Non-degeneracy and quantitative stability of half-harmonic maps from $\mathbb{R}$ to $\mathbb{S}$

Abstract

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 $\mathbf{u}:\mathbb{R}\to \mathbb{S}$ from Möbius transformations can be controlled uniformly by $\Vert \mathbf{u}{\Vert }_{{\dot{H}}^{1/2}(\mathbb{R})}^2-deg\mathbf{u}$. 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 $\mathbf{u}$ is already near to a Blaschke product, then the deviation of $\mathbf{u}$ to Blaschke products can be controlled by $\Vert \mathbf{u}{\Vert }_{{\dot{H}}^{1/2}(\mathbb{R})}^2-deg\mathbf{u}$. Additionally, a striking example is given to show that such quantitative estimate can not be true uniformly for all $\mathbf{u}$ of degree 2. We conjecture similar things happen for harmonic maps ${\mathbb{R}}^2\to {\mathbb{S}}^2$.

36 pages; comments welcome

References in corpus (3)

• Sub-criticality of non-local Schrödinger systems with antisymmetric potentials and applications to half-harmonic maps
• Infinite time blow-up for half-harmonic map flow from $\mathbb{R}$ into ${\mathbb{S}}^1$
• A rigidity estimate for maps from $S^2$ to $S^2$ via the harmonic map flow