NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

HOLOMORPHIC OPEN BOOK DECOMPOSITIONS 33
Hence a is a harmonic function on S and therefore constant. So far, we also know that
the image of u lies on a Reeb trajectory, and it remains to show that this trajectory is
actually periodic.
Let τ : ˜S → S be the universal covering map. The complex structure j lifts to a
complex structure ˜j on ˜S. Now pick smooth functions f, g on ˜S such that
dg = τ ∗ γ =: ˜γ, −df = τ ∗ (γ ◦ j) = ˜γ ◦ ˜j.
Then the map u ◦ τ : ˜S → M satisfies
(u ◦ τ) ∗ λ = df.
Theimageofu◦ τ lies on a trajectory x of the Reeb vector field in view of
D(u ◦ τ)(z)ζ = Df (z)ζ · Xλ (u ◦ τ)(z) ,
hence (u ◦ τ)(z) = x(h(z)) for some smooth function h on ˜S, and it follows that, after
maybe adding a constant to f ,wehave
(u ◦ τ)(z) = x f (z) .
The function f does not descend to S. If it did, it would have to be constant since it
is harmonic. On the other hand, this would imply that u is constant in contradiction to
our assumption that it is not. Therefore, there is a point q ∈ S and two lifts z0,z1 ∈ ˜S
such that f (z0) >f(z1).Letℓ : S 1 → S be a loop which lifts to a path α :[0, 1] → ˜S
with α(0) = z0 and α(1) = z1. Considering the map
v := u ◦ ℓ : S 1 −→ M,
we see that v(t) = (u ◦ τ ◦ α)(t) = x f (α(t)) and x(f (z0)) = x(f (z1)), that is, the
trajectory x is a periodic orbit. Hence the image of u is a periodic orbit for the Reeb
vector field.
The following is the main result of this paper.
THEOREM 1.6
Let M be a closed 3-dimensional manifold, and let λ ′ be a contact form on M. Then
the following holds for a suitable contact form λ = fλ ′ ,wheref is a positive function
on M. There exists a smooth family (S,jτ ,Ɣτ , ũτ = (aτ ,uτ ),γτ )τ∈S 1 of solutions to
(1.1) for a suitable compatible complex structure J :kerλ → ker λ such that
• all maps uτ have the same asymptotic limit K at the punctures, where K is a
finite union of periodic trajectories of the Reeb vector field Xλ;