Constructions of GCM spheres in perturbations of Kerr

Understand

This the first in a series of papers whose ultimate goal is to establish the full nonlinear stability of the Kerr family for $|a|\ll m$.

• The paper builds on the strategy laid out in \cite{KS} in the context of the nonlinear stability of Schwarzschild for axially symmetric polarized perturbations.
• In fact the central idea of \cite{KS} was the introduction and construction of generally covariant modulated (GCM) hypersurfaces on which specific geometric quantities take Schwarzschildian values.
• This was made possible by taking into account the full general covariance of the Einstein vacuum equations.