NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

HOLOMORPHIC OPEN BOOK DECOMPOSITIONS 55
Remark 3.3
We may assume without loss of generality that v0 ≡ u0 and f0 ≡ 0.Ifz ∈ ˙S,thenwe
denote by T (z) > 0 the positive return time of the point u0(z); that is, we have
T (z) := inf T>0 φT (u0(z)) ∈ u0( ˙S) < +∞.
We claim that the return time z ↦→ T (z) extends continuously over the punctures of
the surface, and that therefore there is an upper bound
T := sup T (z) < ∞.
z∈ ˙S
Using (2.1)and(2.6), we note that, asymptotically near the punctures, φT (u0(s, t)) ∈
S 1 × R 2 has the following structure:
φT (u0(s, t)) = t + α(r(s))T, r(s) exp[i(α0 + β(r(s))T )] ,
where r(s) is a strictly decreasing function, α0 is some constant, and α(r),β(r) are
suitable functions for which the limits limr→0 β(r) and limr→0 α(r) exist and are not
zero. Hence, if T = T (u0(s, t)) is the positive return time at the point u0(s, t), then
T u0(s, t) =
and therefore the limit for s →+∞exists.
2π
|β(r(s))| ,
The remainder of this section is devoted to the proof of Theorem 3.2. We recall
that the functions aτ and fτ satisfy the Cauchy-Riemann type equation (2.9)whichis
¯∂jτ (aτ + ifτ ) = u ∗
0 λ ◦ jτ − i(u ∗
0 λ) − γτ − i(γτ ◦ jτ ),
where the complex structure jτ is given by (2.8)or
jτ (z) = πλTu0(z) −1 Tφfτ (z)(u0(z)) −1 · J φfτ (z)(u0(z))
Tφfτ (z) u0(z) πλTu0(z),
and that γτ is a closed 1-form on S with d(γτ ◦ jτ ) = 0.
The following L ∞ -bound is the crucial ingredient for the compactness result. We
claim that
sup
0≤τ