  We study the Sobolev stability of the Positive Mass Theorem (PMT) and the
Riemannian Penrose Inequality (RPI) in the case where a region of a sequence of
manifolds $M^3_i$ can be foliated by a smooth solution of Inverse Mean
Curvature Flow (IMCF) which is uniformly controlled for time $t \in [0,T]$. In
particular, we consider a sequence of regions of manifolds $U_T^i\subset
M_i^3$, foliated by a IMCF, $\Sigma_t$, such that if $\partial U_T^i =
\Sigma_0^i \cup \Sigma_T^i$ and $m_H(\Sigma_T^i) \rightarrow 0$ then $U_T^i$
converges in $W^{1,2}$ to a flat annulus or in the hyperbolic setting it
converges to a annulus portion of hyperbolic space. If instead
$m_H(\Sigma_T^i)-m_H(\Sigma_0^i) \rightarrow 0$ and $m_H(\Sigma_T^i)
\rightarrow m >0$ then we show that $U_T^i$ converges in $W^{1,2}$ to a
topological annulus portion of the Schwarzschild metric or in the Hyperbolic
case to a topological annulus portion of the Anti-de~Sitter Schwarzschild
metric.
