Relative Invariants, Contact Geometry and Open String Invariants

0.65 MB·English

by  An-Min Li

Relative Invariants, Contact Geometry and Open String Invariants
An-Min Li1
Department of Mathematics, Sichuan University
Chengdu, PRC
Li Sheng2
5 Department of Mathematics, Sichuan University
1
Chengdu, PRC
0
2
Abstract
n
a Inthispaperweproposeatheoryofcontactinvariantsandopenstringinvariants,whichare
J
generalizations of the relative invariants. We introduce two moduli spaces A(M+,C,g,m+
6 M
2 ν,y,p,(k,e))andMA(M,L;g,m+ν,y,p,→−µ),provethecompactnessofthemodulispacesand
the existence of the invariants.

]
G
S Contents
.
h
t 1 Introduction 2
a
m
2 Symplectic manifolds with cylindrical ends 4
[
2.1 Contact manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2
2.2 Neighbourhoods of Lagrangian submanifolds. . . . . . . . . . . . . . . . . . . . . . . 4
v
4 2.3 Cylindrical almost complex structures . . . . . . . . . . . . . . . . . . . . . . . . . . 6
9
2.4 J-holomorphic maps with finite energy . . . . . . . . . . . . . . . . . . . . . . . . . . 7
0
1 2.5 Periodic orbits of Bott-type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
0
.
1 3 Weighted sobolev norms 11
0
5
4 Moduli spaces of J-holomorphic maps 12
1
: 4.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
v
4.2 Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
i
X
4.3 Holomorphic blocks in M+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
ar 4.4 Holomorphic blocks in R M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
×
4.5 Local coordinate system of holomorphic blocks . . . . . . . . . . . . . . . . . . . . . 17
f
5 Compactness theorems 17
5.1 Bubble phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5.1.1 Bound of the number of singular points . . . . . . . . . . . . . . . . . . . . . 18
5.1.2 Construction of the bubble tree for (a-2) . . . . . . . . . . . . . . . . . . . . . 19
5.1.3 Construction of bubble tree for (b) . . . . . . . . . . . . . . . . . . . . . . . . 21
5.1.4 For the case of genus 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5.2 - rescaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
T
5.3 Equivalent - rescaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
DT
1partially supported by a NSFCgrant
2partially supported by a NSFCgrant
1
5.4 Procedure of re-scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5.5 Weighted dual graph with a oriented decomposition . . . . . . . . . . . . . . . . . . 29
5.6 Li-Ruan’s Compactification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
6 Gluing theory-Pregluing 34
6.1 Gluing one relative node for M+ (R M) . . . . . . . . . . . . . . . . . . . . . . . 34
∪ ×
6.2 Gluing two relative nodal points for M+ (R M) . . . . . . . . . . . . . . . . . . 35
∪ ×
6.3 Norms on C (Σ ;u TM+ ) . . . . .f. . . . . . . . . . . . . . . . . . . . . . . . . 37
∞ (r) ∗(r) (r)
f
7 Gluing theory–Regularization 38
7.1 Local regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
7.1.1 Top strata. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
7.1.2 Lower strata. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
7.2 Global regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
8 Gluing theory– analysis estimates 42
8.1 Gluing theory for 1-nodal case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
8.2 Estimates of right inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
8.3 Isomorphism between KerD and KerD . . . . . . . . . . . . . . . . . . . . . 43
Sb Sb(r)
8.4 Gluing maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
8.5 Surjectivity and injectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
9 Estimates of differentiations for gluing parameters 48
9.1 Linear analysis on weight sobolov spaces . . . . . . . . . . . . . . . . . . . . . . . . . 48
9.2 Estimates of ∂Qb(r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
∂r
9.3 Estimates of ∂I(r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
∂r
9.4 Estimates of ∂ I (κ,ζ)+Q f (I (κ,ζ)) . . . . . . . . . . . . . . . . . . . . . . 51
∂r r b(r) (r) r
(cid:2) (cid:3)
10 Contact invariants and Open string invariants 55
1 Introduction
Open string invariant theory have been studied by many mathematicians and physicists (see [1,2,
11,16,17]). This theory closely relates to relative Gromov-Witten theory and contact geometry.
