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

For degree \(\pm 1\) $\pm 1$ harmonic maps from \({\mathbb {R}}^2\) ${\mathbb{R}}^2$ (or \({\mathbb {S}}^2\) ${\mathbb{S}}^2$ ) to \({\mathbb {S}}^2\) ${\mathbb{S}}^2$ , Bernand-Mantel et al. (Arch Ration Mech Anal 239(1):219–299, 2021) recently establish a uniformly quantitative stability estimate. Namely, for any map \(u:{\mathbb {R}}^2\rightarrow {\mathbb {S}}^2\) $u:{\mathbb{R}}^2\to {\mathbb{S}}^2$ with degree \(\pm 1\) $\pm 1$ , the discrepancy of its Dirichlet energy and \(4\pi \) $4\pi$ can linearly control the \(\dot{H}^1\) ${\dot{H}}^1$ -difference of u from the set of degree \(\pm 1\) $\pm 1$ harmonic maps. Whether a similar estimate holds for harmonic maps with a higher degree is unknown. In this paper, we prove that a similar quantitative stability result for a higher degree is true only in a local sense. Namely, given a harmonic map, a similar estimate holds if u is already sufficiently near to it (modulo Möbius transforms) and the bound in general depends on the given harmonic map. More importantly, we thoroughly investigate an example of the degree 2 case, which shows that it fails to have a uniformly quantitative estimate like the degree \(\pm 1\) $\pm 1$ case. This phenomenon shows the striking difference between degree \(\pm 1\) $\pm 1$ ones and higher degree ones. Finally, we also conjecture a new uniformly quantitative stability estimate based on our computation.