Quantitative stability of harmonic maps from R2 to S2 with a higher degree.

For degree ± 1 harmonic maps from R 2 (or S 2 ) to 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 : R 2 → S 2 with degree ± 1 , the discrepancy of its Dirichlet energy and 4 π can linearly control the H ˙ 1 -difference of u from the set of degree ± 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 ± 1 case. This phenomenon shows the striking difference between degree ± 1 ones and higher degree ones. Finally, we also conjecture a new uniformly quantitative stability estimate based on our computation. [ABSTRACT FROM AUTHOR]