Einstein four-manifolds of positive sectional curvature and ADM mass

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 $S^4$ or to $\mathbb{CP}^2$ with the Fubini-Study metric. The proof relies on an upper bound on t