Filling by holomorphic discs and its applications. (English) Zbl 0731.53036

Eliashberg, Yakov

Geometry of low-dimensional manifolds. 2: Symplectic manifolds and Jones-Witten-Theory, Proc. Symp., Durham/UK 1989, Lond. Math. Soc. Lect. Note Ser. 151, 45-72 (1990).

This is a very interesting survey which discusses applications of the technique of filling by holomorphic discs to different problems in symplectic and contact geometry. Many of the results concern properties of contact 3-manifolds which are bound symplectic 4-manifolds. For example, the author sketches a proof of the fact that the standard structure is the only contact structure on a 3-sphere which bounds in this way. He also gives conditions under which a Legendrian curve (i.e. a curve everywhere tangent to the contact distribution) is isotopic through Legendrian curves to one in standard position. The applications are discussed in some detail, but unfortunately the proof of the main technical lemma has never been written down. Many of the results can now be sharpened by the use of the author’s new theory of tight contact structures.
[For the entire collection see Zbl 0722.00024.]

Reviewer: Dusa McDuff (Stony Brook)

MSC:

53D05	Symplectic manifolds (general theory)
53D10	Contact manifolds (general theory)
57M25	Knots and links in the \(3\)-sphere (MSC2010)
32F99	Geometric convexity in several complex variables

Keywords:

survey; filling; holomorphic discs; contact 3-manifolds; symplectic 4-manifolds; Legendrian curve; tight contact structures

Citations:

Zbl 0722.00024