Lagrangian Approximation of Totally Real Concordances

Abstract

We show that a two-dimensional totally real concordance can be approximated by a Lagrangian concordance whose Legendrian boundary has been stabilised both positively and negatively sufficiently many times. The main applications that we provide are constructions of knotted Lagrangian concordances in arbitrary four-dimensional symplectiations, as well as of knotted Lagrangian tori in symplectisations of overtwisted contact three-manifolds.

arXiv comment22 pages, 5 figures. The main theorem has been corrected: it is not possible to preserve the totally real isotopy class when approximating a totally real concordance with a Lagrangian, in general we can only ensure that they are related by a smooth isotopy. In the new version the proof of the main theorem in Section 4.3 has been thoroughly expanded, and figures have been added