[@HM11] that the Koiso-Cao soliton is *linearly* unstable.)

Our results in this paper prove that the blowdown soliton is a singularity model attractor for solutions of Ricci flow that originate from a set of Riemannian initial data defined by a structural (isometry) hypothesis and by a (weak) set of pinching conditions that we specify in Section \[sec:evolve\] below. Metrics in this set are not Kähler. These results provide evidence in favor of the conjectured stability of $({\mathcal}L^2_{-1},h)$.

Behavior of Ricci flow near Kähler geometries
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As noted above, the $({\mathcal}L^2_{-1},h)$ shrinker is Kähler. Hence, the study of non-Kähler Ricci flows near the blowdown soliton provides information about the difficult issue of the behavior of Ricci flow solutions that start near, but not in, the subspace of Kähler metrics. Do those solutions stay near or (better) asymptotically approach that subspace, which is of infinite codimension? It is believed by many experts that the subspace of Kähler metrics *should* be dynamically stable for nearby solutions of Ricci flow. Evidence of favor of this conjecture is provided by the work of Streets and Tian [@ST11], who prove that the Kähler subspace is an attractor for Hermitian curvature flow.

While we do not establish 