On stability of type II blow up for the critical NLW on R³⁺¹

We show that the finite time type II blow up solutions for the energy critical nonlinear wave equation \[ \Box u = -u^5 \] on $\R^{3+1}$ constructed in earlier work by Krieger-Schlag-Tataru are stable along a co-dimension three manifold of radial data perturbations in a suitable topology, provided the scaling parameter $\lambda(t) = t^{-1-\nu}$ is sufficiently close to the self-similar rate, i. e. $\nu>0$ is sufficiently small. Our method is based on Fourier techniques adapted to time dependent wave operators of the form \[ -\partial_t^2 + \partial_r^2 + \frac2r\partial_r +V(\lambda(t)r) \] for suitable monotone scaling parameters $\lambda(t)$ and potentials $V(r)$ with a resonance at zero.