Non-Hamiltonian isotopic Lagrangians on the one-point blow-up of CP^2

Abstract

We show that two Hamiltonian isotopic Lagrangians in (CP^2,\omega_\textup{FS}) induce two Lagrangian submanifolds in the one-point blow-up (\widetilde{CP}^2,\widetilde{\omega}_\rho) that are not Hamiltonian isotopic. Furthermore, we show that for any integer k>1 there are k Hamiltonian isotopic Lagrangians in (CP^2,\omega_\textup{FS}) that induce k Lagrangian submanifolds in the one-point blow-up such that no two of them are Hamiltonian isotopic.