In this paper we propose a theory of contact invariants and open string invariants, which are
generalizations of the relative invariants. We outline the idea as follows.
1). Let (M,ω) be a compact symplectic manifold, L M be a compact Lagrangian sub-
⊂
manifold. Let (x , ,x ) be a local coordinate system in O L, there is a canonical coordinates
1 n
··· ⊂
(x , ,x ,y , ,y ) in T L . Suppose that given a Riemannian metric g on L, and consider
1 n 1 n ∗ O
··· ··· |
the unit sphere bundle M. Let Λ be the Liouville form on T∗L, denote λ = −ϑ |Mf. Then (M,λ)
is a contact manifold with contact form λ. The Reeb vector field is given by X = yi ∂ . By
y ∂xi
Lagrangian Neighborhoofd Theorem we can write M L as k k f
− P
M+ = M [0, ) M ,
0
{ ∞ × }
[
f
2
whereM isacompactsymplecticmanifoldwithboundary. Wechooseanalmostcomplexstructure
0
J on M+ such that J is tamed by ω and over the cylinder end J is given by
∂ ∂
J = J˜, JX = , J( )= X,
ξ
| −∂a ∂a
where ξ = kerλ, a is the canonical coordinate in R, and J˜is a dλ-tame almost complex structure
in ξ. Assume that the periodic orbit sets of X are either non-degenerate or of Bott-type, and J˜
can be chosen such that L J = 0 along every periodic orbit.
X
Let Σ◦ be a Riemann surface with a puncture point p. We use the cylinder coordinates (s,t)
near p, i.e.,we consider a neighborhood of p as (s0, ) S1. Let u :Σ◦ M+ be a J-holomorphic
∞ × →
map with finite energy. Suppose that
[lim u(s,S1)] = µi[c ],
i
s
→∞
X
where [c ],i = 1,...,a, is a bases in H (L,Z) and µi Z. Then u(s,t) converges to a periodic
i 1
∈
orbit x M of the Reeb vector field X as s . We can view lim u(s,S1) as a loop in L,
s
represen⊂ting µi[c ]. In this way, we can cont→rol∞the behaviour at infin→it∞y of J-holomorphic maps
i
with finitefenergy.
P
2). To compactify the moduli space we need study the J-holomorphic maps into ( , )
{ −∞ ∞ ×
M . There are two global vector fields on R M: ∂ and X. Similar to the situation of relative
} × ∂a
invariants, there is a R action, which induces a R-action on the moduli space of J-holomorphic
mfaps. We need mod this action. Since there ifs a vector field X with X = 1 on M, the Reeb
| |
vector field, there is a one parameter group ϕ action on M generated by X. In particular, there
θ
−
is a S1-action on every periodic orbit, corresponding to the freedom of the choosingforigin of S1.
Along every periodic orbit we have L λ = 0, on the otherfhand, we assume that on every periodic
X
orbit L J = 0, then we can mod this action.
X
On the other hand, we choose the Li-Ruan’s compactification in [18], that is, we firstly let the
RiemannsurfacesdegenerateinDelingne-MumfordspaceandthenletM+degeneratecompatiblyas
in thesituation of relative invariants. Atany node,theRiemann surfacedegenerates independently
with two parameters, which compatible with those freedoms of choosing the origins ( see section
5 for degeneration and section 6 for gluing). Then both blowups at interior and at infinity lead
§ §
boundaries of codimension 2 or more in the moduli space.
3). Another core technical issue in this paper is to define invariants using virtual techniques.
