We consider half-harmonic maps from $\mathbb{ R}$ (or $\mathbb{ S}$) to $\mathbb{ S}$. We prove that all (finite energy) half-harmonic maps are non-degenerate. In other words, they are integrable critical points of the energy functional. A full description of the kernel of the linearized operator around each half-harmonic map is given. The second part of this paper devotes to studying the quantitative stability of half-harmonic maps. When its degree is $\pm 1$, we prove that the deviation of any map $\boldsymbol{ u}:\mathbb{ R}\to \mathbb{ S}$ from M\"obius transformations can be controlled uniformly by $\|\boldsymbol{ u}\|_{ \dot H^{ 1/2}(\mathbb{ R})}^2-deg \boldsymbol{ u}$. This result resembles the quantitative rigidity estimate of degree $\pm 1$ harmonic maps $\mathbb{ R}^2\to \mathbb{ S}^2$ which is proved recently. Furthermore, we address the quantitative stability for half-harmonic maps of higher degree. We prove that if $\boldsymbol{ u}$ is already near to a Blaschke product, then the deviation of $\boldsymbol{ u}$ to Blaschke products can be controlled by $\|\boldsymbol{ u}\|_{ \dot H^{ 1/2}(\mathbb{ R})}^2-deg \boldsymbol{ u}$. Additionally, a striking example is given to show that such quantitative estimate can not be true uniformly for all $\boldsymbol{ u}$ of degree 2. We conjecture similar things happen for harmonic maps ${ \mathbb R}^2\to { \mathbb S}^2$.
Categories: math.AP math.DG