A note on rigidity of Einstein four-manifolds with positive sectional curvature

Abstract

In this paper, we first prove a topological obstruction for a four-dimensional manifold carrying an Einstein metric. More precisely, assume (M, g) is a closed Einstein four-manifold with $Ric=\rho g$. Denote by K the sectional curvature of M. If $K\ge \delta \ge \frac{2\rho -\sqrt{5}\left|\rho \right|}{6}$ (or $K\le \delta \le \frac{2\rho +\sqrt{5}}{6}$) for some constant $\delta$, then the Euler characteristic $\chi$ and the signature $\tau$ of M satisfy

$$\begin{aligned}\chi \ge \left( \dfrac{3}{8\left( 1-3\delta /\rho \right) ^2} +\dfrac{3}{2}\right) \left|\tau \right|.\end{aligned}$$

Our second result is a rigidity theorem for closed oriented Einstein four-manifolds with positive sectional curvature. Assume $\lambda _1$ is the first eigenvalue of the Laplacian of an oriented closed Einstein four-manifold (M, g) with $Ric = g$. We show that M must be isometric to a round 4-sphere or $\mathbb {CP}^2$ with the (normalized) Fubini-Study metric if the sectional curvature bounded above by $1-\frac{4}{9\lambda _1+12}$ (or bounded below by $\frac{2}{9\lambda _1+12}$).