We study the stability of the positive mass theorem and the Riemannian Penrose inequality in the case where a region of an asymptotically hyperbolic 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 hyperbolic manifolds U-T(i) subset of M-i(3), foliated by a smooth solution to IMCF which is uniformly controlled, and if partial derivative U-T(i) = Sigma(i)(0) boolean OR Sigma(i)(T) m(H)(Sigma(i)(T)) - 0, then U-T(i) converges to a topological annulus portion of the hyperbolic space with respect to L-2 metric convergence. If instead m(H)(Sigma(i)(T)) - m(H)(Sigma(i)(0)) - 0 and m(H)(Sigma(i)(T)) - m 0, then we show that U-T(i) converges to a topological annulus portion of the anti-de Sitter Schwarzschild metric with respect to L-2 metric convergence. Published by AIP Publishing.
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