In this paper we prove that Einstein four-manifolds of 3-positive curvature operator are isometric to (S4,g0)\documentclass[12pt]{minimal}
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\begin{document}$$(S^4, g_0)$$\end{document} or (CP2,gFS)\documentclass[12pt]{minimal}
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\begin{document}$$({\mathbb {C}}P^2, g_{FS})$$\end{document}, and Einstein four-manifolds of 3-nonnegative curvature operator are isometric to (S4,g0)\documentclass[12pt]{minimal}
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\begin{document}$$(S^4, g_0)$$\end{document}, (CP2,gFS)\documentclass[12pt]{minimal}
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\begin{document}$$({\mathbb {C}}P^2, g_{FS})$$\end{document}, or (S2×S2,g0⊕g0)\documentclass[12pt]{minimal}
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\begin{document}$$(S^2\times S^2, g_0\oplus g_0)$$\end{document}, up to rescaling. We also prove that the first eigenvalue of the Laplace operator for Einstein four-manifolds with Ric=g\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Ric}=g$$\end{document} and nonnegative sectional curvature is bounded above by 43+413\documentclass[12pt]{minimal}
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\begin{document}$$\frac{4}{3}+4^{\frac{1}{3}}$$\end{document}. The basic idea of the proofs is to construct an “integrated subharmonic function”, and the main ingredients of the proofs are curvature decompositions (in particular Berger decomposition), the Weitzenböck formula, and the refined Kato inequality. Along with the proofs, we also discover an alternative proof for the Weitzenböck formula using Berger decomposition, and an alternative proof for the refined Kato inequality using Derdziński’s argument.
