Excess decay for minimizing hypercurrents mod $2Q$
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866915357774577664 |
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| author | De Lellis, Camillo Hirsch, Jonas Marchese, Andrea Spolaor, Luca Stuvard, Salvatore |
| author_facet | De Lellis, Camillo Hirsch, Jonas Marchese, Andrea Spolaor, Luca Stuvard, Salvatore |
| contents | We consider codimension $1$ area-minimizing $m$-dimensional currents $T$ mod an even integer $p=2Q$ in a $C^2$ Riemannian submanifold $Σ$ of the Euclidean space. We prove a suitable excess-decay estimate towards the unique tangent cone at every point $q\in \mathrm{spt} (T)\setminus \mathrm{spt}^p (\partial T)$ where at least one such tangent cone is $Q$ copies of a single plane. While an analogous decay statement was proved in arXiv:2111.11202 as a corollary of a more general theory for stable varifolds, in our statement we strive for the optimal dependence of the estimates upon the second fundamental form of $Σ$. This technical improvement is in fact needed in arXiv:2201.10204 to prove that the singular set of $T$ can be decomposed into a $C^{1,α}$ $(m-1)$-dimensional submanifold and an additional closed remaining set of Hausdorff dimension at most $m-2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_08704 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Excess decay for minimizing hypercurrents mod $2Q$ De Lellis, Camillo Hirsch, Jonas Marchese, Andrea Spolaor, Luca Stuvard, Salvatore Analysis of PDEs Differential Geometry 49Q15, 49Q20, 49Q05, 53A10 We consider codimension $1$ area-minimizing $m$-dimensional currents $T$ mod an even integer $p=2Q$ in a $C^2$ Riemannian submanifold $Σ$ of the Euclidean space. We prove a suitable excess-decay estimate towards the unique tangent cone at every point $q\in \mathrm{spt} (T)\setminus \mathrm{spt}^p (\partial T)$ where at least one such tangent cone is $Q$ copies of a single plane. While an analogous decay statement was proved in arXiv:2111.11202 as a corollary of a more general theory for stable varifolds, in our statement we strive for the optimal dependence of the estimates upon the second fundamental form of $Σ$. This technical improvement is in fact needed in arXiv:2201.10204 to prove that the singular set of $T$ can be decomposed into a $C^{1,α}$ $(m-1)$-dimensional submanifold and an additional closed remaining set of Hausdorff dimension at most $m-2$. |
| title | Excess decay for minimizing hypercurrents mod $2Q$ |
| topic | Analysis of PDEs Differential Geometry 49Q15, 49Q20, 49Q05, 53A10 |
| url | https://arxiv.org/abs/2308.08704 |