  We consider rotationally symmetric spaces with low regularity, which we
regard as integral currents spaces or manifolds with Sobolev regularity and are
assumed to have nonnegative scalar curvature. Relying on the flat distance and
on Sobolev norms, we establish several nonlinear stability estimates about the
``distance'' between a rotationally symmetric manifold and the Euclidian space,
which are stated in terms of the ADM mass of the manifold. Importantly, we make
explicit the dependencies and scales involved in this problem, particularly the
ADM mass, the depth, and the CMC reference hypersurface. Several notions of
independent interest are introduced in the course of our analysis, including
the notion of depth of a manifold and a scaled version of the flat-distance,
the D-flat distance as we call it, which involves the diameter of the manifold.
Finally we prove a compactness theorem for sequences of regions with uniformly
bounded depth, whose outer boundaries have fixed area and an upper bound on
Hawking mass.
