A conformally invariant gap theorem characterizing CP² via the Ricci flow

We extend the sphere theorem of Chang et al. (Publ Math Inst Ht Études Sci 98:105–434, 2003) to give a conformally invariant characterization of (CP², g_FS). In particular, we introduce a conformal invariant β(M⁴, [g]) ≥ 0 defined on conformal four-manifolds satisfying a ‘positivity’ condition; it follows from Chang et al.(2003) that if 0 ≤ β(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 ≤ β(M⁴, [g]) < 4 (1 + ϵ) for ϵ> 0 small enough, then M⁴ is diffeomorphic to CP². The Ricci flow is used in a crucial way to pass from the bounds on β to pointwise curvature information.