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

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-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 $m\le 2$.

Final version, to appear in JFA

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• Alexandrov-Fenchel inequalities for convex hypersurfaces with free boundary in a ball
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Cited by in corpus (3)

• Quantitative isoperimetric inequalities for classical capillarity problems
• Willmore-type inequality in unbounded convex sets
• A Minkowski type inequality with free boundary in space forms