A rigidity theorem for Einstein $4$-manifolds with semi-definite sectional curvature, and its consequences

Any oriented $4$-dimensional Einstein metric with semi-definite sectional curvature satisfies the pointwise inequality $$\frac{\mathrm{\mid }s\mathrm{\mid }}{\sqrt{6}}\ge \mathrm{\mid }{W}^{+}\mathrm{\mid }+\mathrm{\mid }{W}^{-}\mathrm{\mid },$$ 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 metrics with semi-definite sectional curvature saturating this pointwise inequality. We then present further consequences of this circle of ideas, in particular to the study of the geography of non-positively curved closed Einstein and Kaehler-Einstein $4$-manifolds. In the Kaehler-Einstein case, we obtain a sharp Gromov-Lueck type inequality.