Threshold phenomenon for the quintic wave equation in three dimensions

For the critical focusing wave equation $\square u=u^5$ on $\mathbb{R}^{3+1}$ in the radial case, we establish the role of the “center stable” manifold $\Sigma$ constructed in [18] near the ground state $(W,0)$ as a threshold between blowup and scattering to zero, establishing a conjecture going back to numerical work by Bizo´n, Chmaj, Tabor [3]. The underlying topology is stronger than the energy norm.