Inverse Mean Curvature Flow and the Stability of the Positive Mass Theorem and Riemannian Penrose Inequality Under $L^2$ Metric Convergence

TL;DR

This paper demonstrates the stability of the Positive Mass Theorem and Riemannian Penrose Inequality under $L^2$ metric convergence for regions foliated by inverse mean curvature flow, showing convergence to flat or Schwarzschild geometries.

Contribution

It establishes new stability results for PMT and RPI under $L^2$ convergence using inverse mean curvature flow in asymptotically flat manifolds.

Findings

Regions with vanishing Hawking mass converge to flat annuli.

Regions with Hawking mass approaching a positive limit converge to Schwarzschild geometry.

Results apply to sequences of manifolds foliated by IMCF with uniform control.

Abstract

We study the stability of the Positive Mass Theorem (PMT) and the Riemannian Penrose Inequality (RPI) in the case where a region of an asymptotically flat manifold $M^3$ can be foliated by a smooth solution of Inverse Mean Curvature Flow (IMCF) which is uniformly controlled. We consider a sequence of regions of asymptotically flat manifolds $U_T^i\subset M_i^3$, foliated by a smooth solution to IMCF which is uniformly controlled, and 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 to a flat annulus with respect to $L^2$ metric convergence. 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 to a topological annulus portion of the Schwarzschild metric with respect to $L^2$ metric convergence.