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

TL;DR

This paper investigates the stability of the Positive Mass Theorem by analyzing the convergence of regions foliated by Inverse Mean Curvature Flow, showing they tend to flat annuli under certain conditions.

Contribution

It demonstrates that under specific conditions, regions with positive scalar curvature foliated by IMCF converge to flat annuli, advancing understanding of PMT stability.

Findings

Regions with positive scalar curvature converge to flat annuli.

Convergence is established with respect to Sormani-Wenger Intrinsic Flat distance.

Results hold even when IMCF is not uniformly controlled near boundaries.

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.