# IMCF and the Stability of the PMT and RPI Under $$L^2$$ L 2 Convergence

IMCF and the Stability of the PMT and RPI Under $$L^2$$ L 2 Convergence 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 $$M^3$$ 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 $$U_T^i\subset M_i^3$$ U T i ⊂ M i 3 , foliated by a smooth solution to IMCF which is uniformly controlled, and if $$\partial U_T^i = \Sigma _0^i \cup \Sigma _T^i$$ ∂ U T i = Σ 0 i ∪ Σ T i and $$m_H(\Sigma _T^i) \rightarrow 0$$ m H ( Σ T i ) → 0 then $$U_T^i$$ U T i converges to a flat annulus with respect to $$L^2$$ L 2 metric convergence. If instead $$m_H(\Sigma _T^i)-m_H(\Sigma _0^i) \rightarrow 0$$ m H ( Σ T i ) - m H ( Σ 0 i ) → 0 and $$m_H(\Sigma _T^i) \rightarrow m >0$$ m H ( Σ T i ) → m > 0 then we show that $$U_T^i$$ U T i converges to a topological annulus portion of the Schwarzschild metric with respect to $$L^2$$ L 2 metric convergence. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annales Henri Poincaré Springer Journals

# IMCF and the Stability of the PMT and RPI Under $$L^2$$ L 2 Convergence

, Volume 19 (4) – Dec 8, 2017
24 pages

/lp/springer_journal/imcf-and-the-stability-of-the-pmt-and-rpi-under-l-2-l-2-convergence-DOilyX86K8
Publisher
Springer International Publishing
Copyright © 2017 by Springer International Publishing AG, part of Springer Nature
Subject
Physics; Theoretical, Mathematical and Computational Physics; Dynamical Systems and Ergodic Theory; Quantum Physics; Mathematical Methods in Physics; Classical and Quantum Gravitation, Relativity Theory; Elementary Particles, Quantum Field Theory
ISSN
1424-0637
eISSN
1424-0661
D.O.I.
10.1007/s00023-017-0641-7
Publisher site
See Article on Publisher Site

### 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 $$M^3$$ 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 $$U_T^i\subset M_i^3$$ U T i ⊂ M i 3 , foliated by a smooth solution to IMCF which is uniformly controlled, and if $$\partial U_T^i = \Sigma _0^i \cup \Sigma _T^i$$ ∂ U T i = Σ 0 i ∪ Σ T i and $$m_H(\Sigma _T^i) \rightarrow 0$$ m H ( Σ T i ) → 0 then $$U_T^i$$ U T i converges to a flat annulus with respect to $$L^2$$ L 2 metric convergence. If instead $$m_H(\Sigma _T^i)-m_H(\Sigma _0^i) \rightarrow 0$$ m H ( Σ T i ) - m H ( Σ 0 i ) → 0 and $$m_H(\Sigma _T^i) \rightarrow m >0$$ m H ( Σ T i ) → m > 0 then we show that $$U_T^i$$ U T i converges to a topological annulus portion of the Schwarzschild metric with respect to $$L^2$$ L 2 metric convergence.

### Journal

Annales Henri PoincaréSpringer Journals

Published: Dec 8, 2017

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