2021/07/18 by Deng, Bin, Sun, Liming, Wei, Juncheng
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2107.08521
We consider half-harmonic maps from ℝ (or \mathbbS) to \mathbbS. 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 ± 1, we prove that the deviation of any map \boldsymbolu:ℝ→ \mathbbS from Möbius transformations can be controlled uniformly by ‖\boldsymbolu‖ H1/2(ℝ)2-deg \boldsymbolu. This result resembles the quantitative rigidity estimate of degree ± 1 harmonic maps ℝ2→ \mathbbS2 which is proved recently. Furthermore, we address the quantitative stability for half-harmonic maps of higher degree. We prove that if \boldsymbolu is already near to a Blaschke product, then the deviation of \boldsymbolu to Blaschke products can be controlled by ‖\boldsymbolu‖ H1/2(ℝ)2-deg \boldsymbolu. Additionally, a striking example is given to show that such quantitative estimate can not be true uniformly for all \boldsymbolu of degree 2. We conjecture similar things happen for harmonic maps \mathbb R2→ \mathbb S2.