Curvature Varifolds with Orthogonal Boundary
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866914876627091456 |
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| author | Kuwert, Ernst Müller, Marius |
| author_facet | Kuwert, Ernst Müller, Marius |
| contents | We consider the class $S^m_\perp(Ω)$ of $m$-dimensional surfaces in $\barΩ \subset {\mathbb R}^n$ which intersect $S = \partial Ω$ orthogonally along the boundary. A piece of an affine $m$-plane in $S^m_\perp(Ω)$ is called an orthogonal slice. We prove estimates for the area by the $L^p$-integral of the second fundamental form in three cases: first when $Ω$ admits no orthogonal slices, second for $m = p = 2$ if all orthogonal slices are topological disks, and finally for all $Ω$ if the surfaces are confined to a neighborhood of $S$. The orthogonality constraint has a weak formulation for curvature varifolds. We classify those varifolds of vanishing curvature. As an application, we prove for any $Ω$ the existence of an orthogonal $2$-varifold which minimizes the $L^2$ curvature in the integer rectifiable class. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_08045 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Curvature Varifolds with Orthogonal Boundary Kuwert, Ernst Müller, Marius Differential Geometry Analysis of PDEs We consider the class $S^m_\perp(Ω)$ of $m$-dimensional surfaces in $\barΩ \subset {\mathbb R}^n$ which intersect $S = \partial Ω$ orthogonally along the boundary. A piece of an affine $m$-plane in $S^m_\perp(Ω)$ is called an orthogonal slice. We prove estimates for the area by the $L^p$-integral of the second fundamental form in three cases: first when $Ω$ admits no orthogonal slices, second for $m = p = 2$ if all orthogonal slices are topological disks, and finally for all $Ω$ if the surfaces are confined to a neighborhood of $S$. The orthogonality constraint has a weak formulation for curvature varifolds. We classify those varifolds of vanishing curvature. As an application, we prove for any $Ω$ the existence of an orthogonal $2$-varifold which minimizes the $L^2$ curvature in the integer rectifiable class. |
| title | Curvature Varifolds with Orthogonal Boundary |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2203.08045 |