A conformally invariant gap theorem characterizing CP2 via the Ricci flow

We extend the sphere theorem of Chang et al. (Publ Math Inst Ht Etudes Sci 98:105-434, 2003) to give a conformally invariant characterization of (CP2,gFS). In particular, we introduce a conformal invariant beta(M4,[g])>= 0 defined on conformal four-manifolds satisfying a 'positivity' condition; it follows from Chang et al. (2003) that if 0 <=beta(M4,[g])<4, then M4 is diffeomorphic to S4. Our main result of this paper is a 'gap' result showing that if b2+(M4)>0 and 4 <=beta(M4,[g])<4(1+E) for E>0 small enough, then M4 is diffeomorphic to CP2. The Ricci flow is used in a crucial way to pass from the bounds on beta to pointwise curvature information.