Stability of the PMT and RPI for asymptotically hyperbolic manifolds foliated by IMCF

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.