As we know, there had been several different approaches, such as Fukaya-Ono [10], Li-Tian [21],
Liu-Tian( [22]), Ruan( [27]), Siebert( [29]) and etc. In [18], Li and Ruan provide a completely new
approach to this issue: they show that the invariants can be defined via the integration on the top
stratum virtually. In order to achieve this goal, they provide refined estimates of differentiations
for gluing parameters: ∂/∂r. In [18] the estimates for ∂ is of order r 2 when r , that is
∂r − → ∞
enough to define invariants. In this paper these estimates achieve to be of exponential decay order
exp( cr). Then we can use the estimates to define the invariants and prove the smoothness of the
−
moduli space. The same method can be applied to GW-invariant for compact symplectic manifold.
Inthispaperweintroducetwomodulispaces (M+,C,g,m+ν,y,p,(k,e)) and (M,L;g,m+
A A
M M
ν,y,p, µ), prove the compactness of the moduli spaces and the existence of the contact invariants
−→
3
(C) (L)
Ψ (α ,...,α ;β ,...,β )andtheopenstringinvariantsΨ (α ,...,α ). In
(A,g,m+ν,k,e) 1 m m+1 m+ν (A,g,m+ν,−→µ) 1 m
our next paper [20] we will prove the smoothness of the two moduli spaces. Our open string in-
(L)
variants Ψ can be generalized to L which is a disjoint union of compact Lagrangian
(A,g,m+ν,−→µ)
sub-manifolds L ,...,L .
1 d
We consider a neighborhood of Lagrangian sub-manifold L as R M. By the same method
×
above we can define a local open string invariant. We will discuss this problem and calculate some
examples in our next paper. f
2 Symplectic manifolds with cylindrical ends
2.1 Contact manifolds
Let ( ,λ) be a (2n 1)-dimensional compact manifold equipped with a contact form λ. We recall
Q −
thata contact formλ is a 1-form on suchthat λ (dλ)n 1 is avolumeform. Associated to ( ,λ)
−
Q Q
we have the contact structure ξ = ker(λ), which is a (2n 2)-dimensional subbundle of T , and
V − Q
(ξ,dλ ) defines a symplectic vector bundle. Furthermore, there is a unique nonvanishing vector
ξ
|
field X = X , called the Reeb vector field, defined by the condition
λ
i λ = 1, i dλ = 0.
X X
We have a canonical splitting of TQ,
T = RX ξ,
Q ⊕
where RX is the line bundle generated by X.
2.2 Neighbourhoods of Lagrangian submanifolds
Let (M,ω) bea compact symplectic manifold, L M bea compact Lagrangian sub-manifold. The
⊂
following Theorem is well-known.
Theorem 2.1. Let (M,ω) be a symplectic manifold of dimension 2n, and L be a compact La-
grangian submanifold. Then there exists a neighbourhood U T L of the zero section, a neigh-
∗
⊂
bourhood V M of L, and a diffeomorphism φ :U V such that
⊂ →
φ ω = dΛ, φ = id, (1)
∗ L
− |
where Λ is the canonical Liouville form.
Let (x , ,x ) be a local coordinate system on O L, there is a canonical coordinates
1 n
··· ⊂
(x , ,x ,y , ,y )
1 n 1 n
··· ···
on T O = T L . In terms of this coordinates the Liouville form can be written as
∗ ∗ O
|
Λ= y dx .
i i
Let π : T L L be the canonical projection. TXhere is a global defined vector field W in T L such
∗ ∗
→
that in the local coordinates of π 1(O), W can be written as
−
n
∂
W|π−1(O) = − yi∂y . (2)
i
i=1
X
4
Suppose that given a Riemannian metric on L, in terms of the coordinates x ,...,x , g =
1 n L
n
g dx dx . It naturally induced a metric on T L. Let
ij i j ∗
i,j=1
P
W
V = .
W
k k
V is a global defined vector field on T L L.
