Uniqueness of Lagrangians in $T^*RP2$

TL;DR

This paper provides a simplified proof that any Lagrangian real projective plane in its cotangent bundle is Hamiltonian isotopic to the zero section, extending known results with a new geometric identification.

Contribution

It offers a new, simpler proof of Lagrangian uniqueness in $T^*\mathbb{R}P^2$ and details the identification of the complement of a symplectic quadric with the cotangent disc bundle.

Findings

Any Lagrangian $\mathbb{R}P^2$ in $T^*\mathbb{R}P^2$ is Hamiltonian isotopic to the zero section.

The complement of a symplectic quadric in $\mathbb{C}P^2$ can be identified with the unit cotangent disc bundle of $\mathbb{R}P^2$.

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 $\mathbb{R}P^2$.