Emily Schaal
We formulate the almost rigidity statement associated with the spacetime version of Penrose inequality, and establish it under the assumption of spherical symmetry in all dimensions. In particular, it is shown that a sequence of spherically symmetric asymptotically flat initial data satisfying the dominant energy condition, and with ADM masses tending to half the area radius of the outermost apparent horizon, must arise from isometric embeddings into a sequence of static spacetimes converging to the exterior region of a Schwarzschild spacetime. More precisely, the bases converge in the volume preserving intrinsic flat sense to the corresponding Schwarzschild time slice, and the static potentials converge in the L2 loc sense. Furthermore, the difference of the second fundamental forms must converge to zero in the L2 norm.