∗
−
Denote by Sn 1(1) (resp.B (0)) theEuclidean unitsphere(resp. theEuclidean unitball). Con-
− 1
sider the coordinates transformation between the sphere coordinates and the Cartesian coordinate
Ψ : (0,1] Sn 1(1) B (0)
− 1
× →
(r,θ , ,θ ) (y , ,y ). (3)
1 n 1 1 n
··· − → ···
Consider the unit sphere bundle M and the unit ball bundle D (T L) in T L, in terms of the
1 ∗ ∗
coordinates (x , ,x ,y , ,y )
1 n 1 n
··· ···
f
n
M π−1(O) = (x1, ,xn,y1, ,yn) π−1(O) gij(x)yiyj = 1 , (4)
| { ··· ··· ∈ | }
i,j=1
X
f n
D1(T∗L)π−1(O) = (x1, ,xn,y1, ,yn) π−1(O) gij(x)yiyj 1 . (5)
| { ··· ··· ∈ | ≤ }
i,j=1
X
Denote λ = Λ f. We have
− |M
Λ= y λ.
−k k
λ is a contact form, i.e., (M,λ) is a contact manifold. Put ξ = ker(λ). Then X |Mf= − gijyi∂∂xj
is the Reeb vector field, anfd V |Mf= ni=1yi∂∂yi. P
ThemapΨinducedamapΨ˜ :(0,P1] M D1(T∗L).ThroughΨ˜ weconsiderD1(T∗L)π−1(O) L
× → | −
as (0,1] M. By Theorem 2.1 we consider M L as
× f −
f M+ = M+ (0,1] M
0 { × }
[
with the symplectic form f
ω = dΛ= y dλ+d y λ, (6)
φ
− k k k k∧
where M+ := M L and M+ is a compact symplectic manifold with boundary.
− 0
We choose the neck stretching technique.
Let φ: [0, ) (0,ℓ] be a smooth function satisfying, for any k > 0,
∞ →
(1) φ < 0, φ(0) = ℓ, φ(a) 0 as a ,
′
→ → ∞
(2) lim ∂kφ = 0.
a 0+ ∂ak
→
Through φ we consider M+ to be M+ = M+ [0, ) M with symplectic form ω = ω,
0 { ∞ × } φ|M0+
and over the cylinder [0, ) M
S
∞ × f
f ωφ = dΛ= φdλ+φ′da λ. (7)
− ∧
5
Moreover, if we choose the origin of R tending to , we obtain R M in the limit.
∞ ×
Choose ℓ < ℓ and denote
0 f
Φ+ = φ: [0, ) (0,ℓ ]φ < 0 .
0 ′
∞ → |
(cid:8) (cid:9)
Let ℓ < ℓ be two real numbers satisfying 0 < ℓ < ℓ ℓ . Let Φ be the set of all smooth
1 2 1 2 ≤ 0 ℓ1,ℓ2
functions φ: R (ℓ ,ℓ ) satisfying
1 2
→
φ < 0, φ(a) ℓ as a , φ(a) ℓ as a .
′ 1 2
→ → ∞ → → −∞
To simplify notations we use Φ to denote both Φ+ and Φ , in case this does not cause confusion.
ℓ1,ℓ2
We fixed φ Φ and consider the symplectic manifold M+ [0, ) M ,ω . For any different
∈ 0 { ∞ × } φ
φ Φ, we have ω = ω and φ φ 1 is a(cid:16)symplectic diffeomorphic(cid:17)over cylinder part
1 ∈ φ|M0+ φ1|M0+ ◦ −1 S f
(0,1] M .
{ × }
2.3 Cyflindrical almost complex structures
Let
M+ = M+ [0, ) M (8)
0 ∞ ×
[n o
beasymplecticmanifoldwithcylindricalend,whereM beafcompactcontactmanifoldwithcontact
form λ. Denote by ω the symplectic form of M+ such that ω = ω, and over the cylinder
φ f φ|M0+
[0, ) M
∞ ×
ω = dΛ= φdλ+φda λ. (9)
φ ′
− ∧
f
We also consider R M. Denote by N one of M+ and R M.
× ×
Put ξ = ker(λ), and denote by X the Reeb vector field defined by
f f
λ(X) = 1, dλ(X) = 0.
