NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

HOLOMORPHIC OPEN BOOK DECOMPOSITIONS 31
dλ ′ induces an area form on each fiber Fϑ with K consisting of closed orbits of the
Reeb vector field Xλ ′,andλ′ orients K as the boundary of (Fϑ,dλ ′ ).
We refer to a contact form λ ′ above as a Giroux contact form. Note that λ ′ is not unique
and that it is in general different from the original contact form λ. The following
theorem by Giroux guarantees existence of such open book decompositions, and it
contains a uniqueness statement as well (see also [16, Proposition 2]).
THEOREM 1.4 ([16, Theorem 3])
Every co-oriented contact structure ξ = ker λ on a closed 3-dimensional manifold is
supported by some open book. Conversely, if two contact structures are supported by
the same open book, then they are diffeomorphic.
In the topological category, it is possible to modify an open book decomposition
such that the pages of the new decomposition have lower genus at the expense of
increasing the number of connected components of K. It was not known for some
time whether a similar statement could also be made in the context of supporting open
book decompositions. In particular, it was unclear whether every contact structure
was supported by an open book decomposition whose pages were punctured spheres
(planar pages). The author and his collaborators could resolve the Weinstein conjecture
for contact forms inducing a planar contact structure in 2005 (see [4]). So the question
of whether all contact structures are planar became a priority, which prompted Etnyre
to address it in [14]. He showed that overtwisted contact structures always admit
supporting open book decompositions with planar pages, but many contact structures
do not. Since then, planar open book decompositions have become an important tool
in contact geometry.
In this paper, we will prove that every contact structure has a supporting open book
decomposition such that the pages solve a homological perturbed Cauchy-Riemann
type equation which we now describe after introducing some notation. We write
πλ = π : TM → ξ for the projection along the Reeb vector field Xλ. Fix a complex
multiplication J : ξ → ξ so that the map ξ ⊕ ξ → R, definedby
(h, k) → dλ(h, J k),
defines a positive definite metric on the fibers. We call such complex multiplications
compatible (with dλ). The equation of interest here is the following nonlinear firstorder
elliptic system. The solutions consist of 5-tuplets (S,j,Ɣ,ũ, γ ) where (S,j)
is a closed Riemann surface with complex structure j, Ɣ ⊂ S is a finite subset,
ũ = (a,u) : ˙S → R × M is a proper map with ˙S = S \ Ɣ,andγ is a 1-form on S so