Inverse nean curvature flow and the stability of the positive mass theorem

Abstract

We study the stability of the Positive Mass Theorem (PMT) in the case where a sequence of regions of manifolds with positive scalar curvature $U_{T}^{i}\subset M_{i}^{3}$ are foliated by a smooth solution to Inverse Mean Curvature Flow (IMCF) which may not be uniformly controlled near the boundary. Then if $\partial U_{T}^{i} = \Sigma _{0}^{i} \cup \Sigma _{T}^{i}$, $m_{H}(\Sigma _{T}^{i}) \rightarrow 0$ and extra technical conditions are satisfied we show that $U_{T}^{i}$ converges to a flat annulus with respect to Sormani-Wenger Intrinsic Flat (SWIF) convergence.