Einstein Four-Manifolds with Sectional Curvature Bounded from Above

Abstract

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 \({\bar{K}} \le M_2 \approx 0.750912\).