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| Format: | Preprint |
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2021
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| Online Access: | https://arxiv.org/abs/2111.07630 |
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| _version_ | 1866929313973010432 |
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| author | Deng, Bin Sun, Liming Wei, Juncheng |
| author_facet | Deng, Bin Sun, Liming Wei, Juncheng |
| contents | 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 $4π$ 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 $\pm1$ ones and higher degree ones. Finally, we also conjecture a new uniformly quantitative stability estimate based on our computation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2111_07630 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Quantitative stability of harmonic maps from $\mathbb{R}^2$ to $\mathbb{S}^2$ with higher degree Deng, Bin Sun, Liming Wei, Juncheng Analysis of PDEs Differential Geometry 58E20, 35B35, 35B38 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 $4π$ 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 $\pm1$ ones and higher degree ones. Finally, we also conjecture a new uniformly quantitative stability estimate based on our computation. |
| title | Quantitative stability of harmonic maps from $\mathbb{R}^2$ to $\mathbb{S}^2$ with higher degree |
| topic | Analysis of PDEs Differential Geometry 58E20, 35B35, 35B38 |
| url | https://arxiv.org/abs/2111.07630 |