Toric constructions of monotone Lagrangian submanifolds in $\mathbb{CP}^2$ and $\mathbb{CP}^1 \times \mathbb{CP}^1$

TL;DR

This paper develops methods to construct and control monotone Lagrangian submanifolds in toric symplectic manifolds, recovering known examples and introducing new ones in $\

Contribution

It introduces a toric construction approach for monotone Lagrangian submanifolds, expanding the known examples in $\

Findings

Recovered known monotone Lagrangians in $\

Constructed new monotone Lagrangians in $\

Provided methods controlling monotonicity in toric settings

Abstract

We extract from a toric model of the Chekanov-Schlenk exotic torus in $\mathbb{CP}^2$ methods of construction of Lagrangian submanifolds in toric symplectic manifolds. These constructions allow for some control of the monotonicity. We recover this way some known monotone Lagrangians in the toric symplectic manifolds $\mathbb{CP}^2$ and $\mathbb{CP}^1 \times \mathbb{CP}^1$ as well as new examples.