  This is a follow-up of our paper \cite{KS-Kerr1} on the construction of
general covariant modulated (GCM) spheres in perturbations of Kerr, which we
expect to play a central role in establishing their nonlinear stability. We
reformulate the main results of that paper using a canonical definition of
$\ell=1$ modes on a $2$-sphere embedded in a $1+3$ vacuum manifold. This is
based on a new, effective, version of the classical uniformization theorem
which allows us to define such modes and prove their stability for spheres with
comparable metrics. The reformulation allows us to prove a second, intrinsic,
existence theorem for GCM spheres, expressed purely in terms of geometric
quantities defined on it. A natural definition of angular momentum for such GCM
spheres is also introduced, which we expect to play a key role in determining
the final angular momentum for general perturbations of Kerr.
