A conformally invariant gap theorem characterizing ${\mathbb{CP}}^2$ via the Ricci flow

Abstract

We extend the sphere theorem of \cite{CGY03} to give a conformally invariant characterization of $({\mathbb{CP}}^2,g_{FS})$. In particular, we introduce a conformal invariant $\beta (M^4,[g])\ge 0$ defined on conformal four-manifolds satisfying a `positivity' condition; it follows from \cite{CGY03} that if $0\le \beta (M^4,[g])<4$, then $M^4$ is diffeomorphic to $S^4$. Our main result of this paper is a `gap' result showing that if $b_2^{+}(M^4)>0$ and $4\le \beta (M^4,[g])<4(1+\epsilon )$ for $\epsilon >0$ small enough, then $M^4$ is diffeomorphic to ${\mathbb{CP}}^2$. The Ricci flow is used in a crucial way to pass from the bounds on $\beta$ to pointwise curvature information.

26 pages