NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

HOLOMORPHIC OPEN BOOK DECOMPOSITIONS 79
The complex structure(s) we chose earlier in (2.7) had the following form near the
binding with respect to the basis {η1,η2}:
J (θ,r,φ) =
0 −rγ1(r)
0
.
1
rγ1(r)
The induced complex structure jτ on the surface is then given by
jτ (z) = [πλTv0(z)] −1
[Tφτ v0(z) ] −1 J φτ (v0(z))
Tφτ v0(z) πλTv0(z).
With v0(s, t) = (t,r(s),α),wefindthat
so that
and
jτ =
πλTv0(s, t) =
r ′ (s) 0
0
γ ′ 1 (s)
µ(r(s))
−τA(r)rγ1(r) − rγ1(r)γ ′ 1 (r)
r ′ µ(r)
r ′ µ(r)
rγ1(r)γ ′ 1 (r)(1 + τ 2 A 2 (r)r 2 γ 2 1
(r)) τA(r)rγ1(r)
−τA(r)rγ1(r) −1
=
1 + τ 2A2 (r)r2γ 2
1 (r) τA(r)rγ1(r)
−1 0
= j0 + τA(r) γ1(r)
τA(r)γ1(r) 1
jτ − jσ = A(r)rγ1(r)(τ − σ )
using the fact that r(s) satisfies the differential equation
With v ∗ 0 λ = γ1(r) dt, we obtain
r ′ (s) = γ ′ 1 (r(s))γ1(r(s))r(s)
.
µ(r(s))
−1 0
(τ + σ )A(r)rγ1(r) 1
v ∗
0λ ◦ (jτ − jσ )|(s,t) = (τ − σ )A r(s) r(s)γ 2
1 r(s)
· (τ + σ )A r(s)
r(s)γ1 r(s) ds + dt .
,