On Positively Curved Einstein Four-Manifolds with Euler Characteristic at Most Three

It is conjectured that a complete, orientable Einstein four-manifold with strictly positive sectional curvature must be isometric to either the round four-sphere $S^4$ or the complex projective plane ${\mathbb{C}\mathbb{P}}^2$ with the Fubini–Study metric. In this paper, we confirm this conjecture when the manifold is homeomorphic to $S^4$ or ${\mathbb{C}\mathbb{P}}^2$. More precisely, we show that a complete, orientable Einstein four-manifold $M$ with strictly positive sectional curvature and Euler characteristic $\chi (M)\le 3$ (equivalently, by Freedman’s classification theorem, homeomorphic to $S^4$ or ${\mathbb{C}\mathbb{P}}^2$) is isometric to either the round $S^4$ or the Fubini–Study ${\mathbb{C}\mathbb{P}}^2$. As an application, we obtain that a complete, orientable Einstein four-manifold with strictly positive sectional curvature admitting a nontrivial Killing field is isometric to either the round $S^4$ or the Fubini–Study ${\mathbb{C}\mathbb{P}}^2$.