a general stability principle for Kähler geometries, the results of this paper do provide the first rigorous examples of non-Kähler solutions of Ricci flow that become asymptotically Kähler, in suitable space-time neighborhoods of developing singularities, at rates that break scaling invariance. We hope that these examples provide motivation for further study of the general question, particularly in (real) dimension $n=4$.

Organization and main results
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The general class of Riemannian geometries that we study in this paper are smooth cohomogeneity-one metrics on the closed manifold ${\mathcal}S^2\tilde\times{\mathcal}S^2$ (the “twisted bundle" of ${\mathcal}S^2$ over ${\mathcal}S^2$). We describe these geometries (which we label “\[${\mathcal}S^2\tilde\times{\mathcal}S^2$\]-warped Berger geometries") in detail below in Section \[sec:evolve\]. Here, for the purposes of stating our main theorem, we note that for these metrics, there are two distinguished fibers ${\mathcal}S^2_\pm$ (at either “pole”); by contrast, a generic fiber is diffeomorphic to ${\mathcal}S^3$.

In Section \[sec-assume\], we identify an open subset of the \[${\mathcal}S^2\tilde\times{\mathcal}S^2$\]-warped Berger geometries by means of five pinching inequalities. These inequalities constitute our Closeness Assumptions, which we require the initial data for our Ricci flow solutions to satisfy. These assumptions ensure that our initial data, while not Kähler, are “not too far” from the subspace of Kähler metrics. 