NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

HOLOMORPHIC OPEN BOOK DECOMPOSITIONS 37
Definition 2.1
Let θ ∈ S 1 = R/2πZ denote polar coordinates on the unit disk D ⊂ R 2 by (r, φ),
and let γ1,γ2 :[0, +∞) → R be smooth functions. A 1-form
λ = γ1(r) dθ + γ2(r) dφ
is called a local model near the binding if the following conditions are satisfied:
(1) the functions γ1,γ2,andγ2(r)/r 2 are smooth if considered as functions on the
disk D (in particular, γ ′ 1 (0) = γ ′ 2 (0) = γ2(0) = 0);
(2) µ(r) := γ1(r)γ ′ 2 (r) − γ ′ 1 (r)γ2(r) > 0 if r>0;
(3) γ1(0) > 0 and γ ′ 1 (r) < 0 if r>0;
(4) limr→0(µ(r)/r) = γ1(0)γ ′′
2 (0) > 0;
(5) κ := (γ ′′ ′′
1 (0)/γ 2 (0)) /∈ Z and κ ≤−1/2;
(6) A(r) = (1/µ 2 (r))(γ ′′
2 (r)γ ′ ′′
1 (r) − γ 1 (r)γ ′ 2
We explain some of the conditions above. First, since
(r)) is of order r for small r>0.
λ ∧ dλ = µ(r)dθ ∧ dr ∧ dφ = µ(r)
dθ ∧ dx ∧ dy,
r
the form λ is a contact form on S 1 × D. The Reeb vector field is given by
X(θ,r,φ) = γ ′ 2 (r)
µ(r)
∂
∂θ − γ ′ 1 (r)
µ(r)
∂
∂φ
The trajectories of X all lie on tori Tr = S 1 × ∂Dr:
We compute
and
=: α(r) ∂
∂θ
+ β(r) ∂
∂φ .
θ(t) = θ0 + α(r) t, φ(t) = φ0 + β(r) t. (2.1)
γ
lim α(r) = lim
r→0 r→0
′′
2 (r)
µ ′ (r)
γ
lim β(r) =−lim
r→0 r→0
′′
1 (r)
µ ′ (r)
= γ ′′
2 (0)
Recalling that ∂ ∂ ∂
= x − y , we obtain for r = 0
∂φ ∂y ∂x
X = 1 ∂
γ1(0) ∂θ ,
γ1(0)γ ′′
1
=
(0) γ1(0)
2
=− γ ′′
1 (0)
γ1(0)γ ′′
2 (0).