论文标题
在欧几里得空间中具有自由边界的最小次曼叶的相对等级不等式
The relative isoperimetric inequality for minimal submanifolds with free boundary in the Euclidean space
论文作者
论文摘要
在本文中，我们主要考虑具有自由边界的最小亚曼叶的相对等级不平等现象。我们首先将Choe-Ghomi-Ritoré引入的限制正常锥体的概念概述\ CITE {CGR06}，并获得广义限制正常锥的最佳面积估计。该区域的估计以及\ cite {cabre2008}中的cabré的ABP方法，提供了一个新的证明，证明了Choe-Ghomi-Roritoré在\ cite {cgr07}中获得的相对等含量不平等。此外，我们在他最近的工作中使用了这个估计和概念的想法\ cite {brendle2019}，以获得最小的submanifolds在$ \ mathbb {r}^{r}^{n+m} $中的相对等级submanifolds，在凸面支撑面上自由边界，并给出了一个最佳的答案，以便开放一个问题，以置于公正的问题上，以置于公正的问题，以置于公正的问题上，以置于公正的问题。 \ cite {choe2005}，打开问题12.6，当编码$ m \ m \ leq 2 $时。
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 $m\leq 2$.
