NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

HOLOMORPHIC OPEN BOOK DECOMPOSITIONS 67
The integral
d(fγ) vanishes by Stokes’s theorem since fγ is a smooth 1-form on
S
the closed surface S. The form da∧γ ◦jf is not smooth on S, but the integral vanishes
anyway for the following reason. As we have proved in the appendix, the form γ ◦ jf
is bounded near the punctures, and hence in local coordinates near a puncture it is of
the form
σ = F (w1,w2)dw1 + F2(w1,w2)dw2, w1 + iw2 ∈ C,
where F1,F2 are smooth (except possibly at the origin) but bounded. Passing to polar
coordinates via
φ :[0, ∞) × S 1 −→ C\{0}
φ(s, t) = e −(s+it) = w1 + iw2,
we see that φ ∗ σ has to decay at the rate e −s for large s. The form da has γ1(r(s)) ds
as its leading term. Computing the integral
Ɣ a(γ ◦ jf ) over small loops Ɣ around
the punctures and using Stokes’s theorem, we conclude that the contribution from
neighborhoods
of the punctures can be made arbitrarily small. Therefore, the integral
˙S da ∧ γ ◦ jf must vanish.
If is a volume form on S, then we may write u∗ 0λ ∧ γ = g · for a suitable
smooth function g. Defining
|u
˙S
∗
0λ ∧ γ | := |g| ,
˙S
we have
γ 2
L2 ,jf =
u
˙S
∗
0λ ∧ γ
≤ |u ∗
0λ ∧ γ |
˙S
≤u ∗
0 λL 2 ,jf γ L 2 ,jf ,
which implies the assertion.
We resume the proof of the compactness result, Theorem 3.2. All the considerations
which follow are local. The task is to improve the regularity of the limit fτ0 and the
nature of the convergence fτ → fτ0 . Because the proof is somewhat lengthy, we
organize it in several steps. For τ