Fine Structure of Singularities in Area-Minimizing Currents Mod$(q)$
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913405983522816 |
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| author | De Lellis, Camillo Minter, Paul Skorobogatova, Anna |
| author_facet | De Lellis, Camillo Minter, Paul Skorobogatova, Anna |
| contents | We study fine structural properties related to the interior regularity of $m$-dimensional area minimizing currents mod$(q)$ in arbitrary codimension. We show: (i) the set of points where at least one tangent cone is translation invariant along $m-1$ directions is locally a connected $C^{1,β}$ submanifold, and moreover such points have unique tangent cones; (ii) the remaining part of the singular set is countably $(m-2)$-rectifiable, with a unique flat tangent cone (with multiplicity) at $\mathcal{H}^{m-2}$-a.e. flat singular point. These results are consequences of fine excess decay theorems as well as almost monotonicity of a universal frequency function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_15889 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fine Structure of Singularities in Area-Minimizing Currents Mod$(q)$ De Lellis, Camillo Minter, Paul Skorobogatova, Anna Analysis of PDEs Differential Geometry 49Q15, 49Q05, 49N60, 35B65, 35J47 We study fine structural properties related to the interior regularity of $m$-dimensional area minimizing currents mod$(q)$ in arbitrary codimension. We show: (i) the set of points where at least one tangent cone is translation invariant along $m-1$ directions is locally a connected $C^{1,β}$ submanifold, and moreover such points have unique tangent cones; (ii) the remaining part of the singular set is countably $(m-2)$-rectifiable, with a unique flat tangent cone (with multiplicity) at $\mathcal{H}^{m-2}$-a.e. flat singular point. These results are consequences of fine excess decay theorems as well as almost monotonicity of a universal frequency function. |
| title | Fine Structure of Singularities in Area-Minimizing Currents Mod$(q)$ |
| topic | Analysis of PDEs Differential Geometry 49Q15, 49Q05, 49N60, 35B65, 35J47 |
| url | https://arxiv.org/abs/2403.15889 |