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arXiv · 2002.00914

The relative isoperimetric inequality for minimal submanifolds with free boundary in the Euclidean space

Lei Liu·Guofang Wang·Liangjun Weng
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Abstract

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-Ritor\'e in \cite{CGR06} and obtain an optimal area estimate for generalized restricted normal cones. This area estimate, together with the ABP method of Cabr\'e in \cite{Cabre2008}, provides a new proof of the relative isoperimetric inequality obtained by Choe-Ghomi-Ritor\'e 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 $m\leq 2$.

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Lei Liu, Guofang Wang, Liangjun Weng. 2020-02-03. The relative isoperimetric inequality for minimal submanifolds with free boundary in the Euclidean space. https://doi.org/10.1016/j.jfa.2023.109945

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