Construction of GCM hypersurfaces in perturbations of Kerr

Abstract: This is a follow-up of \cite{KS:Kerr1} on the general covariant modulated (GCM) procedure in perturbations of Kerr. In this paper, we construct GCM hypersurfaces, which play a central role in extending GCM admissible spacetimes in \cite{KS:main} where decay estimates are derived in the context of nonlinear stability of Kerr family for $|a|\ll m$. As in \cite{KS}, the central idea of the construction of GCM hypersurfaces is to concatenate a $1$--parameter family of GCM spheres of \cite{KS:Kerr1} by solving an ODE system. The goal of this paper is to get rid of the symmetry restrictions in the GCM procedure introduced in \cite{KS} and thus remove an essential obstruction in extending the results to a full stability proof of the Kerr family.