With Gustav Holzegel and Igor Rodnianski, we have proven the linear stability of the Schwarzschild solution to gravitational perturbations. See here. The main result can be stated as follows:

Theorem.Solutions of the system of gravitational perturbations around Schwarzschild (expressed in a double null gauge), arising from regular initial data, remain uniformly bounded, and in fact decay at a sufficiently fast polynomial rate, up to and including the event horizon, to a linearised Kerr solution, after subtraction of a pure gauge solution which itself is controlled by initial data.

This result includes *a fortiori* boundedness and decay statements for solutions of the Teukolsky equation (satisfied by the gauge invariant extremal linearised curvature components of the double null frame). The proof makes use of transformations, defined purely in physical space, associating to a solution of the Teukolsky equation a solution of the Regge–Wheeler equation. The latter equation can then be understood via methods similar to that used to understand the scalar wave equation, as described here. Although these transformations are not in general invertible, one can still recover control on Teukolsky from control on Regge–Wheeler *and initial data* by integrating transport equations. A similar transport equation hierarchy allows one to estimate the full system of gravitational perturbations, not just the gauge invariant quantities. See the paper for more details.