IMCF and the Stability of the PMT and RPI Under \(L^2\) Convergence
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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.
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I would like to thank Christina Sormani for bringing this problem to my attention and for many useful suggestions and discussions. I would also like to thank her for organizing seminars at the CUNY Graduate Center which were important in shaping this paper.
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