On ill-posedness for the one-dimensional periodic cubic Schrodinger equation

We prove the ill-posedness in $H^s(\T)$, $s<0$, of the periodic cubic Schr\"odinger equation in the sense that the flow-map is not continuous from $H^s(\T)$ into itself for any fixed $t\neq 0$. This result is slightly stronger than the one obtained by Christ-Colliander-Tao where the discontinuity of the solution map is established. Moreover our proof is different and clarifies the ill-posedness phenomena. Our approach relies on a new result on the behavior of the associated flow-map with respect to the weak topology of $L^2(\T)$.