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 Schoens 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 $Sigma$.