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

##### MSC:
 53D05 Symplectic manifolds, general 53D10 Contact manifolds, general 57M25 Knots and links in the $$3$$-sphere (MSC2010) 32F99 Geometric convexity in several complex variables