NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

34 CASIM ABBAS
• uτ ( ˙S) ∩ uτ ′( ˙S) =∅ if τ = τ ′ ;
• M\K =
τ∈S 1 uτ ( ˙S);
• the projection P onto S 1 defined by p ∈ uτ ( ˙S) ↦→ τ is a fibration;
• the open book decomposition given by (P,K) supports the contact structure
ker λ, and λ is a Giroux form.
Here is a very brief outline of the argument. The reader is invited to skip forward to
Section 4 to see in more detail how all the partial results of this paper are tied together
to prove the main result. In Section 2, we find a Giroux contact form which has a
certain normal form near the binding. Following an argument by Wendl ([39], [38]),
we will then almost be able to turn the Giroux leaves into solutions of (1.1) without
harmonic form except for the fact that we have to accept a confoliation form instead
of a contact form. Pick one of these Giroux leaves as a starting point. The next step is
to prove a result which permits us to perturb the Giroux leaf into a genuine solution
of (1.1) while simultaneously perturbing the confoliation form slightly into a contact
form. This is where the harmonic form in (1.1) is required. We actually obtain a local
family of nearby solutions, not just one. In Section 3, we prove a compactness result
which extends the local family of solutions into a global one. The remarkable fact is
that there is a compactness result in the context of this paper, although there is none
in general for the perturbed holomorphic curve equation. The special circumstances
in this paper imply a crucial a priori bound which implies that a sequence of solutions
has a pointwise convergent subsequence with a measurable limit. The objective is then
to show that the regularity of this limit is much better, that it is actually smooth.
We consider two solutions (S,j,Ɣ,ũ, γ ) and (S ′ ,j ′ ,Ɣ ′ , ũ ′ ,γ ′ ) equivalent if there
exists a biholomorphic map φ :(S,j) → (S ′ ,j ′ ) mapping Ɣ to Ɣ ′ (preserving the
enumeration) so that ũ ′ ◦ φ = ũ. We will often identify a solution (S,j,Ɣ,ũ, γ ) of
(1.1) with its equivalence class [S,j,Ɣ,ũ, γ ].WenotethatwehaveanaturalR-action
on the solution set by associating to c ∈ R and [S,j,Ɣ,ũ, γ ] the new solution
c + [S,j,Ɣ,ũ, γ ] = [S,j,Ɣ,(a + c, u),γ], ũ = (a,u).
A crucial concept for our discussion is the notion of a finite energy foliation F .
Definition 1.7 (Finite energy foliation)
A foliation F of R × M is called a finite energy foliation if every leaf F is the image
of an embedded solution [S,j,Ɣ,ũ, γ ] of the equations (1.1), that is,
F = ũ( ˙S),