Rigidity in the Ginzburg--Landau approximation of harmonic spheres
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912566866870272 |
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| author | Gianocca, Matilde |
| author_facet | Gianocca, Matilde |
| contents | We prove that not every harmonic map from $S^{2}$ to $S^{2}$ can arise as a limit of Ginzburg--Landau critical points. More precisely, we show that the only degree-one harmonic maps that can be approximated in this way are rotations.
This conclusion follows from a rigidity theorem: we show that for every $γ>0$ and $\varepsilon$ small enough, the only critical points $u_\varepsilon:S^{2}\to\mathbb R^{3}$ of the Ginzburg--Landau energy $E_\varepsilon$ with energy below $8π-γ$ are (up to conjugation) rotations, that is $u_\varepsilon(x)=\sqrt{1-2\varepsilon^{2}}\;R_\varepsilon\,x$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_02389 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rigidity in the Ginzburg--Landau approximation of harmonic spheres Gianocca, Matilde Differential Geometry Analysis of PDEs 53C43, 58E20, 35J60, 35Q56 We prove that not every harmonic map from $S^{2}$ to $S^{2}$ can arise as a limit of Ginzburg--Landau critical points. More precisely, we show that the only degree-one harmonic maps that can be approximated in this way are rotations. This conclusion follows from a rigidity theorem: we show that for every $γ>0$ and $\varepsilon$ small enough, the only critical points $u_\varepsilon:S^{2}\to\mathbb R^{3}$ of the Ginzburg--Landau energy $E_\varepsilon$ with energy below $8π-γ$ are (up to conjugation) rotations, that is $u_\varepsilon(x)=\sqrt{1-2\varepsilon^{2}}\;R_\varepsilon\,x$. |
| title | Rigidity in the Ginzburg--Landau approximation of harmonic spheres |
| topic | Differential Geometry Analysis of PDEs 53C43, 58E20, 35J60, 35Q56 |
| url | https://arxiv.org/abs/2509.02389 |