solitons of lower density. (Compare [@CM12].)

Motivated partly by [@CHI04], it is conjectured by experts (see, e.g., [@HHS14]) that the only candidates to be generic singularity models in dimension $n=4$ are ${\mathcal}S^4$, ${\mathcal}S^3\times{\mathbb}R$, ${\mathcal}S^2\times{\mathbb}R^2$, and ${\mathbb}{CP}^2$ (all with their canonical metrics), and $({\mathcal}L^2_{-1},h)$. (Although the Fubini–Study metric on ${\mathbb}{CP}^n$ is well known to be weakly linearly stable, an argument for its dynamic instability is presented Kröncke [@Kro13].) The manifold $({\mathcal}L^2_{-1},h)$, which is constructed and studied in [@FIK03], is a ${\mathrm}U(2)$-invariant gradient Kähler shrinker on the complex line bundle[^2] ${\mathbb}C\hookrightarrow{\mathcal}L_{-1}^2\twoheadrightarrow{\mathbb}{CP}^1$, which is the complex bundle ${\mathcal}O(-1)$; i.e., it is the blow-up of ${\mathbb}C^2$ at the origin. (We note that Perelman’s $\nu$ functional is not defined on the noncompact manifold $({\mathcal}L^2_{-1},h)$. This is one of the reasons why the claim that the list above includes all $4$-dimensional generic singularity models is conjectural.)

While the construction of the $({\mathcal}L^2_{-1},h)$ shrinker involves the blowup of a point on ${\mathbb}C^2$, following the authors of  [@CHI04]), we call $({\mathcal}L^2_{-1},h)$ the *blowdown soliton*. We do this because, as shown in Theorem 1.6 of [@FIK03], there is a family of Riemannian manifolds ${\mathcal}N_t$, $-\infty<t<\infty$, with the following features: for $t<0$, ${\mathcal}N_t$ is $({\mathcal}L^2_{-1},h(t))$; for $t=0$, ${\mathcal}N_0$ is a 