On the regularity of area minimizing currents at boundaries with arbitrary multiplicity
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916737637679104 |
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| author | Fleschler, Ian Resende, Reinaldo |
| author_facet | Fleschler, Ian Resende, Reinaldo |
| contents | In this paper, we consider an area minimizing integral $m$-current $T$ within a submanifold $Σ$ of $\mathbb{R}^{m+n}$, taking a boundary $Γ$ with arbitrary multiplicity $Q \in \mathbb{N} \setminus \{0\}$, where $Γ$ and $Σ$ are $C^{3, κ}$. We prove a sharp generalization of Allard's boundary regularity theorem to a higher multiplicity setting. Precisely, we prove that the set of density $Q/2$ singular boundary points of $T$ is $\mathcal{H}^{m-3}$-rectifiable. As a consequence, we show that the entire boundary regular set, without any assumptions on the density, is open and dense in $Γ$ which is also dimensionally sharp. Moreover, we prove that if $p \in Γ$ admits an open neighborhood in $Γ$ consisting of density $Q/2$ points with a tangent cone supported in a half $m$-plane, then $p$ is regular.
Furthermore, we show that if the convex barrier condition is satisfied-namely, if $Γ$ is a closed manifold that lies at the boundary of a uniformly convex set and $Σ= \mathbb{R}^{m+n}$-then the entire boundary singular set is $\mathcal{H}^{m-3}$-rectifiable. Additionally, we investigate certain assumptions on $Γ$ that enable us to provide further information about the singular boundary set. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_04566 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the regularity of area minimizing currents at boundaries with arbitrary multiplicity Fleschler, Ian Resende, Reinaldo Analysis of PDEs Differential Geometry 49Q15, 49Q05, 53A10, 49Q20 In this paper, we consider an area minimizing integral $m$-current $T$ within a submanifold $Σ$ of $\mathbb{R}^{m+n}$, taking a boundary $Γ$ with arbitrary multiplicity $Q \in \mathbb{N} \setminus \{0\}$, where $Γ$ and $Σ$ are $C^{3, κ}$. We prove a sharp generalization of Allard's boundary regularity theorem to a higher multiplicity setting. Precisely, we prove that the set of density $Q/2$ singular boundary points of $T$ is $\mathcal{H}^{m-3}$-rectifiable. As a consequence, we show that the entire boundary regular set, without any assumptions on the density, is open and dense in $Γ$ which is also dimensionally sharp. Moreover, we prove that if $p \in Γ$ admits an open neighborhood in $Γ$ consisting of density $Q/2$ points with a tangent cone supported in a half $m$-plane, then $p$ is regular. Furthermore, we show that if the convex barrier condition is satisfied-namely, if $Γ$ is a closed manifold that lies at the boundary of a uniformly convex set and $Σ= \mathbb{R}^{m+n}$-then the entire boundary singular set is $\mathcal{H}^{m-3}$-rectifiable. Additionally, we investigate certain assumptions on $Γ$ that enable us to provide further information about the singular boundary set. |
| title | On the regularity of area minimizing currents at boundaries with arbitrary multiplicity |
| topic | Analysis of PDEs Differential Geometry 49Q15, 49Q05, 53A10, 49Q20 |
| url | https://arxiv.org/abs/2410.04566 |