We show that S^4, S^2 X S^2, C P^2 and C P^2 # -C P^2 admit
metrics of non-negative (sectional) curvature which under the Ricci flow
immediately acquire some negatively curved two-planes. Although this was
previously known for compact manifolds of dimension greater than five
and for non-compact manifolds, these are the first compact
four-dimensional examples showing such behaviour. Our proof makes use of
the fact that these four-manifolds M admit cohomogeneity one actions,
i.e. an isometric action by a Lie group G such that the orbit space M/G
is one-dimensional. This talk is based on joint work with Renato G. Bettiol.