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

TL;DR

This paper introduces a conformally invariant invariant to characterize $\,\mathbb{CP}^2$ among four-manifolds, using Ricci flow to connect curvature bounds with topological classification.

Contribution

It extends sphere theorems to a conformally invariant setting, providing a gap theorem that characterizes $\,\mathbb{CP}^2$ via a new invariant and Ricci flow techniques.

Findings

Defines a conformal invariant $eta$ for four-manifolds.

Establishes a gap theorem for $eta$ near 4 implying $\,\mathbb{CP}^2$.

Uses Ricci flow to relate curvature bounds to topology.

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 $\beta(M^4,[g]) \geq 0$ defined on conformal four-manifolds satisfying 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 result of 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 to pass from the bounds on $\beta$ to pointwise curvature information.