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

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 $m_H({\Sigma }_T^i)\to 0$ then $U_T^i$ converges to a flat annulus with respect to $L^2$ metric convergence. If instead $m_H({\Sigma }_T^i)-m_H({\Sigma }_0^i)\to 0$ and $m_H({\Sigma }_T^i)\to m>0$ then we show that $U_T^i$ converges to a topological annulus portion of the Schwarzschild metric with respect to $L^2$ metric convergence.

31 pages, Lemma 2.10 added, typos corrected. Final version to appear in Annales Henri Poincare

References in corpus (2)

• Stability of the Positive Mass Theorem and Riemannian Penrose Inequality for Asymptotically Hyperbolic Manifolds Foliated by Inverse Mean Curvature Flow
• ODE Maximum Principle at Infinity and Non-Compact Solutions of IMCF in Hyperbolic Space

Cited by in corpus (7)

• Properties of the Null Distance and Spacetime Convergence
• Contrasting Various Notions of Convergence in Geometric Analysis
• On the stability of the positive mass theorem for asymptotically hyperbolic graphs
• Stability of the Positive Mass Theorem and Riemannian Penrose Inequality for Asymptotically Hyperbolic Manifolds Foliated by Inverse Mean Curvature Flow
• Sobolev stability of the PMT and RPI using IMCF
• Stability of the Positive Mass Theorem for Axisymmetric Manifolds
• Long Time Existence of IMCF on Metrics Conformal to Warped Product Manifolds