Ricci flow does not preserve positive sectional curvature in dimension four

Abstract

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.

Extremality and rigidity for scalar curvature in dimension four

“…, and H ′ = Z 2 , including the explicit computation of their curvature operator, see [BK21,Prop. 3.5].…”

“…Namely, by the so-called Finsler-Thorpe trick, an orientable Riemannian 4-manifold has sec ≥ 0 if and only if there exists a function τ such that R + τ * 0, where R denotes the curvature operator, and * the Hodge star operator, each acting on 2-vectors, see Proposition 2.2 for details. Other geometric applications of the Finsler-Thorpe trick have recently appeared in [BM,BKM21a,BKM21b,BK21].…”