The fine structure of the singular set of area-minimizing integral currents II: rectifiability of flat singular points with singularity degree larger than $1$
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914942046699520 |
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| author | De Lellis, Camillo Skorobogatova, Anna |
| author_facet | De Lellis, Camillo Skorobogatova, Anna |
| contents | We consider an area-minimizing integral current $T$ of codimension higher than $1$ in a smooth Riemannian manifold $Σ$. In a previous paper we have subdivided the set of interior singular points with at least one flat tangent cone according to a real parameter, which we refer to as ``singularity degree''. This parameter determines the infinitesimal order of contact at the point in question between the ``singular part'' of $T$ and its ``best regular approximation''. In this paper we show that the set of points for which the singularity degree is strictly larger than $1$, is $(m-2)$-rectifiable. In a subsequent work we prove that the remaining flat singular points form an $(m-2)$-null set, thus concluding that the singular set of $T$ is $m-2$-rectifiable. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2304_11555 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The fine structure of the singular set of area-minimizing integral currents II: rectifiability of flat singular points with singularity degree larger than $1$ De Lellis, Camillo Skorobogatova, Anna Analysis of PDEs Differential Geometry 49Q15, 49Q05, 49N60, 35B65, 35J47 We consider an area-minimizing integral current $T$ of codimension higher than $1$ in a smooth Riemannian manifold $Σ$. In a previous paper we have subdivided the set of interior singular points with at least one flat tangent cone according to a real parameter, which we refer to as ``singularity degree''. This parameter determines the infinitesimal order of contact at the point in question between the ``singular part'' of $T$ and its ``best regular approximation''. In this paper we show that the set of points for which the singularity degree is strictly larger than $1$, is $(m-2)$-rectifiable. In a subsequent work we prove that the remaining flat singular points form an $(m-2)$-null set, thus concluding that the singular set of $T$ is $m-2$-rectifiable. |
| title | The fine structure of the singular set of area-minimizing integral currents II: rectifiability of flat singular points with singularity degree larger than $1$ |
| topic | Analysis of PDEs Differential Geometry 49Q15, 49Q05, 49N60, 35B65, 35J47 |
| url | https://arxiv.org/abs/2304.11555 |