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.
Annales Henri Poincaré – Springer Journals
Published: Dec 8, 2017
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