Lagrangian antisurgery

TL;DR

This paper introduces a new Lagrangian antisurgery operation in symplectic geometry that modifies Lagrangian submanifolds, producing immersed cobordisms and examples connecting different types of tori.

Contribution

It defines a novel antisurgery operation on Lagrangian submanifolds, extending classical surgery, and constructs explicit immersed Lagrangian cobordisms with applications to Clifford and Chekanov tori.

Findings

Introduces a Lagrangian antisurgery operation.

Constructs immersed Lagrangian cobordisms with two ends.

Provides examples connecting Clifford and Chekanov tori.

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.