Homologically area-minimizing surfaces mod $v$ have at worst codimension 2 singular sets asymptotically
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866916111357837312 |
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| author | Liu, Zhenhua |
| author_facet | Liu, Zhenhua |
| contents | De Lellis and coauthors have proved a sharp regularity theorem for area-minimizing currents in finite coefficient homology. They prove that area-minimizing mod $v$ currents are smooth outside of a singular set of codimension at least $1.$ Classical examples like triple junctions demonstrate that their result is sharp.
Surprisingly, even though their regularity theorem cannot be improved for any fixed $v$, if one instead fixes the homology class, then $v$ asymptotically one can always achieve more regularity. For any integral homology class $[Σ]$ on any Riemannian manifold, we show that for $v$ large, any area-minimizing mod $v$ current in $[Σ\mod v]$ must be an integral current, thus having a singular set of codimension at least $2$ in general and of codimension at least $7$ in the hypersurface case. Similar results are obtained for Plateau problems in Euclidean space. Our work is inspired by Morgan's work and based on De Lellis' and coauthors' work. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_18074 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Homologically area-minimizing surfaces mod $v$ have at worst codimension 2 singular sets asymptotically Liu, Zhenhua Differential Geometry De Lellis and coauthors have proved a sharp regularity theorem for area-minimizing currents in finite coefficient homology. They prove that area-minimizing mod $v$ currents are smooth outside of a singular set of codimension at least $1.$ Classical examples like triple junctions demonstrate that their result is sharp. Surprisingly, even though their regularity theorem cannot be improved for any fixed $v$, if one instead fixes the homology class, then $v$ asymptotically one can always achieve more regularity. For any integral homology class $[Σ]$ on any Riemannian manifold, we show that for $v$ large, any area-minimizing mod $v$ current in $[Σ\mod v]$ must be an integral current, thus having a singular set of codimension at least $2$ in general and of codimension at least $7$ in the hypersurface case. Similar results are obtained for Plateau problems in Euclidean space. Our work is inspired by Morgan's work and based on De Lellis' and coauthors' work. |
| title | Homologically area-minimizing surfaces mod $v$ have at worst codimension 2 singular sets asymptotically |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2401.18074 |