Rigidity in the Ginzburg--Landau approximation of harmonic spheres

Abstract

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 $\varepsilon$ small enough, the only critical points $u_\varepsilon:S^{2}\to\mathbb R^{3}$ of the Ginzburg--Landau energy $E_\varepsilon$ with energy below $8\pi-\gamma$ are (up to conjugation) rotations, that is $u_\varepsilon(x)=\sqrt{1-2\varepsilon^{2}}\;R_\varepsilon\,x$.