Lagrangian antisurgery

Abstract

We describe an operation which modifies a Lagrangian submanifold $L$ in a symplectic manifold $(M, \omega)$ such as to produce a new immersed Lagrangian submanifold $L'$, which as a smooth manifold is obtained by surgery along a framed sphere in $L$. Intuitively, this can be described as collapsing an isotropic disc with boundary on $L$ to a point. The inverse operation generalizes classical Lagrangian surgery. We also describe corresponding immersed Lagrangian cobordisms between $L$ and $L'$ . After removal of their singular locus, we obtain examples of embedded Lagrangian cobordisms with precisely two ends. As an application, we use this construction to produce interesting examples of Lagrangian cobordisms between Clifford and Chekanov tori.