Low-energy $α$-harmonic maps into the round sphere
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913224558903296 |
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| author | Sharp, Ben |
| author_facet | Sharp, Ben |
| contents | We classify low-energy $α$-harmonic maps from a closed non-spherical Riemannian surface $Σ$ of constant curvature to the round sphere via their bubble scales and centres. In particular we show that as $1<α\downarrow 1$ and assuming $E_α$ is close to $|
Σ|+4π$ then degree-one $α$-harmonic maps blow a bubble based at a critical point $a_c$ of a an explicit function $\mathcal{J}$ and at scale $\sqrt{ |\mathcal{J}(a_c)|^{-1}(α-1)}$. Up to a constant, $\mathcal{J}$ is the sum of the squares of any $L^2$-orthonormal basis of holomorphic one-forms on the domain. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_02875 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Low-energy $α$-harmonic maps into the round sphere Sharp, Ben Analysis of PDEs Differential Geometry 35R01, 53C42, 58E30 We classify low-energy $α$-harmonic maps from a closed non-spherical Riemannian surface $Σ$ of constant curvature to the round sphere via their bubble scales and centres. In particular we show that as $1<α\downarrow 1$ and assuming $E_α$ is close to $| Σ|+4π$ then degree-one $α$-harmonic maps blow a bubble based at a critical point $a_c$ of a an explicit function $\mathcal{J}$ and at scale $\sqrt{ |\mathcal{J}(a_c)|^{-1}(α-1)}$. Up to a constant, $\mathcal{J}$ is the sum of the squares of any $L^2$-orthonormal basis of holomorphic one-forms on the domain. |
| title | Low-energy $α$-harmonic maps into the round sphere |
| topic | Analysis of PDEs Differential Geometry 35R01, 53C42, 58E30 |
| url | https://arxiv.org/abs/2402.02875 |