# Uniqueness of real Lagrangians up to cobordism

## TL;DR

This paper establishes the uniqueness of real Lagrangian submanifolds up to cobordism in closed symplectic manifolds and classifies them in specific cases like P^2 and S^2S^2, highlighting their isotopy types.

## Contribution

It proves the cobordism uniqueness of real Lagrangians and classifies their isotopy types in key symplectic manifolds.

## Key findings

- Real Lagrangians are unique up to cobordism in closed symplectic manifolds.
- In P^2, real Lagrangians are unique up to Hamiltonian isotopy.
- In S^2S^2, real Lagrangians are either Hamiltonian isotopic to the antidialgonal sphere or Lagrangian isotopic to the Clifford torus.

## Abstract

We prove that a real Lagrangian submanifold in a closed symplectic manifold is unique up to cobordism. We then discuss the classification of real Lagrangians in $\mathbb{C} P^2$ and $S^2\times S^2$. In particular, we show that a real Lagrangian in $\mathbb{C} P^2$ is unique up to Hamiltonian isotopy and that a real Lagrangian in $S^2\times S^2$ is either Hamiltonian isotopic to the antidialgonal sphere or Lagrangian isotopic to the Clifford torus.