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

TL;DR

This paper explores the stability of harmonic maps from a0b2 to a0b2 with higher degree, showing local stability results and illustrating the failure of uniform estimates for degrees greater than one.

Contribution

It extends stability analysis from degree b1 1 to higher degrees, demonstrating local stability and providing a detailed example for degree 2.

Findings

Local stability estimate holds near a harmonic map for higher degrees.

Uniform stability estimate fails for degrees greater than one.

Degree 2 case exhibits distinct behavior from degree b1 1 cases.

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 $4\pi$ 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 M\"{o}bius 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 $\pm1$ ones and higher degree ones. Finally, we also conjecture a new uniformly quantitative stability estimate based on our computation.