Rigidity in the Ginzburg--Landau approximation of harmonic spheres

We prove that not every harmonic map from $S^2$ to $S^2$ can arise as a limit of Ginzburg–Landau critical points. More precisely, we show that the only degree-one harmonic maps that can be approximated in this way are rotations. This conclusion follows from a rigidity theorem: we show that for every $\gamma >0$ and $\epsilon$ small enough, the only critical points $u_{\epsilon }:S^2\to {\mathbb{R}}^3$ of the Ginzburg–Landau energy $E_{\epsilon }$ with energy below $8\pi -\gamma$ are (up to conjugation) rotations, that is $u_{\epsilon }(x)=\sqrt{1-2{\epsilon }^2}{R}_{\epsilon }x$.