solutions illuminates two outstanding issues concerning four-dimensional Ricci flow: i) the stability of certain singularity models in such flows, and ii) the behavior of Ricci flows that start at non-Kähler Riemannian geometries which are nonetheless close to Kähler geometries. To motivate our work here, we discuss each of these issues in turn.

Behavior of “generic Ricci flow”
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One of the keys to understanding the nature of singularities that develop in solutions of $n$-dimensional Ricci flow is to adequately classify the set of singularity models that may arise. Singularity formation in $3$-dimensional Ricci flow has been fairly well-understand since the work of Hamilton [@Ha1] and of Perelman [@Pe]. Indeed, it follows from the pinching estimate derived by Ivey [@Ivey93] and improved by Hamilton [@Ha1] that the only possible $3$-dimensional singularity models have nonnegative sectional curvature, which is a highly restrictive condition. By contrast, Máximo’s results [@Maximo14] imply that, starting in dimension $n=4$, models of finite-time singularity formation can have Ricci curvature of mixed sign (even for Kähler solutions). This leaves nonnegative scalar curvature as the only known restriction on singularity models for $n\geq4$. This restriction is too weak to be useful.

In dimensions $n\geq4$, therefore, a classification of all singularity 