Inverse Mean 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 $\partial U_T^i={\Sigma }_0^i\cup {\Sigma }_T^i$, $m_H({\Sigma }_T^i)\to 0$ 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.

arXiv commentComments welcome, 51 pages, 10 figures. v2: Adjustments made throughout for final version to appear in Communications in Analysis and Geometry