We choose a dλ-tame almost complex structure J for the symplectic vector bundle (ξ,dλ) M
→
such that
1
ge (h,k) = dλ(x)(h,J(ex)k)+dλ(x)(k,J(x)h) , (1f0)
J(x) 2
(cid:16) (cid:17)
for all x M, h,k ξ , defines a smooth fibrewiese metric for ξ. Wee assume that we can choose J
x
∈ ∈
such that on every periodic orbit L J = 0. Denote by Π : TM ξ the projection along X. We
X
→
define a Riefmannian metric , on M by e
h i
e f
h,fk = λ(h)λ(k)+ge(Πh,Πk) (11)
h i J
for all h,k TM.
∈
Given a J as above there is an associated almost complex structure J on R M defined by
f ×
∂ ∂
e J = J˜, JX = , J( )= X, f (12)
ξ
| ∂a ∂a −
6
where a is the canonical coordinate in R. It is easy to check that J defined by (12) is ω -tame over
ϕ
the cylinder end. We can choose an almost complex structure J on M+ such that J is tamed by ω
and over the cylinder end J is given by (12).
There is a canonical coordinate system for R M and for cylinder end of M+, but still there
×
are some freedom of choosing the coordinates:
f
When we write M+ as (8) we have chosen a coordinate a over the cylinder part. We can choose
different coordinate aˆ over the cylinder part such that
a = aˆ+C (13)
for some constant C > 0. Similarly, for R M we can choose aˆ such that
×
a = aˆ C (14)
f ±
for some constant C > 0.
For any φ Φ
∈ 1
v,w = (ω (v,Jw)+ω (w,Jv)) v,w TN (15)
h iωφ 2 φ φ ∀ ∈
defines a Riemannian metric on N. Note that , is not complete. We choose another metric
h iωφ
, on N such that
h i
, = , on M+ (16)
h i h iωφ 0
and over the tubes
(a,v),(b,w) =ab+λ(v)λ(w)+ge(Πv,Πw), (17)
h i J
where we denote by Π : TM ξ the projection along X. It is easy to see that , is a complete
→ h i
metric on N.
f
2.4 J-holomorphic maps with finite energy
Let (Σ,i) be a compact Riemann surface and P Σ be a finite collection of puncture points.
⊂
Denote Σ◦=Σ P. Let u:Σ◦ N be a J-holomorphic map, i.e., u satisfies
\ →
du i= J du. (18)
◦ ◦
Following [14] we impose an energy condition on u. For any J-holomorphic map u:Σ◦ N and any
→
φ Φ the energy E (u) is defined by
φ
∈
E (u) = u ω . (19)
φ ∗ φ
ZΣ
Let z = es+2π√ 1t. One computes over the cylindrical part
−
u ω = (φdλ((πu) ,(πu) )) φ(a2+a2))ds dt, (20)
∗ φ s t − ′ s t ∧
which is a nonnegative integrand. A J-holomorphic map u :Σ◦ N is called a finite energy J-
e e
→
holomorphic map if over the cylinder end
sup u ω + u dλ < . (21)
∗ φ ∗
∞
φ∈Φ(cid:26)ZΣ (cid:27) ZΣ
7
For a J-holomorphic map u: Σ R M we write u = (a,u) and define
→ ×
fE(u) = u∗dλ. e (22)
ZΣ
Denote e e
∞
E(s) = u (dλ).
∗
Zs ZS1
Then
e e
E(s) = ∞ Πu 2dsdt,
t
| |
Zs ZS1
edE(s) = Πeu 2dt. (23)
t
ds − | |
ZS1
e
Here and later we use denotes the norm with respect to the metric defined by (17).
|·| e
The following two lemmas are well-known ( see [13]):
Lemma 2.2. (1) Let u = (a,u) : C R M be a J-holomorphic map with finite energy. If
→ ×
u (π dλ) = 0, then u is a constant.
