Hausdorff measure bounds for density-$Q$ flat singularities of minimizing integral currents
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909595770814464 |
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| author | Caldini, Gianmarco Skorobogatova, Anna |
| author_facet | Caldini, Gianmarco Skorobogatova, Anna |
| contents | In this article we prove that the set of flat singular points of locally highest density of area-minimizing integral currents of dimension $m$ and general codimension in a smooth Riemannian manifold $Σ$ has locally finite $(m-2)$-dimensional Hausdorff measure. In fact, the set of such flat singular points can be split into a union of two sets, one of which we show is locally $\mathcal{H}^{m-2}$-negligible, while for the other we obtain local $(m-2)$-dimensional Minkowski content bounds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_19234 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hausdorff measure bounds for density-$Q$ flat singularities of minimizing integral currents Caldini, Gianmarco Skorobogatova, Anna Differential Geometry Analysis of PDEs In this article we prove that the set of flat singular points of locally highest density of area-minimizing integral currents of dimension $m$ and general codimension in a smooth Riemannian manifold $Σ$ has locally finite $(m-2)$-dimensional Hausdorff measure. In fact, the set of such flat singular points can be split into a union of two sets, one of which we show is locally $\mathcal{H}^{m-2}$-negligible, while for the other we obtain local $(m-2)$-dimensional Minkowski content bounds. |
| title | Hausdorff measure bounds for density-$Q$ flat singularities of minimizing integral currents |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2504.19234 |