Einstein Four-Manifolds with Sectional Curvature Bounded from Above

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Given an Einstein structure $${\bar{g}}$$ with positive scalar curvature on a four-dimensional Riemannian manifold, that is $${\bar{R}}ic=\lambda {\bar{g}}$$ for some positive constant $$\lambda $$, a basic problem is to classify such Einstein 4-manifolds with positive or nonnegative sectional curvature. For convenience, the Ricci curvature is always normalized to be $${\bar{R}}ic=1$$. In this paper, we firstly show that if the sectional curvature of $${\bar{g}}$$ satisfies $${\bar{K}}\le \frac{\sqrt{3}}{2}\approx 0.866025$$, then $${\bar{g}}$$ must have nonnegative sectional curvature. Next, we prove a rigidity theorem of Einstein four-manifolds with nonnegative sectional curvature satisfying the additional condition that $${\bar{K}}_{ik}+s{\bar{K}}_{ij}\ge K_s$$ for every orthonormal basis $$\{e_i\}$$ with $${\bar{K}}_{ik}\ge {\bar{K}}_{ij}$$, where s is some nonnegative constant. More precisely, we show that such Einstein manifolds must be isometric to either $$S^4$$, or $$RP^4$$, or $$CP^2$$ (with standard metrics respectively). As a corollary, we obtain a rigidity result of Einstein four-manifolds with $${\bar{R}}ic=1$$ and the sectional curvature satisfying the upper bound $${\bar{K}} \le M_2 \approx 0.750912$$.