Huber Theorem revisited in dimensions 2 and 4
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866908709915983872 |
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| author | Laurain, Paul Martino, Dorian |
| author_facet | Laurain, Paul Martino, Dorian |
| contents | We study the second Huber theorem in dimensions 2 and 4. In dimension 2, we prove a new version assuming that the Gauss curvature lies in a negative Sobolev space using Coulomb frames. In dimension $4$, given a metric having a pointwise singularity with $L^p$-bounds on the Bach tensor, we construct a conformal metric which is regular across the singularity. To do so, we introduce another Coulomb-type condition, similar to the case of Yang--Mills connections. This enables us to obtain a conformal metric satisfying an $\varepsilon$-regularity property. We obtain a generalization of the two-dimensional case that can be applied to study the singularities of Bach-flat metrics and immersions with second fundamental forms in $W^{2,\frac{4}{3}+\varepsilon}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_05541 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Huber Theorem revisited in dimensions 2 and 4 Laurain, Paul Martino, Dorian Differential Geometry 53C18 53C21 We study the second Huber theorem in dimensions 2 and 4. In dimension 2, we prove a new version assuming that the Gauss curvature lies in a negative Sobolev space using Coulomb frames. In dimension $4$, given a metric having a pointwise singularity with $L^p$-bounds on the Bach tensor, we construct a conformal metric which is regular across the singularity. To do so, we introduce another Coulomb-type condition, similar to the case of Yang--Mills connections. This enables us to obtain a conformal metric satisfying an $\varepsilon$-regularity property. We obtain a generalization of the two-dimensional case that can be applied to study the singularities of Bach-flat metrics and immersions with second fundamental forms in $W^{2,\frac{4}{3}+\varepsilon}$. |
| title | Huber Theorem revisited in dimensions 2 and 4 |
| topic | Differential Geometry 53C18 53C21 |
| url | https://arxiv.org/abs/2502.05541 |