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？bius 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. 

　　Publication: 

　　Advances in Mathematics, Volume 420, 1 May 2023, 108979 

　　Author: 

　　Bin Deng 

　　Department of Mathematics and Statistics, Wuhan University, Wuhan, Hubei Province, 430072, PR China 

　　Liming Sun 

　　Academy of Mathematics and Systems Science, the Chinese Academy of Sciences, Beijing 100190, PR China 

　　Email: lmsun@amss.ac.cn 

　　Jun-cheng Wei 

　　Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada