Any oriented 4-dimensional Einstein manifold with semi-definite sectional curvature (that is, everywhere non-positive or non-negative) satisfies the pointwise inequality \( \frac{|s|}{\sqrt{6}}\ge |W^+|+|W^-|, \) where s, \(W^+\) and \(W^-\) are respectively the scalar curvature, the self-dual and anti-self-dual Weyl curvatures. We give a complete characterization of closed 4-dimensional Einstein manifolds with semi-definite sectional curvature realizing the (pointwise) equality case of this inequality. We then present further consequences of this circle of ideas, in particular to the study of the geometry and topology of non-positively curved closed Einstein and Kähler–Einstein 4-manifolds. In the Kähler–Einstein case, we obtain a sharp Gromov–Lück type inequality.
