We exhibit the first examples of closed 4-manifolds with nonnegative sectional curvature that lose this property when evolved via Ricci flow.

We find examples of cohomogeneity one metrics on $$S^4$$ S 4 and $$\mathbb {C}P^2$$ C P 2 with positive sectional curvature that lose this property when evolved via Ricci flow. These metrics are arbitrarily small perturbations of Grove–Ziller metrics with flat planes that become instantly negatively curved under Ricci flow.

We find examples of cohomogeneity one metrics on S 4 and CP 2 with positive sectional curvature that lose this property when evolved via Ricci flow. These metrics are arbitrarily small perturbations of Grove-Ziller metrics with flat planes that become instantly negatively curved under Ricci flow.

We prove an existence result for the prescribed Ricci curvature equation for certain doubly warped product metrics on S d 1 +1 × S d 2 , where di ≥ 2. If T is a metric satisfying certain curvature assumptions, we show that T can be scaled independently on the two factors so as to itself be the Ricci tensor of some metric.

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