Reference Request: "Neck Stretching Procedure" (In Symplectic Field Theory)

Question

I've been reading some papers in Symplectic Geometry which refer to something called "Stretching the neck", and give reference to Eliashberg, Givental and Hofer's SFT paper (http://arxiv.org/abs/math/0010059) I ran a search after the words "neck" and "stretch" and they do not appear in the paper, I guess the theory is there under a different name. Since it's a long and complicated paper and I just want to understand the general idea and results of this "neck stretching" so I can continue reading those other papers i'm currently reading, Could someone please refer me to the part in the the SFT paper which deals with neck stretching? or alternatively, refer me to some other paper/survey/someone's thesis/book in which it is clearly explained?

Thank you

Answer 1

Neck-stretching is a deformation of an almost complex structure in a neighbourhood of a hypersurface. In the Eliashberg-Givental-Hofer paper, neck-stretching is called "splitting along a contact submanifold" (see Section 1.3) - the description of the almost complex structure is sketched at the end of Section 1.4.

A more detailed account of neck-stretching (and the behaviour of punctured pseudoholomorphic curves under this deformation) is given in Bourgeois-Eliashberg-Hofer-Wysocki-Zehnder (e.g. Section 3.4)

http://arxiv.org/abs/math/0308183

and in Cieliebak-Mohnke (e.g. Section 2.7)

http://projecteuclid.org/euclid.jsg/1154467631

Answer 2

You'll find an extensive discussion with applications of the "neck stretching" technique in Jonathan Evans' thesis, Symplectic topology of some Stein and rational surfaces (chapter 5).