I ask this question because I want to better understand the argument of neck elongation in symplectic geometry and the possible conclusions in my environment.

Suppose I have a Lefschetz fibrillation $ E $ about $ M $, Where $ M $ is an exact symplectic manifold (possibly with ends). To let $ C $ be a contact type codimension $ 1 $ Subtype of $ M $, As far as I understand, when I stretch my neck $ C $ I create a family of manifolds $ M_t $. $ t in Bbb R ^ + $ (With $ M_0 = M $) all symplectomorph too $ M $"Showing the stretching of the neck" around $ C $ at the time $ t geq 0 $,

According to the thesis of Evans **(E)**This process tends to a limit $ M _ { infty} $ which is obtained from $ M $ by cutting $ C $ and sticking the positive / negative part of the symplectization of $ C $ (In the work it is called as $ Bbb S _ { pm} (C) $, I believe that this process should similarly lead to a family of Lefschetz fibrations $ E_t $ about $ M_t $ and a limiting vibration $ E _ { infty} $ (I believe we put something trivial on the ends we introduce).

I am interested in understanding the moduli space of pseudo-holomorphic portions of such a vibration. Let's say I have chosen a generic compatible AC. structure $ J $ and I remember that I have no bubbling due to my accuracy assumption. From my stretching process, a family of manifolds arises $ mathcal {M} _t $, What can I say about this family of module spaces? How do I deal with the borderline case? $ mathcal {M} _ { infty} $? Is it somehow a limit to the family of spaces that are parametrized by the positive realities?

I think I can come up with a naïve rationale for my claim, but trying to push it to the limit, I expect many technical issues and I do not know them yet.

Can somebody point out to me if there are obvious obstacles / difficulties in what I am trying to understand and if there is literature that somehow sheds light on what I am trying to understand?

Thanks in advance for all comments!

**references**

**(E)** Jonathan David Evans – Symplectic topology of some stone and rational surfaces