Uniqueness of Lagrangians in \(T^*{\mathbb {R}}P^2\)

Abstract

We present a new and simpler proof of the fact that any Lagrangian \({\mathbb {R}}P^2\) in \(T^*{\mathbb {R}}P^2\) is Hamiltonian isotopic to the zero section. Our proof mirrors the one given by Li and Wu for the Hamiltonian uniqueness of Lagrangians in \(T^*S^2\), using surgery to turn Lagrangian spheres into symplectic ones. The main novel contribution is a detailed proof of the folklore fact that the complement of a symplectic quadric in \({\mathbb {C}}P^2\) can be identified with the unit cotangent disc bundle of \({\mathbb {R}}P^2\).