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Non-degeneracy and quantitative stability of half-harmonic maps from {\mathbb R} to {\mathbb S} (Liming Sun)

Time:2023-06-26  Source:Font Size:[Large | Medium | Small]  [Print]
 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.



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