Constructions of GCM spheres in perturbations of Kerr

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. The goal of this paper is to get rid of the symmetry restriction in the construction of GCM spheres and thus remove an essential obstruction in extending the result of \cite{KS} to a full stability proof of the Kerr family.