The classification of closed Einstein four-manifolds with positive sectional curvature

Theorem 1.1: a connected smooth closed Einstein four-manifold with strictly positive sectional curvature, with no orientability assumption, is after positive rescaling isometric to round $S^4$, Fubini-Study $\mathbb{CP}^2$ or round $\mathbb{RP}^4$. The core step is half-conformal flatness ($W^+\equiv0$ or $W^-\equiv0$) from a weighted Hessian estimate and moment balance of the two Weyl blocks. Companions: a closed Einstein four-manifold with $\mathrm{Ric}=3g$, $K\ge0$ and one zero-curvature plane has universal cover $S^2(1/\sqrt3)\times S^2(1/\sqrt3)$, giving the full nonnegative classification; and a simply connected closed four-manifold with $K\ge0$ and small scale-invariant $L^2$ trace-free Ricci energy is diffeomorphic to $S^4$, $\mathbb{CP}^2$ or $S^2\times S^2$. Not shown: anything in higher dimensions or for complete noncompact metrics.

Theorem 1.1: a connected smooth closed Einstein four-manifold with strictly positive sectional curvature, with no orientability assumption, is after positive rescaling isometric to round $S^4$, Fubini-Study $\mathbb{CP}^2$ or round $\mathbb{RP}^4$. The core step is half-conformal flatness ($W^+\equiv0$ or $W^-\equiv0$) from a weighted Hessian estimate and moment balance of the two Weyl blocks. Companions: a closed Einstein four-manifold with $\mathrm{Ric}=3g$, $K\ge0$ and one zero-curvature plane has universal cover $S^2(1/\sqrt3)\times S^2(1/\sqrt3)$, giving the full nonnegative classification; and a simply connected closed four-manifold with $K\ge0$ and small scale-invariant $L^2$ trace-free Ricci energy is diffeomorphic to $S^4$, $\mathbb{CP}^2$ or $S^2\times S^2$. Not shown: anything in higher dimensions or for complete noncompact metrics.