Sobolev stability of the PMT and RPI using IMCF

Abstract

We study the Sobolev stability of the Positive Mass Theorem (PMT) and the Riemannian Penrose Inequality (RPI) in the case where a region of a sequence of manifolds $M_i^3$ can be foliated by a smooth solution of Inverse Mean Curvature Flow (IMCF) which is uniformly controlled for time $t\in [0,T]$. In particular, we consider a sequence of regions of manifolds $U_T^i\subset M_i^3$, foliated by a IMCF, ${\Sigma }_t$, such that 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 in $W^{1,2}$ to a flat annulus or in the hyperbolic setting it converges to a annulus portion of hyperbolic space. 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 in $W^{1,2}$ to a topological annulus portion of the Schwarzschild metric or in the Hyperbolic case to a topological annulus portion of the Anti-de~Sitter Schwarzschild metric.

37 pages, comments welcome. Some definitions and important estimates are repeated from arXiv:1705.00591, arXiv:1707.09388 for the convenience of the reader. The additional $L^{\infty }$ estimates of the first version have been removed due to issues with the proof. Final version to appear in General Relativity and Gravitation