Effective results on uniformization and intrinsic GCM spheres in perturbations of Kerr

Understand

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.