The relative isoperimetric inequality for minimal submanifolds with free boundary in the Euclidean space
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2020
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| _version_ | 1866910386890997760 |
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| author | Liu, Lei Wang, Guofang Weng, Liangjun |
| author_facet | Liu, Lei Wang, Guofang Weng, Liangjun |
| contents | In this paper, we mainly consider the relative isoperimetric inequalities for minimal submanifolds with free boundary. We first generalize ideas of restricted normal cones introduced by Choe-Ghomi-Ritoré in \cite{CGR06} and obtain an optimal area estimate for generalized restricted normal cones. This area estimate, together with the ABP method of Cabré in \cite{Cabre2008}, provides a new proof of the relative isoperimetric inequality obtained by Choe-Ghomi-Ritoré in \cite{CGR07}. Furthermore, we use this estimate and the idea of Brendle in his recent work \cite{Brendle2019} to obtain a relative isoperimetric inequality for minimal submanifolds with free boundary on a convex support surface in $\mathbb{R}^{n+m}$, which is optimal and gives an affirmative answer to an open problem proposed by Choe in \cite{Choe2005}, Open Problem 12.6, when the codimension $m\leq 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2002_00914 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | The relative isoperimetric inequality for minimal submanifolds with free boundary in the Euclidean space Liu, Lei Wang, Guofang Weng, Liangjun Differential Geometry Analysis of PDEs Primary: 53C21 Secondary: 53C40 In this paper, we mainly consider the relative isoperimetric inequalities for minimal submanifolds with free boundary. We first generalize ideas of restricted normal cones introduced by Choe-Ghomi-Ritoré in \cite{CGR06} and obtain an optimal area estimate for generalized restricted normal cones. This area estimate, together with the ABP method of Cabré in \cite{Cabre2008}, provides a new proof of the relative isoperimetric inequality obtained by Choe-Ghomi-Ritoré in \cite{CGR07}. Furthermore, we use this estimate and the idea of Brendle in his recent work \cite{Brendle2019} to obtain a relative isoperimetric inequality for minimal submanifolds with free boundary on a convex support surface in $\mathbb{R}^{n+m}$, which is optimal and gives an affirmative answer to an open problem proposed by Choe in \cite{Choe2005}, Open Problem 12.6, when the codimension $m\leq 2$. |
| title | The relative isoperimetric inequality for minimal submanifolds with free boundary in the Euclidean space |
| topic | Differential Geometry Analysis of PDEs Primary: 53C21 Secondary: 53C40 |
| url | https://arxiv.org/abs/2002.00914 |