Einstein Four-Manifolds with Sectional Curvature Bounded from Above

  • Zhuhong ZhangEmail author


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


Einstein manifold Ricci flow Sectional curvature Curvature pinching estimate 

Mathematics Subject Classification

53C24 53C25 



The author was partially supported by NSFC 11301191. He is grateful to Professor Huai-Dong Cao for encouragement and very helpful discussions.


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Authors and Affiliations

  1. 1.School of Mathematical SciencesSouth China Normal UniversityGuangzhouPeople’s Republic of China

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