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 $${\mathrm{\partial }}_{t}u=u\times \mathrm{\Delta }u,u:[0,T_{+})\times {\mathbb{R}}^2\to {\mathbb{S}}^2,E(u_0)<4\pi \mathrm{.}$$ Working directly with the original equation through the complex tangential fields $W_m={\mathrm{\partial }}_{m}u+\mathrm{i}u\times {\mathrm{\partial }}_{m}u$ and the ambient connections ${\mathrm{\Gamma }}_{\alpha }$, we prove frequency-envelope bounds governed by the critical norm $\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_{{\mathrm{\ell }}^2{L}_{t,x}^4}$. As consequences, $T_{+}=\mathrm{\infty }$ if $\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_{{\mathrm{\ell }}^2{L}^4([0,T_{+})\times {\mathbb{R}}^2)}<\mathrm{\infty }$, and also if a terminal one-window high-frequency tail ${\mathcal{A}}_{\mathrm{w}\mathrm{i}\mathrm{n}}(t)$ satisfies ${\mathrm{lim\hspace{0.17em}inf}}_{t\uparrow T_{+}}{\mathcal{A}}_{\mathrm{w}\mathrm{i}\mathrm{n}}(t)<{\epsilon }_{\ast }$. Thus any finite-time failure must force persistent high-frequency concentration on all sufficiently late critical parabolic windows.