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-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 $m\leq 2$.