Einstein four-manifolds of positive sectional curvature and ADM mass

Abstract: We prove that closed Einstein four-manifolds $(M,g)$ with positive sectional curvature and Euler characteristic and signature satisfying $2χ(M)-3|τ(M)|\le4$ must be homothetically isometric to the round $S<sup>4$ or to $\mathbb{CP}<sup>2$ with the Fubini-Study metric. The proof relies on an upper bound on the ADM mass of conformal blow-ups of $(M,g)$, combined with a lower one from \cite{GM}.