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

Abstract

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 $\mathbb{CP}^2$ with the Fubini--Study metric. In this paper, we confirm this conjecture when the manifold is homeomorphic to $S^4$ or $\mathbb{CP}^2$. More precisely, we show that a complete, orientable Einstein four-manifold $M$ with strictly positive sectional curvature and Euler characteristic $χ(M) \le 3$ (equivalently, by Freedman's classification theorem, homeomorphic to $S^4$ or $\mathbb{CP}^2$) is isometric to either the round $S^4$ or the Fubini--Study $\mathbb{CP}^2$. 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 $\mathbb{CP}^2$.