Einstein four-manifolds of three-nonnegative curvature operator

In this paper we prove that Einstein four-manifolds of 3-positive curvature operator are isometric to \((S^4, g_0)\) or \(({\mathbb {C}}P^2, g_{FS})\), and Einstein four-manifolds of 3-nonnegative curvature operator are isometric to \((S^4, g_0)\), \(({\mathbb {C}}P^2, g_{FS})\), or \((S^2\times S^2, g_0\oplus g_0)\), up to rescaling. We also prove that the first eigenvalue of the Laplace operator for Einstein four-manifolds with \(\mathrm {Ric}=g\) and nonnegative sectional curvature is bounded above by \(\frac{4}{3}+4^{\frac{1}{3}}\). 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 Weitzenbock formula, and the refined Kato inequality. Along with the proofs, we also discover an alternative proof for the Weitzenbock formula using Berger decomposition, and an alternative proof for the refined Kato inequality using Derdzinski’s argument.