  We prove the nonlinear stability of the Schwarzschild spacetime under axially
symmetric polarized perturbations, i.e. solutions of the Einstein vacuum
equations for asymptotically flat $1+3$ dimensional Lorentzian metrics which
admit a hypersurface orthogonal spacelike Killing vectorfield with closed
orbits. While building on the remarkable advances made in last 15 years on
establishing quantitative linear stability, the paper introduces a series of
new ideas among which we emphasize the general covariant modulation (GCM)
procedure which allows us to construct, dynamically, the center of mass frame
of the final state. The mass of the final state itself is tracked using the
well known Hawking mass relative to a well adapted foliation itself connected
to the center of mass frame. Our work here is the first to prove the nonlinear
stability of Schwarzschild in a restricted class of nontrivial perturbations.
To a large extent, the restriction to this class of perturbations is only
needed to ensure that the final state of evolution is another Schwarzschild
space. We are thus confident that our procedure may apply in a more general
setting.
