NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

46 CASIM ABBAS
and
Jδε2 = xγ1(r)h(r)
η2 +
r
y
r2h(r) η1
= − 1
r xyγ1(r)h(r)
xy
+
r3
ε2
h(r)γ1(r)
1
+
r x2 y
γ1(r)h(r) +
2
r3
ε1.
γ1(r)h(r)
Inserting x = r cos φ, y = r sin φ, and demanding that the limit is φ-independent as
r → 0, we arrive at the condition that rh(r) γ1(r) ≡±1 for small r. Recalling that
we need h>0, we obtain
h(r) = 1
rγ1(r)
for small r.
As usual, we continue Jδ to an almost-complex structure ˜Jδ on R × M by setting
˜Jδ(θ,r,φ) ∂
∂τ
:= Xδ(θ,r,φ),
where τ denotes the coordinate in the R-direction. We emphasize that ˜Jδ also makes
sense for δ = 0. We now arrange r(s) in (2.6) such that the Giroux leaves ũα = (aα,uα)
become ˜J0-holomorphic curves. ∗ We compute for r ≤ 1 − ε0
∂sũα + ˜J0(uα)∂tũα = γ1(r) ∂ ∂
+ r′
∂τ ∂r +
˜J0(uα)
∂
∂θ
= γ1(r) ∂ ∂
+ r′
∂τ ∂r + ˜J0(uα) γ1(r) Xδ(uα)
+ ˜J0(uα)
∂
− γ1(r)Xδ(uα)
∂θ
′ ∂
= r
∂r + ′
˜J0(uα)
γ 1 (r)
γ1(r)
µ(r)
∂ ∂
− γ2(r)
∂φ ∂θ
= r ′ − γ ′ 1 (r)
∂
µ(r)h(r) ∂r ,
hence the Giroux leaves satisfy the equation if we choose r to be a solution of the
ordinary differential equation
r ′ (s) =
γ ′ 1 (r(s))
µ(r(s)) h(r(s)) .
∗ The calculation shows that we can make them Jδ-holomorphic for all δ ≥ 0 near the binding.