C ∗ ∗
e f
(2) LRetu= (a,u) :R S1 R M˜ beaJ-holomorphic mapwithfiniteenergy. If u (π dλ) =
e × → × R S1 ∗ ∗
0, then (a,u) = (kTs+c,kt+d), where k Z+, c and d are constants. ×
∈ R
e e
Lemma 2.3. Leet u = (a,u) : C D R M be a nonconstant J-holomorphic map with finite
1
− → ×
energy. Put z = es+2π√ 1t. Then for any sequence s , there is a subsequence, still denoted
− i
→ ∞
by si, such that e f
lim u(s ,t)= x(kTt)
i
i
→∞
in C (S1) for some kT-periodic orbit x(kTt).
∞ e
2.5 Periodic orbits of Bott-type
Let M bethe locus of minimalperiodicorbits with = ℓ , whereeach is a connected
F ⊂ F i=1Fi Fi
component of with minimal periodic T . We assume that
i
F S
f
Assumption 2.4. (1) every is either non-degenerate or of Bott-type;
i
F
(2) let be of Bott-type, then there exists a free S1-action on such that Z = /S1 is a
i i i i
F F F
closed, smooth manifold. Set n = dim( );
i i
F
(3) for every periodic orbit x, there is a smooth submanifold M of dimension > 2 such that
x
ℜ ⊂
dλ = 0 and x , and the almost complex structure J˜can be chosen such that L J˜= 0
|ℜx ⊂ ℜx X
along x. f
Let u= (a,u): C D R M be a J-holomorphic map with finite energy. Put z = es+2πit.
1
− → ×
Assume that there exists a sequence s such that u˜(s ,t) x(kTt) in C (S1,M) as
i i ∞
→ ∞ −→
i for someek Z, where T is fthe minimal periodic. Following Hofer (see [14]) we introduce a
→ ∞ ∈
convenient local coordinates near the periodic orbit x. Since S1 = R/Z, we work in the cofvering
space R S1.
→
8
Lemma 2.5. Let (M,λ) be a (2n 1)-dimensional compact manifold, and let x(kt) be a k-periodic
−
orbit with the minimal periodic T. Then there is an open neighborhood U S1 R2n 2 of S1 0
−
⊂ × ×{ }
with coordinates (ϑf,w , ,w ) and an open neighborhood M of x(t)t R and a
1 2n 2
diffeomorphism ψ : U ···mapp−ing S1 0 onto x(t)t R sℑuch⊂that { | ∈ }
→ ℑ ×{ } { | ∈ }
f
ψ λ = gλ , (24)
∗ 0
where λ = dϑ+ w dw and g : U R is a smooth function satisfying
0 i n+i 1
− →
P
g(ϑ,0) = T, dg(ϑ,0) = 0 (25)
for all ϑ S1.
∈
Remark 2.6. We call the coordinate system (ϑ,w) in Lemma 2.5 a pseudo-Darboux coordinate
system, and call the following transformation of two local pseudo-Darboux coordinate systems
(ϑ,w) (ϑˇ,w), ϑˇ= ϑ+ϑ (26)
0
−→
a canonical coordinate transformation, where ϑ is a constant.
0
The following theorem is well-known (see [5,18,32])
Theorem 2.7. Suppose that satisfies Assumption 2.4. Let u : C D R M˜ be a J-
1
F − → ×
holomorphic map with finite energy. Put z = es+2π√ 1t. Then
−
lim u(s,t) = x(kTt)
s
→∞
in C∞(S1) for some kT-periodic orbit x, aned there are constants ℓ0, ϑ0 such that for any 0 < c <
min{21, C212} and for all n = (n1,n2) ∈Z20
≥
∂n[a(s,t) kTs ℓ ] e cs (27)
0 n − | |
| − − | ≤ C
∂n[ϑ(s,t) kt ϑ ] e cs (28)
0 n − | |
| − − | ≤ C
∂nw(s,t) e cs, (29)
n − | |
| | ≤ C
where C are constants. Here (ϑ,w) is a pseudo-Darboux coordinate near the periodic orbit x.
