Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Optimal isoperimetric inequalities for complete proper minimal
  submanifolds in hyperbolic space   Let $\Sigma$ be a $k$-dimensional complete proper minimal submanifold in the
Poincar\'{e} ball model $B^n$ of hyperbolic geometry. If we consider $\Sigma$
as a subset of the unit ball $B^n$ in Euclidean space, we can measure the
Euclidean volumes of the given minimal submanifold $\Sigma$ and the ideal
boundary $\partial_\infty \Sigma$, say $\rvol(\Sigma)$ and
$\rvol(\partial_\infty \Sigma)$, respectively. Using this concept, we prove an
optimal linear isoperimetric inequality. We also prove that if
$\rvol(\partial_\infty \Sigma) \geq \rvol(\mathbb{S}^{k-1})$, then $\Sigma$
satisfies the classical isoperimetric inequality. By proving the monotonicity
theorem for such $\Sigma$, we further obtain a sharp lower bound for the
Euclidean volume $\rvol(\Sigma)$, which is an extension of Fraser and Schoen's
recent result \cite{FS} to hyperbolic space. Moreover we introduce the
M\"{o}bius volume of $\Sigma$ in $B^n$ to prove an isoperimetric inequality via
the M\"{o}bius volume for $\Sigma$.

24
The categories of this paper are: dg-ga
37
The categories of this paper are: supr-con
32
The categories of this paper are: atom-ph
7
The categories of this paper are: hep-th
33
The categories of this paper are: plasm-ph
5
The categories of this paper are: gr-qc
17
The categories of this paper are: q-fin
16
The categories of this paper are: stat
34
The categories of this paper are: chem-ph
4
The categories of this paper are: cond-mat
21
The categories of this paper are: adap-org
19
The categories of this paper are: eess
18
The categories of this paper are: econ
29
The categories of this paper are: funct-an
15
The categories of this paper are: nucl-ex
1
The categories of this paper are: cs