  We investigate the asymptotic expansion and the renormalized volume of
minimal submanifolds, $Y^m$ of arbitrary codimension in Poincare-Einstein
manifolds, $M^{n+1}$. In particular, we derive formulae for the first and
second variations of renormalized volume for $Y^m \subseteq M^{n+1}$ when $m <
n + 1$. We apply our formulae to the codimension $1$ and the $M =
\mathbb{H}^{n+1}$ case. Furthermore, we prove the existence of an asymptotic
description of our minimal submanifold, $Y$, over the boundary cylinder
$\partial Y \times \mathbb{R}^+$, and we further derive an $L^2$-inner-product
relationship between $u_2$ and $u_{m+1}$ when $M = \mathbb{H}^{n+1}$. Our
results apply to a slightly more general class of manifolds, which are
conformally compact with a metric that has an even expansion up to high order
near the boundary.
