Lagrangian Approximation of Totally Real Concordances

TL;DR

This paper demonstrates that totally real concordances can be approximated by Lagrangian concordances with stabilized Legendrian boundaries, enabling the construction of knotted Lagrangian structures in symplectic and contact manifolds.

Contribution

It introduces a method to approximate totally real concordances by Lagrangian ones with stabilized Legendrian boundaries, facilitating new constructions in symplectic topology.

Findings

Construction of knotted Lagrangian concordances in four-dimensional symplectizations

Creation of knotted Lagrangian tori in overtwisted contact manifolds

Approximation technique for totally real to Lagrangian 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.