Quantitative stability of harmonic maps from ${\mathbb{R}}^2$ to ${\mathbb{S}}^2$ with higher degree

Abstract

For degree $\pm 1$ harmonic maps from ${\mathbb{R}}^2$ (or ${\mathbb{S}}^2$) to ${\mathbb{S}}^2$, Bernand-Mantel, Muratov and Simon \cite{bernand2021quantitative} recently establish a uniformly quantitative stability estimate. Namely, for any map $u:{\mathbb{R}}^2\to {\mathbb{S}}^2$ with degree $\pm 1$, the discrepancy of its Dirichlet energy and $4\ddot{I}€$ can linearly control the ${\dot{H}}^1$-difference of $u$ from the set of degree $\pm 1$ harmonic maps. Whether a similar estimate holds for harmonic maps with higher degree is unknown. In this paper, we prove that a similar quantitative stability result for higher degree is true only in local sense. Namely, given a harmonic map, a similar estimate holds if $u$ is already sufficiently near to it (modulo Möbius transform) and the bound in general depends on the given harmonic map. More importantly, we investigate an example of degree 2 case thoroughly, which shows that it fails to have a uniformly quantitative estimate like the degree $\pm 1$ case. This phenomenon show the striking difference of degree $\pm 1$ ones and higher degree ones. Finally, we also conjecture a new uniformly quantitative stability estimate based on our computation.

comments are welcome. To appear in Cal. Var. PDE