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We establish the global well-posedness of the derivative nonlinearSchr\"odinger equation with periodic boundary condition in the Sobolev space$H^{\frac12}$, provided that the mass of initial data is less than $4\pi$. Thisresult matches the one by Miao, Wu, and Xu and its recent mass thresholdimprovement by Guo and Wu in the non-periodic setting. Below $H^{\frac12}$, weshow that the uniform continuity of the solution map on bounded subsets of$H^s$ does not hold, for any gauge equivalent equation.

Global well-posedness of the derivative nonlinear Schr\"odinger equation with periodic boundary condition in $H^{\frac12}$