NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

HOLOMORPHIC OPEN BOOK DECOMPOSITIONS 77
spaces, that is,
ϕl − ϕkC k+1,γ (B ′′′ ) ≤ C ( ˆFτl − ˆFτkC k,γ (B ′′ )
+ˆGτl − ˆGτkC k,γ (B ′′ ) +ϕl − ϕkC k,γ (B ′′ )).
The sequence ( ˆFτk) now converges in the C0,α2-norm, and the sequence ( ˆGτk) converges
in any Hölder norm. We obtain with the above regularity estimate C1,α2-convergence of
the sequence (ϕk), and composing with α−1 yields C1,α4-convergence
of (fτk) and (µτl).
τk
Invoking Theorem 3.7 again then improves the convergence of the transformations
ατk, α−1 to C τk 2,α4.
We now iterate the procedure using the regularity estimate for the
Cauchy-Riemann operator in Hölder space and the estimate for the Beltrami equation
in Theorem 3.7.
Theorem 3.2 follows if we apply the implicit function theorem to the limit solution
(S,jτ0, ũτ0 = (aτ0,uτ0),γτ0), hence we obtain the same limit for every sequence {τk},
and we obtain convergence in C∞ .
4. Conclusion
The following remarks tie together the loose ends and prove the main result, Theorem
1.6. We start with a closed 3-dimensional manifold with contact form λ ′ . Giroux’s
theorem, Theorem 1.4, then permits us to change the contact form λ ′ to another
contact form λ such that ker λ = ker λ ′ and such that there is a supporting open book
decomposition with binding K consisting of periodic orbits of the Reeb vector field
of λ. Invoking Proposition 2.4, we construct a family of 1-forms (λδ)0≤δ