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We study the stability of the Positive Mass Theorem (PMT) in the case where asequence of regions of manifolds with positive scalar curvature $U_T^i\subsetM_i^3$ are foliated by a smooth solution to Inverse Mean Curvature Flow (IMCF)which may not be uniformly controlled near the boundary. Then if $\partialU_T^i = \Sigma_0^i \cup \Sigma_T^i$, $m_H(\Sigma_T^i) \rightarrow 0$ and extratechnical conditions are satisfied we show that $U_T^i$ converges to a flatannulus with respect to Sormani-Wenger Intrinsic Flat (SWIF) convergence.

Inverse Mean Curvature Flow and the Stability of the Positive Mass Theorem