  The goal of this paper is to provide a geometric framework for analyzing the
uniform decay properties of solutions to the Teukolsky equation in the fully
nonlinear setting of perturbations of Kerr. It contains the first nonlinear
version of the Chandrasekhar transformation introduced in the linearized
setting in \cite{D-H-R-Kerr} and \cite{Ma} with the intent to use it in our
ongoing project to prove the full nonlinear stability of slowly rotating Kerr
as solution to the Einstein vacuum equations.
