Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The relative isoperimetric inequality for minimal submanifolds in the
  Euclidean space   In this paper, we mainly consider the relative isoperimetric inequalities for
minimal submanifolds in $\mathbb{R}^{n+m}$. We first provide, following Cabr\'e
\cite{Cabre2008}, an ABP proof of the relative isoperimetric inequality proved
in Choe-Ghomi-Ritor\'e \cite{CGR07}, by generalizing ideas of restricted normal
cones given in \cite{CGR06}. Then we prove a relative isoperimetric
inequalities for minimal submanifolds in $\mathbb{R}^{n+m}$, which is optimal
when the codimension $m\le 2$. In other words we obtain a relative version of
isoperimetric inequalities for minimal submanifolds proved recently by Brendle
\cite{Brendle2019}. When the codimension $m\le 2$, our result gives an
affirmative answer to an open problem proposed by Choe in \cite{Choe2005}, Open
Problem 12.6. As another application we prove an optimal logarithmic Sobolev
inequality for free boundary submanifolds in the Euclidean space following a
trick of Brendle in \cite{Brendle2019b}.

37
The categories of this paper are: supr-con
8
The categories of this paper are: hep-ex
31
The categories of this paper are: ao-sci
32
The categories of this paper are: atom-ph
27
The categories of this paper are: comp-gas
15
The categories of this paper are: nucl-ex
0
The categories of this paper are: hep-ph
29
The categories of this paper are: funct-an
12
The categories of this paper are: nucl-th
36
The categories of this paper are: cmp-lg
28
The categories of this paper are: alg-geom
25
The categories of this paper are: patt-sol
7
The categories of this paper are: hep-th
26
The categories of this paper are: bayes-an
1
The categories of this paper are: cs
19
The categories of this paper are: eess