Large-Data Global Criteria for Two-Dimensional Critical-Energy Schrödinger Maps

We study large-data continuation below the critical energy for the two-dimensional sphere-valued Schrödinger map \[ \partial_t u=u\timesΔu, \qquad u:[0,T_+)\times\mathbb{R}^2\to\mathbb{S}^2, \qquad E(u_0)<4π. \] Working directly with the original equation through the complex tangential fields $W_m=\