NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

56 CASIM ABBAS
Restricting any of the solutions to a simply connected subset U ⊂ ˙S, we can
write γτ = dhτ for a suitable function hτ : U → R, and the maps
ũτ : U → R × M, ũτ = (aτ + hτ ,uτ )
are ˜J -holomorphic curves. If two such curves ũτ and ũτ ′ have an isolated intersection,
then the corresponding intersection number is positive (see [28], [5], or [27] for
positivity of (self-)intersections for holomorphic curves). We claim that
u0( ˙S) ∩ uτ ( ˙S) =∅, ∀ 0 0 is different from u0( ˙S), we conclude that if uτ and u0 intersect,
then the intersection point of the corresponding holomorphic curves ũτ and ũ0 must
be isolated. But on the other hand, this implies that uτ ′ and u0 would also intersect for
all τ ′ sufficiently close to τ by positivity of the intersection number showing that the
set O is open.
We conclude from the above that we have a sequence τk ↘ ˜τ and points pk,qk ∈ ˙S
such that uτk(pk) = u0(qk). Passing to a suitable subsequence, we may assume
convergence of the sequences (pk)k∈N and (qk)k∈N to points p, q ∈ S. Because of
u˜τ ( ˙S) ∩ u0( ˙S) =∅the points p, q must be punctures, and they have to be equal
z0 = p = q ∈ S\ ˙S. The reason for this is the following. The maps uτk,u0 are
asymptotic near the punctures to a disjoint union of finitely many periodic Reeb
orbits which are not iterates of other periodic orbits. Also, different punctures always
correspond to different periodic orbits. This follows from Giroux’s result and our
constructions in Section 2 of this paper.