On certain Lagrangian submanifolds of $S^2\times S^2$ and $\mathbb{C}\mathrm{P}^n$

Abstract

We consider various constructions of monotone Lagrangian submanifolds of $ℂP^n$, $S^2\times S^2$, and quadric hypersurfaces of $ℂP^n$. In $S^2\times S^2$ and $ℂP^2$ we show that several different known constructions of exotic monotone tori yield results that are Hamiltonian isotopic to each other, in particular answering a question of Wu by showing that the monotone fiber of a toric degeneration model of $ℂP^2$ is Hamiltonian isotopic to the Chekanov torus. Generalizing our constructions to higher dimensions leads us to consider monotone Lagrangian submanifolds (typically not tori) of quadrics and of $ℂP^n$ which can be understood either in terms of the geodesic flow on $T^{\ast }{S}^n$ or in terms of the Biran circle bundle construction. Unlike previously known monotone Lagrangian submanifolds of closed simply connected symplectic manifolds, many of our higher-dimensional Lagrangian submanifolds are provably displaceable.