LAUSR

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 $$Ric=\rho g$$ R i c = ρ g . Denote by K the sectional curvature of M . If $$K\ge \delta \ge \frac{2\rho -\sqrt{5}\left|\rho \right|}{6}$$ K ≥ δ ≥ 2 ρ - 5 ρ 6 (or $$K\le \delta \le \frac{2\rho +\sqrt{5}}{6}$$ K ≤ δ ≤ 2 ρ + 5 6 ) for some constant $$\delta $$ δ , then the Euler characteristic $$\chi $$ χ and the signature $$\tau $$ τ of M satisfy $$\begin{aligned}\chi \ge \left( \dfrac{3}{8\left( 1-3\delta /\rho \right) ^2} +\dfrac{3}{2}\right) \left|\tau \right|.\end{aligned}$$ χ ≥ 3 8 1 - 3 δ / ρ 2 + 3 2 τ . Our second result is a rigidity theorem for closed oriented Einstein four-manifolds with positive sectional curvature. Assume $$\lambda _1$$ λ 1 is the first eigenvalue of the Laplacian of an oriented closed Einstein four-manifold ( M ,  g ) with $$Ric = g$$ R i c = g . We show that M must be isometric to a round 4-sphere or $$\mathbb {CP}^2$$ CP 2 with the (normalized) Fubini-Study metric if the sectional curvature bounded above by $$1-\frac{4}{9\lambda _1+12}$$ 1 - 4 9 λ 1 + 12 (or bounded below by $$\frac{2}{9\lambda _1+12}$$ 2 9 λ 1 + 12 ).