n
Let x be a minimal periodic orbit. We choose a local pseudo-Darboux coordinate on an open
o
set M nearx . Let beanopensetsuchthat iscompactandx .Wefixapositive
o o
ℑ ⊂ O O ⊂ ℑ ⊂ O
constantC.DenotebyStheclassofJ-holomorphicmapswithfiniteenergyu :[s , ) S1 R M˜
′0 ∞ × → ×
satisfyingf
(i) sup E (u) 1~, where ~ is the constant in Theorem 5.2 and Lemma 5.3,
φ∈Φ φ ≤ 2
(ii) lim u(s,t)= x(kTt) in C (S1) with x ,
s ∞
→∞ ⊂ O
(iii) u([s , ) S1) lie in the pseudo-Darboux coordinate system on ,
′0 ∞e× ℑ
(iv) there is a ball D (s ,t ) [s , ) S1 such that u(D (s ,t )) and a(D (s ,t )) C.
e 1 0 0 ⊂ ′0 ∞ × 1 0 0 ⊂ O | 1 0 0 | ≤
In our next paper [19] we will prove the following theorem.
e
9
Theorem 2.8. Let u S. Then the constants C in Theorem 2.7 depend only on ,s ,C, n, ~,
n 1 0
∈ C
c and . Moreover, we have
O
ℓ C , (30)
0 1
| | ≤
where C depends only on ~, C ,c,s ,C and .
1 n 0
O
Following [14] we introduce functions
a (s,t) = a(s,t) ks, ϑ (s,t) = ϑ(s,t) kt. (31)
⋄ ⋄
− −
Denote
£ = (a ,ϑ ). (32)
⋄ ⋄
Set b = 0, b (w) = w , i = 1, ,n 1, and
i n 1+i i
− ∀ ··· −
∂ ∂
e = b , i = 1, ,2n 2.
i i
∂w − ∂ϑ ··· −
i
Thenξ =span e , ,e . Denote J˜e = J˜ e .Inthebasis ∂ ,e , ,e ,theReeb vector
1 2n 2 i ij i ϑ 1 2n 2
{ ··· − } ··· −
field can be re-written as
P
1 1
X = ∂ + e (g)e e (g)e . (33)
g ϑ g2  n−1+i i − i−n+1 i
i n 1 i n
≤X− X≥
 
Let X˜ = 1 (e g, ,e g, e g, , e g). By (18), we have
g2 n ··· 2n−2 − 1 ··· − n−1
£ +J£ = h, w +Jw +a X˜ a J˜X˜ = 0.
s t s t t s
−
where
e
h = (Σb (w)(w ) +(ϑ +Σb (w)(w ) )(Σf w ), Σb (w)(w ) (ϑ +Σb (w)(w ) )(Σf w ))
i i t t i i t i i i i s t i i s i i
− −
1
and f = ∂ g(ϑ,τw)dτ. Let V = £ and g = h . Denote
i 0 wi t t
R
Eˆ(£) = £ 2+ £ 2dsdt. (34)
s t
k k k k
ZΣ
In our next paper [19] we will prove the following theorem.
Theorem 2.9. Suppose that M satisfies the Assumption 2.4. Let u: [ R,R] S1 R M be a
− × → ×
J-holomorphic maps with finite energy. Assume that
f f
(i) supE (u, R s R)+Eˆ(£) 1~,
φ − ≤ ≤ ≤ 2
φ Φ
∈
(ii) u([ R,R] S1) lie in a pseudo-Darboux coordinate system (ϑ,w) on ,
− × ℑ
(iii) nu( R, ) C , nu(R, ) C , where n = (n ,n ),
e k∇ − · kL2(S1) ≤ 2 k∇ · kL2(S1) ≤ 2 1 2
n1,n2 2 n1,n2 2
≤ ≤
P P
Then there exist constants > 0 and 0 < c< 1 depending only on J˜and C such that
C1 2 2
w (s,t) e c(R s), £ e c(R s), s R 1, (35)
1 − −| | 1 − −| |
|∇ | ≤ C |∇ | ≤ C ∀| | ≤ −
10