the case N = 0:

Since ¯∂_JH˜(h, u) = ( ¯∂h,∂¯_J,H,s˜ ₀u) with

¯

∂_J,H,s˜ ₀u = du+XH˜(j, z, h0₁+s0, u)⊗dh02

+ J(u)·(du+XH˜(j, z, h0₁+s0, u)⊗dh02)·i,

where XH˜_{(j, z, s, u) denotes the symplectic gradient of ˜H(j, z, s,·) : M → R, it follows}

that the linearization Dh,u of ¯∂_JH˜ is again of diagonal form.

It follows that for n = 2 we get transversality from lemma 1.3.2 and lemma 1.4.1 by the special choice of ˜H(2)_.

For n ≥ 3 let us describe the setup for the underlying universal Fredholm prob- lem:

As before the Cauchy-Riemann operator extends to a C∞_{-section in a Banach space}

bundle ˜Ep,d → Bp,d×H˜`. Here Bp,d = Bp,d(R×S1 _×M;P+_{, P}−_{) denotes the manifold of}

maps from section 5, which is given by the product

Bp,d(R×S1×M; (x±_k, T_k±)) =H_const1,p,d( ˙S,C)× Bp(M; (x±_k))× M0,n,

while the set of coherent Hamiltonian perturbations H`

n(M; (H(k))nk=2−1) is now replaced by

the set of coherent Hamiltonian homotopies ˜

H` = ˜H`_n(M;H; ( ˜H(k))n_k=2−1)

for fixed coherent Hamiltonian H :`_nMn+1×M →R and ˜H(2), ...,H˜(n−1):

Any ˜H(n) _∈H˜` <sub>is a C</sub>`_-map

˜

H(n) :M0,n+1×R×M →R,

which extends to a C`_{-map on M}

• on¡(M0,n+1− M0,n+1)∪(M0,n×N0) ¢ ×R×M it is given by ˜ H(2)_{, ...,H˜}(n−1)_, • H˜(n) _=H(n)_{/2 on M} 0,n+1×(−∞,−2N)×M, • and ˜H(n)_=H(n) _onM 0,n+1×(+2N,+∞)×M,

where N0 ⊂S˙ again denotes the fixed neighborhood of the punctures. It follows that the

tangent space at ˜H = ˜H(n) _∈H˜` _{is given by}

T_H˜H˜

`

n = ˜H `

n(M; 0; (0)nk=2−1).

Since the linearization of ¯∂_JH˜ at (¯h, u, j,H˜)∈ B

p,d_×H˜` _{is again of diagonal form,}

D¯_h,u,j,H˜ = Dj+ diag( ¯∂, Du,H˜) :

TjM0,n⊕Hconst1,p,d( ˙S,R2)⊕H1,p(u∗T M)⊕T_H˜H˜

` →Lp,d_(T∗_{S˙ ⊗}

j,iR2)⊕Lp(T∗S˙ ⊗j,J u∗T M)

it remains by lemma 1.3.2 to prove surjectivity of D_u,H˜, which is the linearization

of the perturbed Cauchy-Riemann operator ¯∂J,s0(u,H˜) = ¯∂J,H,s˜ 0(u). Since the proof is

in the central arguments completely similar to lemma 1.3.3, we just sketch the main points: Assume for some η ∈ Lp_(T∗_{S˙ ⊗}

j,J u∗T M) that hDu,H˜(ξ,G˜), ηi = 0 for all (ξ,G˜) ∈

H1,p_(u∗_{T M)⊕T} ˜

HH˜ `

. Fromhη, Duξi= 0 for all ξ we already know that it suffices to show

that η vanishes on an open and dense subset.

Now observe that it follows from the same arguments used to prove lemma 1.3.3 that 0 = hD_H˜G, η˜ i = Z ˙ S−B kdh0₁(z)k2 ·dG˜(j, z, h₀1(z) +s0, u(z))·η1(z)dz for all ˜G∈T_H˜H˜ `

, whereB is the set of branch points of the map

h0 _{: S}˙ _{→ R×S}1_{, we again write η(z) = η}

1(z) ⊗ dh01 + η2(z) ⊗ dh02 with

η2(z) + J(u)η1(z) = 0 for z ∈ S˙ − B and where dG˜(j, z, h01(z) + s0,·) denotes the

differential of ˜G(j, z, h1

0(z) +s0,·) : M →R. But with this we can prove as before that η

vanishes identically on the open and dense subset ˙S−B:

Assume to the contrary that η(z0)6= 0, i.e., η1(z0)6= 0 for some z0 ∈S˙ −B. As in the

proof of lemma 1.3.3 we findG0 ∈C∞(M) so that

dG0(u(z0))·η1(z0)>0.

Settingj0 :=j, observe that we can organize all fixed mapsh0 : ˙S →R×S1 for different

˜

ϕ∈C∞_(M

0,n+1×R,[0,1]) be a smooth cut-off function around

(j0, z0, h10(j0, z0) +s0)∈ M0,n+1×R withϕ(j0, z0, h10(j0, z0) +s0) = 1 and ϕ(j, z, h10(j, z) +

s) = 0 for (j, z, s) 6∈ U(j0, z0, s0). Here the neighborhood U(j0, z0, s0) ⊂ M0,n+1 ×R is

chosen so small that

U(j0, z0, s0) ∩ ³¡ (M0,n+1− M0,n+1) ∪ (M0,n+1×N0) ¢ × R´ = ∅, U(j0, z0, s0) ∩ ³ M0,n+1 × ¡(−∞,−1)∪(+1,+∞)¢´ = ∅,

and dG0(z, u(z))·η1(z)≥0 for all (z, j, h10(j, z) +s)∈U(j0, z0, s0).

Defining ˜G:M0,n+1×R×M →R by ˜G(j, z, s, p) :=ϕ(j, z, s)·G0(p), this leads to the

desired contradiction since we found ˜G∈T_H˜H˜

` = ˜H`_n(M; 0; 0, ...,0) with hD_H˜ ·G, η˜ i= Z ˙ S−B kdh0₁(z)k2 _{· dG˜(j} 0, z, h10(j0, z) +s0, u(z))·η1(z)dz >0.

So we have shown that the corresponding universal moduli space

M(R×S1 _{× M;P}+_{, P}−_;JH_{; (J}H,˜(k)₎n−1

k=2) is again transversally cut out by the Cauchy-

Riemann operator ¯∂J. Further it follows by the same arguments as in section 4 that we

can choose a (smooth) coherent Hamiltonian homotopy ˜

H : `_nM0,n+1×R → C∞(M) such that for all N ∈ N and P+, P− the moduli spaces

M(R×S1_×M;P+_{, P}−_;JH˜

N) are transversally cut out by the Cauchy-Riemann operator.

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