Global well-posedness of the derivative nonlinear Schrödinger equation with periodic boundary condition in H12

We establish the global well-posedness of the derivative nonlinear Schrödinger equation with periodic boundary condition in the Sobolev space H12, provided that the mass of initial data is less than 4π. This result matches the one by Miao, Wu, and Xu and its recent mass threshold improvement by Guo and Wu in the non-periodic setting. Below H12, we show that the uniform continuity of the solution map on bounded subsets of Hs does not hold, for any gauge equivalent equation.