On Certain Lagrangian Submanifolds of $S^2\times S^2$ and $C{P}^n$

We consider various constructions of monotone Lagrangian submanifolds of $C P^n, S^2\times S^2$,andquadrichypersurfacesof$C P^n$\mathrm{.}In$S^2\times S^2$and$C P^2$ 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 $C{P}^2$ 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 $C P^n$whichcanbeunderstoodeitherintermsofthegeodesicflowon$T^*S^n$ 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.