Improved generic regularity of codimension-1 minimizing integral currents
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913360125100032 |
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| author | Chodosh, Otis Mantoulidis, Christos Schulze, Felix |
| author_facet | Chodosh, Otis Mantoulidis, Christos Schulze, Felix |
| contents | Let $Γ$ be a smooth, closed, oriented, $(n-1)$-dimensional submanifold of $\mathbb{R}^{n+1}$. We show that there exist arbitrarily small perturbations $Γ'$ of $Γ$ with the property that minimizing integral $n$-currents with boundary $Γ'$ are smooth away from a set of Hausdorff dimension $\leq n-9-\varepsilon_n$, where $\varepsilon_n \in (0, 1]$ is a dimensional constant.
This improves on our previous result (where we proved generic smoothness of minimizers in $9$ and $10$ ambient dimensions). The key ingredients developed here are a new method to estimate the full singular set of the foliation by minimizers and a proof of superlinear decay of closeness (near singular points) that holds even across non-conical scales. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_13191 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Improved generic regularity of codimension-1 minimizing integral currents Chodosh, Otis Mantoulidis, Christos Schulze, Felix Differential Geometry Analysis of PDEs Let $Γ$ be a smooth, closed, oriented, $(n-1)$-dimensional submanifold of $\mathbb{R}^{n+1}$. We show that there exist arbitrarily small perturbations $Γ'$ of $Γ$ with the property that minimizing integral $n$-currents with boundary $Γ'$ are smooth away from a set of Hausdorff dimension $\leq n-9-\varepsilon_n$, where $\varepsilon_n \in (0, 1]$ is a dimensional constant. This improves on our previous result (where we proved generic smoothness of minimizers in $9$ and $10$ ambient dimensions). The key ingredients developed here are a new method to estimate the full singular set of the foliation by minimizers and a proof of superlinear decay of closeness (near singular points) that holds even across non-conical scales. |
| title | Improved generic regularity of codimension-1 minimizing integral currents |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2306.13191 |