Harmonic maps to the circle with higher dimensional singular set
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866915029633204224 |
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| author | Badran, Marco |
| author_facet | Badran, Marco |
| contents | In a closed, oriented ambient manifold $(M^n,g)$ we consider the problem of finding $\mathbb{S}^1$-valued harmonic maps with prescribed singular set. We show that the boundary of any oriented $(n-1)$-submanifold can be realised as the singular set of an $\mathbb{S}^1$-valued map, which is classically harmonic away from the singularity and distributionally harmonic across. If the singular set $Γ$ is also embedded and $C^{1,1}$, we consider three variational relaxations of the same problem and show that the energy of minimisers converges, after renormalisation, to the volume $\mathcal{H}^{n-2}(Γ)$ plus a lower-order "renormalised energy" -- common to all relaxations -- describing an energetic interaction between different components of the singular set. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_14186 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Harmonic maps to the circle with higher dimensional singular set Badran, Marco Differential Geometry Analysis of PDEs 58E20, 53C43 In a closed, oriented ambient manifold $(M^n,g)$ we consider the problem of finding $\mathbb{S}^1$-valued harmonic maps with prescribed singular set. We show that the boundary of any oriented $(n-1)$-submanifold can be realised as the singular set of an $\mathbb{S}^1$-valued map, which is classically harmonic away from the singularity and distributionally harmonic across. If the singular set $Γ$ is also embedded and $C^{1,1}$, we consider three variational relaxations of the same problem and show that the energy of minimisers converges, after renormalisation, to the volume $\mathcal{H}^{n-2}(Γ)$ plus a lower-order "renormalised energy" -- common to all relaxations -- describing an energetic interaction between different components of the singular set. |
| title | Harmonic maps to the circle with higher dimensional singular set |
| topic | Differential Geometry Analysis of PDEs 58E20, 53C43 |
| url | https://arxiv.org/abs/2411.14186 |