Non-degeneracy and quantitative stability of half-harmonic maps from [formula omitted] to [formula omitted

Abstract

We consider half-harmonic maps from R (or S) to 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 ±1, we prove that the deviation of any map u:R→S from Möbius transformations can be controlled uniformly by ‖u‖H˙1/2(R)2−|degu|. This result resembles the quantitative rigidity estimate of degree ±1 harmonic maps R2→S2 which was proved recently. Furthermore, we address the quantitative stability for half-harmonic maps with higher degree. We prove that if u is already near to a Blaschke product, then the deviation of u to Blaschke products can be controlled by ‖u‖H˙1/2(R)2−|degu|. Additionally, a striking example shows that such a quantitative estimate can not be true uniformly for all u of degree 2. One can prove that similar things happen for harmonic maps R2→S2 (see our paper [17]).