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

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