Blowdowns along Ricci flow in dimension four

We show that the blowdown shrinking Ricci soliton of Feldman, Ilmanen and Knopf is strictly dynamically stable in the following strong local sense: (1) The Ricci flow from every metric on a closed 4-manifold containing a region close to a rescaling of a sufficiently large compact subset of the soliton in either orientation and with controlled geometry elsewhere develops a singularity with unique tangent flow modeled on the soliton. (2) The singular time slice has an isolated conical singularity, and by Gianniotis-Schulze the flow can be continued by blowing down a sphere of self-intersection $\pm 1$. (3) On a closed 4-manifold, the set of metrics whose Ricci flows have their first singularity modeled on the soliton is $C^{14}$-open. It is nonempty if and only if the manifold contains a sphere of self-intersection $\pm 1$. (4) The soliton is strongly rigid among shrinking Ricci solitons. As applications, we construct $C^{14}$-open sets of initial data whose Ricci flows perform any prescribed ordered sequence of blowdowns. (1) Every closed simply connected smooth 4-manifold, including spin and exotic examples, can be reached by Ricci flow from a standard $k{\mathbb{CP}}^2\#\ell \overline{\mathbb{CP}}{}^2$, through finitely many blowdown singularities. (2) On the standard $2{\mathbb{CP}}^2\#9\overline{\mathbb{CP}}{}^2$, there are infinitely many open sets of initial metrics whose Ricci flows, after a single blowdown, reach pairwise nondiffeomorphic manifolds. The proof uses the spectral analysis of Naff and the second author and deals with the gauge-fixing problem along Ricci flow through a Ważewski argument inspired by Stolarski. We also use Bamler-Lai's PDE-ODI principle. Except for continuation (requiring nonnegative curvature of the asymptotic cone), all results apply to linearly stable nonflat asymptotically conical shrinkers in all dimensions.