  In this paper, we consider the initial value problem for the quintic,
defocusing nonlinear Schr\"odinger equation on $\Bbb T^2$ with general data in
the critical Sobolev space $H^{\frac{1}{2}} (\Bbb T^2)$. We show that if a
solution remains bounded in $H^{\frac{1}{2}} (\Bbb T^2)$ in its maximal
interval of existence, then the solution is globally well-posed in $\Bbb T^2$.

Analysis of PDEs