NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

44 CASIM ABBAS
the standard complex structure on the cylinder [0, +∞) × S 1 after introducing polar
coordinates near the punctures. Moreover, all the punctures are positive ∗ the family
(ũα)0≤α≤2π is a finite energy foliation, and the curves ũα are ˜Jδ-holomorphic near the
punctures.
Proof
We parameterize the leaves of the open book decomposition uα : ˙S → M, 0 ≤ α<
2π, and we assume that they look as follows near the binding:
uα :[0, +∞) × S 1 −→ S 1 × D1,
uα(s, t) = t,r(s)e iα ,
(2.6)
where r are smooth functions with lims→∞ r(s) = 0 to be determined shortly. We
use the notation (r, φ) for polar coordinates on the disk D = D1. We identify some
neighborhood U of the punctures of ˙S with a finite disjoint union of half-cylinders
[0, +∞) × S1 . Recall that the binding orbit is given by
t
x(t) = , 0, 0 , 0 ≤ t ≤ 2πγ1(0)
γ1(0)
and that it has minimal period T = 2πγ1(0). We define smooth functions aα : ˙S → R
by
s
aα(z) := 0 γ1
′ ′ r(s ) ds if z = (s, t) ∈ [0, +∞) × S1 ⊂ U,
0 if z/∈ U
so that
u ∗
α λ0 ◦ j = daα,
where j is a complex structure on ˙S which equals the standard structure i on [0, +∞)×
S 1 (i.e., near the punctures). We want to turn the maps ũα = (aα,uα) : ˙S → R × M
into ˜J0-holomorphic curves for a suitable almost-complex structure ˜J0 on R × M.
Recall that the contact structure is given by
∂ ∂
ker λδ = Span{η1,η2} = Span , −γ2(r)
∂r ∂θ
We define complex structures Jδ :kerλδ → ker λδ by
Jδ(θ,r,φ) − γ2(r) ∂
∂θ
∂
+ γ1(r) := −
∂φ
1
h(r)
∗ A puncture pj is called positive for the curve (aα,uα) if limz→pj aα(z) =+∞.
∂
+ γ1(r) .
∂φ
∂
∂r
(2.7)