NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

HOLOMORPHIC OPEN BOOK DECOMPOSITIONS 43
and, in particular,
Xδ(θ,r,φ) = ∂
∂φ for r ≥ 1 − ε0, (2.5)
which implies that the Reeb vector fields Xδ converge as δ ↘ 0. In addition to λδ being
contact forms for δ>0, we also want the given open book decomposition to support
ker λδ, hence Xδ needs to be transverse to the pages of the open book decomposition
which is equivalent to γ ′ 1 (r) = 0. A curve γ (r) fulfilling these conditions can clearly
be constructed.
The following result shows that we can always assume that a Giroux contact form is
equal to any of the forms provided by Proposition 2.4.
PROPOSITION 2.5
Let M be a closed 3-dimensional manifold with contact structure ξ. Then, for every
δ>0, there is a diffeomorphism ϕδ : M → M such that ker λδ = ϕ∗ξ where λδ is
given by Proposition 2.4.
Proof
Existence of an open book decomposition supporting ξ follows from the existence
part of Giroux’s theorem. On the other hand, Proposition 2.4 yields contact forms λδ
such that ker λδ is also supported by the same open book decomposition as ξ for any
δ>0. By the uniqueness part of Giroux’s theorem, ξ and ker λδ are diffeomorphic.
It follows from our previous construction of the forms λδ that λ0 satisfies λ0 ∧dλ0 > 0
on ∂W ×D1−ε0 and that λ0 = dφ otherwise. For δ → 0, the Reeb vector fields Xδ will
converge to some vector field X0, which is the Reeb vector field of λ0 if r